Chapter 3

Newton’s First and Second Laws

3.1 Lecture

In this lesson, we will study the motion of an object that is falling under the action of gravity,but is constrained to move along a flat surface as it falls. This type of motion is referredto as “constrained linear motion.” We will need to increase our sophistication of analysis inorder to fully analyze this motion.

3.1.1 Formulate

State the Problem

A mass, m, is released from rest and allowed to slide down along an inclined planar surfacedue to the action of gravity. Find the mean acceleration, a, of the object. Given a transducerthat may be used to measure the displacement of the mass along an inclined surface (at aknown angle) as a function of time, we are to develop expressions that can be used tointerpret experimental data and present estimates for the components of position, velocity,and acceleration as a function of time. Estimate the mean friction force between the objectand the planar surface. Estimate the static coe�cient of friction, µs and the dynamiccoe�cient of friction, µd.

State the Known Information

The following information is provided

m1

= [kg] � mass of object (constant) (3.1)

✓ = [degrees] � inclination angle of planar surface (constant) (3.2)

g = 9.81 [m/s2] � standard value of gravitational acceleration (3.3)

We use the symbol � to indicate that the quantity is known prior to the start of the problemsolving method.

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State the Desired Information

Upon conclusion of the experiment and analysis, we shall be required to report:

z(t) = ? [m] ↵ vertical displacement plot (3.4)

Vz(t) = ? [m/s] ↵ vertical velocity plot (3.5)

az(t) = ? [m/s2] ↵ mean vertical acceleration (3.6)

x(t) = ? [m] ↵ horizontal displacement plot (3.7)

Vx(t) = ? [m/s] ↵ horizontal velocity plot (3.8)

ax(t) = ? [m/s2] ↵ mean horizontal acceleration (3.9)

as = ? [m/s2] ↵ mean tangential acceleration (3.10)

µs = ? [�] ↵ static coe�cient of friction (3.11)

µd = ? [�] ↵ dynamic coe�cient of friction (3.12)

f = ? [N ] ↵ sliding friction force (3.13)

We use the symbol ↵ to indicate that this quantity is unknown, and the value must bedetermined as a result of the problem solving method. Uncertainties should be reported onall required quantities, either graphically or in tabular form.

3.1.2 Assume

Identify Assumptions

The following assumptions may be employed during the analysis. We use the term staticcoe�cient of friction, µs, to describe the frictional force between the ramp surface and theobject at the point when motion is incipient. We use the term dynamic coe�cient of friction,µd, to describe the frictional force between the ramp surface and the object when the objectis sliding on the surface.

µs = [�] � constant and uniform but unknown (3.14)

µd = [�] � constant and uniform but unknown (3.15)

f µN [N ] � Amontons’ 1st Law (3.16)

Fr/m1

= [N ] � normal reaction force between mass and ramp (3.17)

FLeg1 = [N ] � reaction force between ramp support and Earth (3.18)

FLeg2 = [N ] � reaction force between ramp support and Earth (3.19)

mR = [kg] � rass of ramp assembly (3.20)

x = 0 [m] � at base of ramp (3.21)

z = 0 [m] � at base of ramp (3.22)

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Justify Assumptions

We need to justify each assumption proposed for use in our analysis.Equations 3.14 and 3.15 say that we are assuming the value of the coe�cients of static

and dynamic friction to be constant (not changing with time), and uniform (the same every-where between the contact surface of the ramp and the object). We have taken great careto have the surface finish of the apparatus be consistent and uniform. If experimental ob-servations indicate a non-constant tangential acceleration, then we may have to re-evaluatethis assumption.

Equation 3.16 says that the frictional force f existing between the object and the rampsurface is linearly proportional to the normal reaction force N (perpendicular to the surface)between the object and the ramp surface. The constant of proportionality is the coe�cientof friction, µ. This relationship, stating that ‘the force of friction is directly proportional tothe applied load” is known as “Amontons’ First Law” and dates back to the 17th century.Incidentally, Amontons’ Second Law states that “the force of friction is independent of theapparent area of contact.”

Equations 3.17 through 3.20 define the reaction forces between surfaces and the massof apparatus, respectively. The sign convention associated with these symbols is noted inthe diagrams. The units associated with these symbols are noted in the equations. Thesesymbols are not necessary to report as desired information, and are presented here for thesake of documenting all nomenclature used in the analysis. If we compute a negative valuefor any of the forces indicated, that simply means that the assumed direction of action (thearrowhead) was pointing in the opposite direction from the actual line of action for the force.However, if we compute a negative value for any force, that should be an indication to us asengineers that something about our intuitive understanding of the problem was inaccurate,or perhaps that we made an algebraic mistake in the solution of the problem.

Equations 3.21 through 3.22 define the origin of the coordinate system used in the anal-ysis. This coordinate system, and its relative orientation, should be consistently indicatedon all diagrams.

3.1.3 Chart

Schematic Diagrams

Last week, we learned about the motion of an object under the action of a single externalforce – gravity. This week, we will study the motion of an object when it moves under theaction of gravity, but is also constrained to move along an inclined plane. Let’s define a localcoordinate system as indicated by the x and z axes in the schematic diagram of Figure 3.1.

In this case, we need to become a bit more sophisticated with the way we describe themotion of the object as it moves down the inclined plane. Let’s define the angle ✓ as theacute angle between the horizontal plane and the inclination of the ramp. We choose to placethe origin of our coordinate system near the bottom of the ramp, and we define the positivez direction as upwards, away from the Earth. The ramp is supported by some legs, and the

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Figure 3.1: Schematic diagram of an object sliding down an inclined plane.

object is initially held at the top of the ramp. As shown in Figure 3.1, we carefully choosethe system boundary so that the leg supports of the ramp are not the focus of our analysis.We are really only interested in the interaction between the ramp and the object. While werecognize that the ramp exerts a force on the Earth, and that the Earth exerts an equal andopposite force on the ramp, that interaction is simply not the focus of our investigation andwe don’t want to be distracted by those details. The system boundary line is our reminderof this fact.

A more detailed schematic of the experiment is shown in Figure 3.2. In this schematic, weshow a transducer at the top of the ramp, looking downwards along the surface inclined atan angle ✓ to the horizontal. The distance from the transducer face to the mass is denoted asd, while the distance from the origin at the bottom of the ramp to the face of the transduceris denoted as L. The tangential position of the mass along the ramp may be computedas s = L � d. The analysis presented here will be given in this context. If you employa di↵erent experimental configuration, you may have to employ an appropriate coordinatetransformation in order to interpret the results from your sensor. Recall, when doing theprevious experiment, the ultrasonic sensor was pointing downwards towards the floor, butwe chose to measure elevation measured vertically upward from the floor. Similarly, you willneed to interpret your sensor data and convert the resulting distance information into theappropriate coordinate system.

Free Body Diagrams

Free body diagrams (FBDs) for the mass and ramp are presented in Figure 3.3. Note thatthese include the three crucial elements that are required for free body diagrams:

1. For each object in the system: the object is drawn with all corresponding forces thatact on the body shown with arrows indicating the direction of the force and identifyinglabels.

2. A coordinate system is show. In this case we have one set for both objects, but

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Figure 3.2: Schematic diagram of an experiment to measure an object sliding down aninclined plane toward the earth.

sometimes di↵erent coordinate systems are used for di↵erent objects.

3. Any special dimensions such as angles or lengths are identified. Here, we show theangle ✓ that the ramp makes to the surface of Earth.

Figure 3.3: Free body diagrams of mass and ramp.

Newton’s Law of Gravity tells us that the Earth will exert a force�!W = m

1

�!g upon theobject. Even though the Earth is outside of the system boundary and is not in contact withthe object, we know from personal observation that the influence of the Earth is still present.Thus, we need to account for this gravity force. The magnitude of the force is W , and itis directed from the object toward the center of the Earth. As we saw in our experimentlast week, an object under the influence of gravity will drop to the Earth. In this case,

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however, the object cannot drop vertically down, since the ramp is in the way. Our previousexperience with objects on ramps tells us that we expect the object to slide down the ramptowards the Earth. Indeed, this is part of the reason the coordinate system includes bothx�z (or ı� k in terms of unit vectors) and tangential-normal (or s� n) systems. We will seelater when performing the force vector analysis, that it is convenient to have a coordinatesystem in which the largest number of the force vectors are aligned along the coordinateaxes.

The ramp exerts a force upon the object, Fr/m1

, which is perpendicular to the surfaceof the ramp. Newton’s Third Law says that the object exerts an equal and opposite force,Fr/m1

, upon the ramp. This force, perpendicular to the surface of the ramp is called the “nor-mal force,” often referred to in generic terminology as simply “N”. Notice that we labeledthese reaction forces as having the same magnitude but with the arrowheads pointing in theopposite directions. An idealized frictionless ramp inhibits the object from fall through itssurface, but it does not inhibit the motion of the object along its surface. Real surfaces thatinteract with one another always exhibit some friction, and this friction is always directedsuch that the friction resists the relative motion (or the incipient motion) between the op-posing surfaces. The friction force is labeled as fm1

in the FBDs. The friction force of theramp upon the mass is directed tangentially upwards along the ramp, opposing the motionof the mass down the ramp. Newton’s Third Law then states that the corresponding frictionforce of the mass upon the ramp is directed tangentially downwards along the ramp - inopposition to the relative motion of the mass. That is, if we were to attach our coordinatesystem origin to the mass, then the ramp would appear to be moving upwards, towards anobserver sitting on the mass. While the Earth and the ramp interact with one another, wereally are not concerned with that interaction, which lies outside of the system boundary.

We hypothesize, from personal experience, that the object, m1

, illustrated in Figure 3.1will slide down the ramp towards the Earth as soon as we release it from our hand. We aregoing to conduct an experiment to measure the motion of the object along the direction ofthe ramp.

3.1.4 Execute

Recall The Governing Equations.

The first step in the execution phase is to recall the relevant governing equations. In thiscase, we recall Newton’s laws.

If :X�!

F = 0 Then : �!a = 0 Newton’s 1st Law (3.23)

X�!F = lim

�t!0

��!p�t

= lim�t!0

�(m�!V )

�tNewton’s 2nd Law (3.24)

�!F Action = ��!

F Reaction Newton’s 3rd Law (3.25)�!F g = g ·m # Newton’s Law of Gravity near Earth (3.26)

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Last week, we studied Newton’s Law of Gravity for a body in free fall near the Earth’ssurface. This week, we continue that investigation, but we add the constraint of a rampwhich limits the motion of the object. We have used Newton’s Third Law (“For everyaction, there is an equal and opposite reaction”) to construct the free body diagrams ofthe object, the ramp, and the Earth. We used Newton’s Law of Gravity to quantify theinfluence of the Earth upon the system. We will use Newton’s First Law and Newton’sSecond Law to study the motion of the object as it moves down the ramp. Remember thatNewton’s First Law says “A body at rest will remain at rest, and a body in motion willremain in motion, unless it is acted upon by a net external force.” We predict that thenormal component of the object’s weight will be balanced by the normal reaction force ofthe ramp upon the object, in accordance with Newton’s First Law. Furthermore, we predictthat once placed upon the ramp, an object will not begin sliding for small angles of inclinefor the ramp. However, beyond some critical angle, the object will slide. This particularangle, when motion is incipient, will be how we define the coe�cient of static friction, µs.The fact that we can increase the angle ✓ above 0[degrees] and not have motion is proof ofthe existence of friction, and confirmation of Newton’s First Law.

Newton’s Second Law, Equation 3.24, should describe the motion of the object as itmoves down the ramp. We expect, from past experience, that if the angle of incline isabove some critical value (the incipient motion threshold), then the mass will slide downthe ramp. Using the known information that the mass of the object is constant, as given byEquation 3.1, we can move the mass outside of the di↵erential, and Newton’s Second Lawfor a constant mass object under the action of some net external force,

P�!F may be written

in the simplified form: X�!F = m�!a (3.27)

In our previous lab, all motion was assumed to be along a single coordinate direction, andwe could do the entire analysis along this axis of motion. In the upcoming lab, we will onceagain measure the displacement of the object along a straight line as a function of time. Wewill call this motion the tangential displacement |s| as a function of time. However, we willalso find it valuable to study the motion in Cartesian coordinates, with the z axis normalto the surface of the Earth, and the x axis tangential to the surface of the Earth. Giventhe tangential displacement, we will compute the x and z components of displacement, andfrom those values, we will estimate the x and z components of velocity (Vx and Vz) andacceleration (ax and az). We will compare both the tangential and x, z quantities to thosethat were computed during the free-fall experiment. While we could conduct this entire Laband Studio using only the “tangential” direction, using the Cartesian coordinate system willhelp us develop skills with vectors that will become essential later. First, we need to developsome additional background about vector mathematics in order to tackle this problem.

Scalars and Vectors

In many physical applications, we deal with quantities that have only a magnitude, and thereis no sense of “directionality” associated with them. Examples of such physical quantities

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are “mass”, “temperature”, “color” and “volume.” All physical quantities that may bedescribed only by their magnitude are defined as “scalars.” In other applications, we dealwith quantities that require both a magnitude and a direction to describe them. Examples ofsuch physical quantities are “position”, “velocity”, “acceleration”, and “force.” All physicalquantities that are described by both their magnitude and direction are defined as “vectors.”This week, we will learn about scalars and vectors, and how to use both of them formallyto better understand physical phenomena, and mathematically use scalars and vectors inexperimental models and computer simulations.

Consider the vector �!s shown in Figure 3.4. The vector points from the origin to the

Figure 3.4: Cartesian space depiction of the vector �!s .

Cartesian point (6, 4). We label the horizontal axis with the symbol ı to indicate the positivex direction, and the symbol k to indicate the positive z direction. We can think of the vector�!s being used to indicate both the direction and magnitude of travel. If we imagine ourselvesstanding at the origin, and beginning to walk along the direction of the line �!s , then aftersome time we would arrive at the point indicated.

The vector �!s can be broken into two component vectors, placed head to tail as illustratedin Figure 3.5. If we started our hypothetical walk from the origin as in the previous example,then we could provide directions to another person along the lines of “starting at the origin,walk six paces in the ı direction, then turn left at a right angle and proceed four paces inthe k direction to arrive at my location.” We refer to the horizontal displacement as x andthe vertical displacement as z. In mathematical terms, we can write the sentence describingthe motion as:

�!s = xı+ zk (3.28)

Let’s make a note of each term in Equation 3.28. The quantities x and z are scalars. Theyhave a magnitude only. Looking at Figure 3.5 we can see that x = 6 and z = 4. Thequantities ı and k are called unit vectors. They have a magnitude, which we assign to beunity, and they have a direction, aligned with an axis. In most engineering disciplines, weadopt the standard convention that ı is the unit vector associated with the x axis, k is the

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Figure 3.5: Cartesian space depiction of the vector �!s with x and z components.

unit vector associated with the z axis, and | is the unit vector associated with the y axis. Themathematical Equation 3.28 can be read as the following sentence: “starting at the origin,the vector �!s has a component x units in the ı direction, and z units in the k direction.”Notice that the end point of vector �!s is independent of the order in which we traversethe components. In Figure 3.5, we show the tail of the vertical component vector attachedto the head of the horizontal component vector. Alternatively, we could align the verticalcomponent vector along the k axis and then connect the tail of the horizontal component tothe head of the vertical component. Either way, we still end up producing the same resultantvector, �!s . The magnitude of the resultant vector �!s is denoted as |�!s |, and it is a scalar.The magnitude |�!s | can be computed from Pythagoras’ theorem, since the ı and k directionsare 90� from one another:

|�!s | =px2 + z2 (3.29)

Let’s look at one more interpretation of the vector as illustrated in Figure 3.6. In many

Figure 3.6: Polar coordinate depiction of the vector �!s with x and z components.

engineering applications, we like to work with the x and z components of motion. In other

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cases, we might like to work with the idea of a “distance and a heading.” In Figure 3.6, theoriginal vector �!s has a magnitude of |s| as given by Equation 3.29. The vector �!s makesan angle ✓ as illustrated in Figure 3.6. From our high school trigonometry class, we candetermine the value of ✓ from the relation:

✓ = tan�1

⇣zx

⌘(3.30)

The vector �!s is expressed in polar coordinates as

�!s = |s|\✓ (3.31)

The mathematical Equation 3.31 can be read as the following sentence: “starting at theorigin, the vector �!s has a magnitude of |s| units at an angle ✓ from the positive horizontalaxis.” Previously, we assumed that we knew the components x and z and wanted to computethe magnitude |�!s | and direction ✓. In other cases, we will know the magnitude and directionand desire to compute the components. Given the magnitude and direction, we can usetrigonometry as indicated by Figure 3.6 to compute the horizontal component as:

x = |�!s | cos(✓) (3.32)

and the vertical component asz = |�!s | sin(✓) (3.33)

Vector Addition

Vectors can be added and subtracted from one another using their components. Considerthe two vectors �!s

1

and �!s2

as illustrated in Figure 3.7. We can add vectors �!s1

and �!s2

Figure 3.7: Two vectors �!s1

and �!s2

with x and z components.

together to obtain the resultant vector �!r , as indicated by Equation 3.34

�!r = �!s1

+�!s2

(3.34)

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Figure 3.8: Vector sum �!r = �!s1

+ �!s2

.

Graphically, this vector addition is illustrated in Figure 3.8 Algebraically, we can find theresult by adding the vector components:

�!r = �!s1

+�!s2

xr ı+ yrk = (x1

+ x2

)ı+ (z1

+ z2

)k (3.35)

Vector addition obeys the commutative law that we learned in algebra. That is, we can saythat:

�!r = �!s1

+�!s2

= (x1

+ x2

)ı+ (z1

+ z2

)k

= �!s2

+�!s1

= (x2

+ x1

)ı+ (z2

+ z1

)k (3.36)

Vector addition also obeys the associative law. For example, given three vectors �!s1

, �!s2

,and �!s

3

, we can write that

(�!s1

+�!s2

) +�!s3

= �!s1

+ (�!s2

+�!s3

) (3.37)

The graphical representation of Equation 3.36 is presented in Figure 3.9 Note that, whilethe resultant �!r is the same by virtue of the commutative law, Figure 3.9 clearly illustratesthat the path taken to get from the origin to the destination is quite di↵erent. From a math-ematical perspective, the result is identical. However, from an engineering perspective, thepath taken may be significant. We can imagine, for example, that if this was an orienteeringor geo-caching exercise, one set of directions might cause us to cross a creek while the otherpath might allow us to keep our feet dry!

Vectors are most easily subtracted through the negation operator. If we want to createa new resultant vector �!r

2

as the di↵erence of two vectors, we can write

�!r2

= �!s1

��!s2

= �!s1

+ (��!s2

) (3.38)

= (x1

� x2

)ı+ (z1

� z2

)k

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Figure 3.9: Vector sum �!r2

= �!s2

+ �!s1

.

Until now, we have assumed that all of the vectors have the same units. For example, wemight measure distance in “walking paces” or “meters” or “feet”. When we work with vec-tors, they must all have the same units. Is it invalid to add one vector expressed in “meters”to another vector expressed in “feet.” Both vectors must be converted to a consistent set ofunits before they can be added together.

Simplify the Governing Equations.

With this background in vectors, we can now write Newton’s Second Law for a body ofconstant mass in its vector components as:

�!F Net = Fx(Net)ı+ Fz(Net)k = m

⇣axı+ azk

⌘(3.39)

To simplify the analysis, it will also be convenient to write the component form of theseequations in terms of the normal n and tangential t components:

�!F Net = Fn(Net)n+ Fn(Net)t = m

�ann+ att

�(3.40)

For the schematic shown in Figure 3.2 the transducer is shown at the bottom of the ramplooking upwards. We position the local x axis at the base of the ramp with the positive xpointed to the right. Using the right hand rule, this suggests that the y axis should bedirected into the page, and that the z axis be directed upwards, away from the center of theEarth. To think of the right hand rule, use your right hand, with your four fingers extendedtogether. Position your right thumb upwards at a 90 degree angle. Point your four fingersin the direction of the x axis. Rotate your hand through a motion so that your fingersnow point in the direction of the positive y axis. When you complete this motion properly,your thumb will be pointed in the direction of the positive z axis. The displacement vector,�!s , is the distance from the surface of the transducer to the nearest point of the object,measured parallel to the ramp surface. We need to use our knowledge of the schematicdiagram to convert the scalar transducer measurement into a vector quantity. One way to

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describe the position of the object as a function of time is to simply state that the object islocated at |s| from the base of the ramp. However, this is not convenient for implementingin computer simulations. Let’s use our understanding of vector terminology to determinethe components of the position vector as a function of time. The angle ✓ in Figure 3.2 ismeasured from the negative x axis, and has a magnitude less than 90� or ⇡/2 radians. Thischoice of ✓ will require us to assign the proper sign to the x and z position components tomaintain consistency with our chosen coordinate system.

Consider the vector diagrams shown in Figure 3.10. Referring to the position vectordiagram in the figure, we note that the distance (magnitude) from the transducer to the ballis labeled s, and is directed at the angle ✓ to the horizontal, along the axis of the ramp.We construct a right triangle from the transducer surface to the ball surface, and label theposition vector components as x in the horizontal direction and z in the vertical direction.Recall from trigonometry that the sum of the interior angles of a triangle is 180�. Since wehave constructed a right triangle, the remaining interior angle is (90��✓). From the position

Figure 3.10: Force Vector for the normal and tangential components of the weight vector.

vector triangle, we can then say that the position vector, �!s is given by the expression

�!s = xı+ zk (3.41)�!s = �|s| cos ✓ı+ |s| sin ✓k (3.42)

where we have defined x = �|s| cos ✓ and z = |s| sin ✓.Next, we consider the force vector diagram shown in Figure 3.10. The mass m

1

experi-ences a force due to gravity in the magnitude m

1

g directed towards the center of the Earth.We wish to break this force into its normal (perpendicular to the ramp) and tangential(parallel to the ramp) components. We can use similar triangles to observe that we havetwo parallel vertical lines in the force and position vector diagrams. Furthermore, the rampincline is parallel to the ramp tangent. Using the trigonometric theorem that alternate inte-rior angles are equal, we can quantify the lower angle in the force vector diagram as being(90��✓). Once again using the knowledge that the force vector diagram forms a right angle,

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and that the sum of interior angles of a triangle is 180�, we can easily find the upper angleof the force vector triangle as ✓.

Using trigonometry on the force vector diagram, we compute the normal and tangentialcomponents of the weight,

�!W

1

, as

WNormal = m1

g cos ✓ (3.43)

WTangential = m1

g sin ✓ (3.44)

This is reassuring. Our force vector diagram indicates that the gravitational forces tendto push the object against the ramp (in the normal direction), and down the ramp (in thetangential direction). This is consistent with our intuitive understanding of the system.

Now, refer back to Figure 3.3, and look at the free body diagram for the object. Weknow that the mass will not fall through the normal surface of the ramp, but rather that itwill move tangentially along the surface of the ramp. Thus, we expect that the ramp willexert a force upon the object that is equal in magnitude and opposite in direction to thenormal component of the object’s weight:

|Fr/m1

| = |WNormal| = m1

g cos ✓ (3.45)

For the moment, consider the condition when motion is incipient – the angle of theinclined ramp is such that the block is on the verge of beginning to slide down the ramp.Consider assumption 3.16 which is an empirical law describing the magnitude of the frictionforce existing between two surfaces. The coe�cient µ is experimentally determined for acombination of surfaces, and it is generally unique to that combination of surfaces. Forexample, the value of µ for a wooden block on an aluminum ramp is significantly di↵erentthan it would be for a wooden block on a ramp constructed of gravel. There exist somecritical value of inclination, ✓s, at which motion is incipient. At any inclination below ✓sthe object does not move, and the friction force is high enough to resist the tangentialforce induced by gravity. At any inclination above ✓s the object will move. In this case,the limiting equality in Amontons’ law applies, and at this particular inclination, we useEquation 3.16 to define the maximum achievable static friction force:

fs = µsm1

g cos ✓s (3.46)

[N ] = [�][kg][m/s2][�]

At this point of equilibrium, Newton’s first law (Equation 3.23) tells us that the up-ramptangential force due to friction perfectly balances the down-ramp component of the object’sweight, as illustrated in the free body diagram of Figure 3.3 and the force vector diagram ofFigure 3.10.

�!F Net = 0 (3.47)

XFtangential = 0 (3.48)

fs �m1

g sin ✓s = 0 (3.49)

µsm1

g cos ✓s �m1

g sin ✓s = 0 (3.50)

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When the angle of inclination, ✓, is greater than the condition of incipient motion, ✓s, theblock will slide down the ramp. Equation 3.16 describes the magnitude of the friction forceexisting between two surfaces. The coe�cient µd is the experimentally determined coe�cientfor this sliding motion condition. In this case, the inequality condition of Amontons’ lawapplies, and at this particular inclination, the dynamic friction force, f (which is less thanthe static friction force) is given by:

f = µdm1

g cos ✓ (3.51)

[N ] = [�][kg][m/s2][�]

We know that the system is not in equilibrium, and that motion is taking place, so Newton’ssecond law (Equation 3.24) tells us that the up-ramp tangential force due to friction is notsu�cient to balance the down-ramp component of the object’s weight, as illustrated in thefree body diagram of Figure 3.3 and the force vector diagram of Figure 3.2. In this case, forthe case of constant mass (by virtue of Equation 3.1), Newton’s second law states that thenet tangential force will result in a tangential acceleration as along the ramp:

XFtangential = m

1

as (3.52)

f �m1

g sin ✓ = m1

as (3.53)

µdm1

g cos ✓ �m1

g sin ✓ = m1

as (3.54)

The friction force f resists the motion of the object, while the tangential component of theweight promotes the motion of the object.

Solve

For the case when the object is in static equilibrium, on the verge of incipient motion, wecan rearrange Equation 3.50 as follows:

µs =m

1

g sin ✓sm

1

g cos ✓s(3.55)

µs =sin ✓scos ✓s

(3.56)

µs = tan ✓s (3.57)

[�] = [�] dimensionless quantity

The angle of incipient motion, ✓s, is sometimes referred to as the “friction angle.” Thecoe�cient µs is called the “static coe�cient of friction” and the friction force fs under theseconditions is sometimes referred to as the “traction.” Equation 3.57 provides the solutionfor one item of desired information, as requested by Equation 3.11.

Since we will be measuring the mean acceleration, as of the block as it slides down theincline, let’s substitute this into Equation 3.54 and then rearrange the equation to solve forthe coe�cient of dynamic friction as follows:

µdm1

g cos ✓ �m1

g sin ✓ = m1

as (3.58)

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µdg cos ✓ � g sin ✓ = as (3.59)

µd � tan ✓ =as

g cos ✓(3.60)

µd = tan ✓ +as

g cos ✓at any ✓ (3.61)

µd = µs +as

g cos ✓sat ✓s (3.62)

[�] = [�] +[m/s2]

[m/s2]dimensionless quantity

Equation 3.61 provides the solution requested by Equation 3.12. The term in the denom-inator on the right hand side, g cos ✓, represents the down-ramp tangential component ofthe gravitational acceleration of the earth. Note that, since the direction �!s is directed up-wards along the ramp by our sign convention in Figure 3.2, the mean tangential acceleration,as, will be a negative number. As the magnitude |�!as | ! 0 in Equation 3.61 the value ofµd ! µs. At the friction angle, ✓ = ✓s, Equation 3.62 shows that µd < µs. Now that we havean estimate for the dynamic coe�cient of friction, µd, we can substitute Equation 3.61 intoEquation 3.51 and use our knowledge of the object weight on the incline to find the desiredsliding friction force from Equation 3.13 with the expression

f = µdm1

g cos ✓ (3.51)

Another manner that we can estimate the mean friction force during the sliding motion isto recall Newton’s Second Law for the tangential component of motion, given by Equation3.53, and solve for the mean friction force f in terms of the observed acceleration and theobject weight. Rearranging this equation to solve for f yields:

f = m1

(as + g sin ✓) (3.63)

Recall that as will be a negative quantity by virtue of our sign convention.The remaining items of desired information, Equations 3.4 through 3.9, will be determined

in Studio, using experimental data obtained in Lab.

3.1.5 Test

Validate

We have validated that the units on each result are correct. Upon completion of the exper-iment and analysis, we should determine if the signs on the results are consistent with thesign convention assumed in the analysis. If any sign discrepancies arise, then these shouldbe reviewed as potential indicators of error.

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Verify

In lab, we will conduct an experiment to measure the motion of the object down an inclinedplane. We will use many of the skills learned during the first two weeks to conduct thisexperiment. Following the lab, we should report our findings of all desired quantities assupported by experimental data. We should compare the observed acceleration of the object,and compare our measured results with those obtained for the free fall experiment.

Apply Intuition

Equation 3.57 is consistent with our intuition. After the mass begins moving, the frictionforce between the object and the surface is lower than the traction, or f < µN . It iscommon practice to introduce a new friction coe�cient, µd, called the “dynamic coe�cientof friction.” The dynamic coe�cient of friction is always less than the static coe�cientof friction, or µd < µs. This intuitive knowledge is confirmed by Equation 3.61 since weanticipate that as < 0. If you have ever tried to push a heavy box across a carpeted floor,you have probably observed that it is “harder to get the box moving and easier to keep itmoving once it has started.” This observation is consistent with Amontons’ law. This isalso the fundamental reason why better braking is achieved in a vehicle with rolling friction,rather than sliding friction. When the wheel of your car is rolling, there is an instantaneouspoint of zero velocity between the wheel and the road, which causes the higher “static”coe�cient of friction, and yields the best “traction.” Conversely, if you have been behindthe wheel of a car that has started to skid, then the wheel is sliding across the road surface,rather than rolling. In this case, the lower “dynamic” coe�cient of friction applies, andyou have reduced “traction.” Amontons’ Law from the 17th century provides the theoreticalfoundation for why anti-lock braking systems work so well! It only took 400 years for controlstechnology to catch up with the theory of Amontons!

3.1.6 Iterate

In lab, we will conduct multiple trials of this experiment. Each student member of thelab group should conduct at least one independent trial, at a unique angle of inclination.Multiple trials per student may be necessary to achieve high quality results.

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3.2 Lab - A Single Body in Constrained 2-d Motion

3.2.1 Scope

This week in lab you will use a transducer to measure the position of a block as a functionof time as it slides along an inclined surface. The experiment will be similar to the one youconducted last week, except that the body will be constrained to move along an inclinedsurface. In addition, we will use a new sensor, called an “angle sensor” or “inclinometer”to measure the angle of inclination of the ramp. Both the ultrasonic transducer and theinclinometer must be calibrated prior to use in the experiment.

3.2.2 Goal

The goals of this laboratory experiment are to

1. acquire voltage vs. time data with an automated data acquisition system,

2. convert the voltage vs. time data into distance vs. time,

3. estimate the instantaneous tangential position and velocity of the object as it moves,

4. estimate the mean tangential acceleration of the object during the motion interval,

5. determine the horizontal and vertical components of motion (instantaneous position,instantaneous velocity, and mean acceleration) for the object,

6. confirm our understanding of Newton’s First and Second Laws of Motion,

7. understand Amontons’ First Law of friction.

3.2.3 Units of Measurement to use

Measurements may be conducted in a combination of customary U.S. units and S.I. unit.All results shall be reported in the SI system of units.

Table 3.1: Units of Measurement to be used for object in constrainted linear motion.Quantity Customary units SI units

Time [s] [s]Voltage [V ] [V ]Length [inch] [m]Velocity [inch/sec] [m/s]

Acceleration [inch/sec2] [m/s

2]Angle [degrees] [radians] or [degrees]

Friction Coe�cient [�] [�]

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3.2.4 Reference Documents

Review the reference material from Chapter 1, especially regarding the ultrasonic transducerand the method of calibration used for the first week?s lab exercise. The procedure outlinedin Chapter 1 will be used to calibrate the transducer this week as well, but only once for theentire group.

3.2.5 Terminology

The following terms must be fully understood in order to achieve the educational objectivesof this laboratory experiment.

Length Voltage Acceleration Speed Velocity UnitHorizontal Vertical Component Normal Tangential VectorMagnitude Direction Scalar Resultant Constraint AxisOrigin Associative Law Incipient Negation Unit Vector AngleInclination Commutative Law Inclinometer Static Dynamic TimeFriction Amontons’ 1st Law Traction Sliding Rolling

3.2.6 Summary of Test Method

On the myCourses site for this course you will find links to one or more videos on YouTubefor this week’s exercise. Watch all of the available videos, and complete the online lab quizfor the week. The videos are your best reference for the specific tasks and procedures tofollow for completing the laboratory exercise.

3.2.7 Calibration and Standardization

Ultrasonic Transducer Calibration

The Lead Technologist and Assistant Technologist shall calibrate the ultrasonic transducerto determine the relationship between position and voltage, using an approach similar tothat of week 1, prior to conducting trials of the current experiment. The same calibrationsoftware VI used during the week 1 experiment shall be used to manually record calibrationdata for the ultrasonic transducer. The calibration shall be conducted using the woodenblock as the calibration surface, rather than ball used previously. It is a good calibrationpractice to calibrate the transducer in a configuration as close as possible to the mannerin which it will be used. The entire team may use one calibration curve for the ultrasonictransducer. The configuration of the apparatus is illustrated in Figure 3.11.

Angle Sensor Calibration

The Lead Technologist is responsible for calibrating the “zero angle” of the angle sensor. Thethree-point surface calibration plate is shown in Figure 3.12. The term “inclinometer” is aspecial name given for an angle sensor that is used to measure an angle of inclination relative

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Figure 3.11: Apparatus configured for transducer calibration.

Figure 3.12: Three-point planar calibration surface.

to a surface which is horizontal to the Earth’s local gravitational field. Since we are using abubble level to determine the “zero angle” for our angle sensor, we can reasonably use thespecialized term of “inclinometer” to describe the angle sensor. The three thumbscrews canbe adjusted to tilt the flat plate in various orientations. When the bubble is centered in thecircular target ring, then the plate is perpendicular to the direction of the local gravitationalforce of the Earth. An outline of the procedure to calibrate the angle sensor is:

1. The Technologists shall take the inclinometer for the lab group to the horizontal surfacecalibration station in the lab.

2. The Technologists shall adjust the thumbscrews on the three-point surface calibrationplate until the level bubble is concentric with the indicator circle on the bubble level.

3. After ensuring that the three-point surface calibration plate is a level plane by adjustingthe bubble, the Technologists shall “zero” the inclinometer by pressing the appropriatebutton on the device.

4. The Technologists shall report the instrument least count inaccuracy for the inclinome-ter to the Scribe and other members of the lab group. All group members shall recordthis information in their respective logbook.

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3.2.8 Apparatus

All required apparatus and equipment components are described and demonstrated in theinstructional videos for this exercise, or will be familiar from common or previous use.

3.2.9 Measurement Uncertainty

In Lab 1, we learned about measurement uncertainty. In Lab 2, we learned about thepropagation of uncertainty due to numerical di↵erentiation (addition and subtraction), andthe influence of data averaging on the standard uncertainty of an average. In Lab 3, wewill learn about the propagation of uncertainty through mathematical operations such asmultiplication and trigonometric functions. This week we will use the central di↵erencingtechnique to estimate the instantaneous velocity and acceleration based on position data. Itwas shown in the Lab section of Chapter 2 (starting at Equation 2.23) that when using thismethod we estimate that the uncertainty in the instantaneous velocity will be

✏V ⌘ ✏z�t

(3.64)

where ✏z is the uncertainty in the elevation data, and we assume that the time measurement ishighly accurate. This week, we are applying the central di↵erencing method to block positiondata, whose uncertainty is dependent only on the ruler’s uncertainty. Thus, the uncertaintyin position may be taken as one half the Instrument Least Count (ILC) inaccuracy of theruler. The instantaneous velocity for this week is then given by:

✏V ⌘ ✏p�t

(3.65)

where ✏p is the uncertainty in the position data. It can be similarly shown that the uncer-tainty in the instantaneous acceleration when applying the central di↵erencing method todiscrete velocity values is given by

✏a =✏V�t

=✏p�t2

(3.66)

We previously observed that the uncertainty in the instantaneous velocity or accelerationmay be quite high, but that the uncertainty in the mean values are more modest. For thisclass, we defined the uncertainty in the mean of a series of instantaneous values as

✏a ⌘✏apNa

(3.67)

where Na represents the number of unique estimates for the instantaneous value that areavailable from the experimental data set and ✏a is the uncertainty associated with each ofthose instantaneous estimates. We can use an expression similar to Equation 3.67 to esti-mate the uncertainty of a data series. Additional, more sophisticated methods for studyinguncertainty will be presented later in the curriculum.

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During Lab, the lab group conducted an assessment of the friction angle, ✓s, and itsuncertainty, ✏✓s is given by the inclinometer’s ILC divided by 2. The static coe�cientof friction, µs, is given by Equation 3.57, which requires us to find the coe�cient µ asa trigonometric function of the angle, ✓s. How can we estimate the uncertainty in thecoe�cient of friction, ✏µs? While tedious, the method is straightforward. First, we computethree values for µs, and denote them as the “nominal value” and the “first” and “second”limits. The limiting values are calculated using the uncertainty bounds on ✓s:

µs = tan(✓s) Nominal value (3.68)

µ+

= tan(✓s + ✏✓s) First limit (3.69)

µ� = tan(✓s � ✏✓s) Second limit (3.70)

We define the uncertainty, ✏µs , of the result as:

✏µs = ±0.5|µ+

� µ�| (3.71)

A similar procedure may be used to estimate the propagation of uncertainty through othertrigonometric functions such as sin and cos.

Let’s study the propagation of uncertainty when we multiply two uncertain values to-gether. Let’s say that we have two variables a, b with uncertainties ✏a, ✏b respectively. Thenominal product of the two variables is

w = ab (3.72)

Now, recognizing that each term has an uncertainty, we expect the product to have anuncertainty.

w ± ✏w = (a± ✏a)(b± ✏b) (3.73)

Applying the FOIL method, we can expand this as

w ± ✏w = ab± a✏b ± b✏a ± ✏a✏b (3.74)

Since w = ab, we can subtract that common term from both sides of the equation, and dividethrough by the product ab

±✏wab

= ±✏bb± ✏a

a± ✏a✏b

ab(3.75)

But, once again we recognize that w = ab so we can write this conveniently as

±✏ww

= ±✏bb± ✏a

a± ✏a✏b

ab(3.76)

In a well designed experiment, we anticipate that ✏a << a and ✏b << b. The quantity ✏a✏b/abshould be very small in comparison to the other terms, and may be neglected. We now definethe relative uncertainty in the product of two values as the sum of the relative uncertainties:

✏ww

⌘ ✏aa+

✏bb

(3.77)

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Repeated application of this rule demonstrates that the relative uncertainty associated witha product of several values w = abcdef... may be approximated by

✏ww

⌘ ✏aa+

✏bb+

✏cc+

✏dd+

✏ee+

✏ff

+ ... (3.78)

At this step in the uncertainty analysis, it is useful to examine the terms on the rightside of the above equation. These terms represent relative uncertainties for the di↵erentvariables�that is, the ratio of a each variable’s uncertainty to its respective magnitude.Often, it will be the case that the relative uncertainty associated with one or more of thevariables will be much smaller than the other ones. If that is the case, then these smallerterms can be dropped from the equation, thereby simplifying the uncertainty analysis.

Pure mathematical constants such as a factor of 2 or physical constants such as g haveno uncertainty associated with them. Therefore, if we apply Equation 3.77, to

w = ka (3.79)

where k is a constant, then we have

✏ww

⌘ ✏aa+

✏kk

(3.80)

Since k is constant, then ✏k = 0. Substituting in for ✏k = 0,

✏ww

⌘ ✏aa+

0

k(3.81)

and then for w = ka, and rearranging

✏ww

⌘ ka✏aa

(3.82)

or finally,

✏ww

⌘ k✏a (3.83)

The uncertainty relationships discussed here and in each Measurement Uncertainty section ofthis text are summarized for your convenience in the Engineering Mechanics Reference Table,which is posted on myCourses under General Content. These uncertainty relationships canbe applied to the friction coe�cient calculations and force calculations to help quantify therange of uncertainty in your analysis.

3.2.10 Preparation of Apparatus

All required equipment for conducting the laboratory exercise is made available either withinone or both of the drawers attached to the lab bench, or available from the laboratory instruc-tor. You are expected to bring all other necessary materials, particularly your logbook and a

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flash drive for storing electronic data as appropriate. You are to follow the general specifica-tions for team roles within the lab. Although there are specific, individual expectations foreach role, you are each responsible overall to ensure that the objectives and requirements ofthe laboratory exercise are met, and that all rules and procedures are followed at all times,especially any that are related to safety in the lab. When finished, all equipment is to bereturned to the proper location, in proper working order.

3.2.11 Sampling, Test Specimens

Before conducting any other testing with the ultrasonic transducer, it should be calibratedby the same method used in previous labs, adjusted for the newer elements in this week’sexercise. Following the calibration, the critical angle for the apparatus is to be determinedaccording to the basic procedure outlined in the lab videos. Then, each member of the labgroup should conduct his/her own block test and associated data collection. Each member ofthe group should test the block?s motion at a di↵erent inclination of the apparatus. If timeallows, it is always a good idea to conduct multiple trials per person (even if those trials donot show appreciable variation�that would be a good thing to demonstrate). Also, if timeallows, it is a good idea to repeat the friction angle determination to see if anything changesduring testing. For this exercise, some of the data will be recorded in the logbooks, whilethe dynamic data will be recorded digitally through the LabVIEW interface to a numberof tab-delimited, ascii data files. Note that you must also measure the mass of theblock and the length of the ramp from the end of the transducer to the blockcatcher (distance L in Figure 3.2) must also be measured at some point.

3.2.12 Procedure - Lab Portion

Record all observations and notes about your lab experiment inyour logbook.

The instructional videos for this exercise cover the specific procedures to follow as youset up the apparatus to make measurements, and then as you actually collect data withthe various devices and software interfaces. More generally, you should always observe thefollowing general procedures as you conduct any of the exercises in this laboratory.

1. Come prepared to lab, having watched the videos in detail, then completing the asso-ciated lab quiz and preparing your logbook before you arrive to class.

2. Follow the basic outline of elements to include in your logbook related to headers,footer, and signatures.

3. As you conduct the exercise, please pay attention to the following safety concerns:

• Watch for tripping hazards, due to cables and moving elements.

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• Watch for pinch points, during assembling and disassembly.

• Be careful of shock hazards while connecting and operating electrical components

4. Every week, for every exercise, your logbook will minimally contain background notesand information that you collect before the lab, at least one schematic of the apparatus,various standard tables for recording the organization of your roles and equipmentused, the actual data collected and/or notes related to the data collected (if doneelectronically for instance), and any other information relevant to the reporting andanalysis of the data and understanding of the exercise itself.

5. All students should create and complete a table indicating the sta�ng plan for theweek (that is, the roles assumed by each group member), as shown in Table 1.2.

6. All students should create and complete a table listing all equipment used for the exer-cise, the location (from where was it obtained: top drawer, bottom drawer, instructor?)and all identifying information that is readily available. If the manufacturer and se-rial number are available, then record both (this would be an ideal scenario). If not,record whatever you can about the component. In some, cases, there will be no specificidentifying information whatsoever either because of the simplicity of the component,or because of its origin. In these cases, just identify the component as best you can,perhaps as “Manufactured by RITME.” The point here is to give as much informationas possible in case someone was to try to reproduce or verify what you did. Refer toTable 1.3.

7. For the Lab Manager only: create a key sign-out/sign-in table for obtaining thekey to the equipment drawers, as shown in Table 1.4.

8. All students should create a table or series of tables as appropriate to collect his/herown data for the exercise, as well as any specific notes related to the data collectionactivities. In those cases where data collection is done electronically, there may not beany data tables required.

9. Many of the laboratory exercises will require the use of a specific software interfacefor measurements and/or control. In all cases, these will be made available on themyCourses site unless stated otherwise.

10. The Scribe (or a designated alternative) should take a photo of each group memberperforming some aspect of the laboratory exercise for inclusion in the lab reportthat will be generated during the studio session. Refer to the example lab report formore details.

11. Record all relevant data and observations in your logbook, even those that may nothave been explicitly requested or indicated by the textbook or videos. If in doubtabout any measurements, it is better to make the measurement rather than not.

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12. When you are finished with all lab activities, make sure that all equipment has beenreturned to the proper place. Log out of the computer, and straighten up everythingon the lab bench as you found it. Put the lab stools back under the bench and out ofthe way.

13. Prepare for the upcoming studio session for the week by carefully read and understandSection 2.3 of the textbook, and complete the Studio pre-work prior to your arrival atStudio.

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3.3 Studio

This week in Studio, you will use the calibration techniques you learned in Week 1 to calibratethe voltage to the block position along the inclined plane. You will then use the calibrationcurve to analyze block position as a function of time and the block velocity and accelerationusing the central di↵erence method. Once the acceleration is calculated, the coe�cient offriction will be determined along with the forces on block. The equations for calculatingforces rely heavily on the FBD’s and Newton’s second law theory discussed in Section 3.1of the text. Before beginning the Studio procedures, please review Section 3.3.1 Calculationand Interpretation of Results, which will provide a summary of equations that you will needto complete the Studio. Also review Section 3.2.9 Measurement Uncertainty, which willdescribe the process for analyzing the experimental errors.

Record all observations and notes about your studio procedures inyour logbook.

3.3.1 Calculation and Interpretation of Results

This week in Studio, we will calculate velocity and acceleration using a “central di↵erenceapproximation.” The central di↵erence approximation follows directly from the definition ofthe derivative, taking a larger time step for the derivative compared to the forward di↵erenceapproximation.

The following mathematical equations may be used in the context of the Studio Proce-dure. These equations follow directly from our work in the previous week, or were developedearlier in this Chapter and are summarized here for convenience.

ps(t) = c0

+ c1

V (t) calibration curve for elevation (3.84)

[m] = [m] +[m]

[v][v] unit validation

px(t) = �ps(t) cos ✓ horizontal component of position (3.85)

pz(t) = +ps(t) sin ✓ vertical component of position (3.86)

�t =1

Sample Ratetime between samples (3.87)

[s]

[Sample]=

1

[Sample/s]unit validation

2�t = (t+�t)� (t��t) time between two samples (3.88)

Vs(t) ⇡ps(t+�t)� ps(t��t)

2�tvelocity by central di↵erence (3.89)

[m/s] =[m]� [m]

[s]Unit validation

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as(t) ⇡Vs(t+�t)� Vs(t��t)

2�tacceleration by central di↵erence (3.90)

[m/s2] =[m/s]� [m/s]

[s]Unit validation

asavg =1

Na

row=NaX

row=1

as(t) mean tangential acceleration (3.91)

µs = tan ✓s static coe�cient of friction (3.92)

[�] = [�] Unit validation

µd = tan ✓ +as

g cos ✓dynamic coe�cient of friction (3.93)

[�] = [�] +[m/s2]

[m/s2]unit validation

f = µdmg cos ✓ friction force from experimental µd (3.94)

[N ] = [�][kg][m/s2][�] Unit validation

f = m1

�asavg + g sin ✓

�friction force from experimental as,avg

[N ] = [kg]([m/s2] + [m/s2][�]) unit validation

(3.95)

3.3.2 Procedure - Studio Portion

Studio Pre-work

Prior to arriving at Studio, each student should have acquired the necessary data in lab,recorded data in your notebook and stored data on a thumb drive. You should also have acorresponding schematic that clearly identifies each measurement that was made in symbolicnotation. Specifically, you should have the following: angle of incipient motion, ✓s and itsuncertainty, angle of each incline trial, ✓ and its uncertainty, mass of the block, m, value forthe distance from the bottom of the ramp to the sensor, L, a data table with three columns,including block position, pB, from the top of the ruler to the block, nominal mean voltagereadings, and standard error in the mean voltage. You will also need the instrument leastcount (ILC) for the ruler.

In addition, you will complete several steps of the Studio exercise. This will allow a moreproductive time with the instructor to discuss the physical meaning of the analysis results.You will upload your studio pre-work to your individual drop-box for the correspondingweek. You will receive a quiz grade based on the completeness of your submission.

Please complete at least steps 1-9 and upload your pre-work spreadsheet toyour individual drop-box, before coming to coming to Studio. You can work on theremaining portions of the exercise during Studio. All steps with the exception of the reportare due within 24 hrs after leaving Studio.

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Videos

There are videos available to help with some of the excel techniques that may be new to you.Please note that the videos are done using Excel 2010 and some elements of theanalysis have been changed since they were produced. In any cases where thereis a discrepancy, you should consider what is in the textbook to be correct. Wehave highlighted steps where videos might be helpful. However, you can also complete thesteps simply by following the written instructions.

Steps to Complete the Analysis

1. LOGBOOK: Before you begin, enter a Studio Week 3 header on a new page in yourlogbook. Use Figure 1.10 as a template. After completing the analysis, you will printout your graphs, answer questions and make observations related to your analysis. Itis important to make it clear that the work entered today is from Studio Week 3.

2. LOGIN: Login to your PC with your RIT account information. Insert your USB driveinto the USB port on your computer.

3. CREATE A NEW FOLDER: On your USB drive, create a working folder called Week 3.Store all of your Lab and Studio files for today’s session in this folder. this week youcan use a good portion of your Week 2 Gravity.xls spreadsheet. Open this file and savewith a new name.

4. CALIBRATION COEFFICIENTS: You can use your calibration worksheet from week2, however you will need to alter it a bit. Please make the necessary corrections sothat you can create a calibration curve for the position ps versus V , referring to Figure3.13. Note from your schematic, which should be drawn in your logbook that ps ismeasured from the bottom of the ramp. Depending on how you recorded the blockposition you may or may not need to account for the sensor o↵set or the location ofthe sensor with respect to the bottom of the ramp. This is for you to work out. Recordthe new calibration equation for block position in your log book and replace the oldcalibration coe�cients in the appropriate cells in your Week3Calibration worksheet.

5. CREATE THE INCLINED RAMP DATA TABLE: You will edit your Week2Gravityspreadsheet. Rename the tab to say Week3Friction. Here you will update the ”BallDrop Data” table to look like that shown in Figure 3.14. Open your data file fromthe inclined ramp experiment lab and copy the two columns of data to your ”InclinedRamp Data from Lab” table. Make sure the data cells remain blue to indicate thatthese data are constants, rather than equations.

6. CREATE THE HEADER, CONSTANTS, CONVERSION FACTORS and CALIBRA-TION COEFFICIENTS TABLES: Edit your table templates so they look like the ta-bles shown in Figure 3.15. Notice that the changes are highlighted in red. Enter thenecessary data and equations. Note, if you have already linked the calibration data to

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Figure 3.13: Screen capture of calibration sheet for this week’s analysis. Note that dependingon how you recorded the block position in lab, your sheet may appear di↵erently.

Figure 3.14: Inclined ramp data measured in lab

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your analysis worksheet by entering a reference equation in the cell, the slope and in-tercept from your calibration curve should automatically update. If you have not donethis, please make the necessary corrections. In addition, you should have an equationin the cell for sample time, so that the value is automatically updated.

VIDEO RESOURCE: Studio 03 Video 1 - Creating Tables

Test: Make sure that your sample time is referencing the new lab data.

Figure 3.15: Header, Table of Constants, Conversion Factors and Calibration Coe�cients.

7. CREATE THE PROCESSED DATA TABLE: Create the table shown in Figure 3.16in your spreadsheet.

VIDEO RESOURCE: Studio 03 Video 1 - Creating Tables

8. APPLY THE CALIBRATION EQUATION TO OBTAIN TANGENTIAL POSITIONFROM THE VOLTAGE READINGS: Enter the necessary equations in column H toconvert voltage, VD[v], to tangential position, ps[m]. Your first equation for ps[m]should be entered into cell H5. Hint: The equation you will use is the trendlinefrom your new calibration curve ps versus V from your Week3Calibration worksheettab. You have these coe�cients referenced in the Calibration Coe�cient Table forconvenience.

VIDEO RESOURCE: Studio 03 Video 2 - Processed Data Table

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Figure 3.16: Template for the Processed Data Table

9. CALCULATE POSITION COMPONENTS: Enter equations in columns I and J tofind the horizontal and vertical components of position. Be careful that your resultyields the correct sign that is consistent with your schematic and assigned coordinateaxis.

10. CALCULATE VELOCITY USING CENTRAL DIFFERENCE METHOD: Enter anequation in column K to calculate the block velocity as a function of time, usinga method called central di↵erence. Note, this method is di↵erent than the forwarddi↵erence method used last week, yet it is derived from the definition of derivative.You may want to refer to Section 3.3.1 for the equation. Drag the formula down to fillthe entire table with velocity values.

VIDEO RESOURCE: Studio 03 Video 2 - Processed Data Table 2:26/4.03

Test: Select a few di↵erent cells that contain your velocity formula, double click andobserve the cell references from the color-highlighted cells. Pay particular attention tothe top and bottom few rows of the table and make sure that you are not referencingempty cells. Delete invalid equations if needed.

11. CALCULATE ACCELERATION USING CENTRAL DIFFERENCE METHOD: En-ter an equation in column L to calculate the block acceleration as a function of time,using the central di↵erence method. Note, that the form of the equation is similarto the previous step when calculating velocity. However, now we are calculating thedi↵erence in velocity over a given time step, rather than the di↵erence in position. Youmay want to refer to Section 3.3.1 for the equation. Drag the formula down to fill theentire table with acceleration values.

Test: Select a few di↵erent cells that contain your acceleration formula, double clickand observe the cell references from the color-highlighted cells. Pay particular attentionto the top and bottom few rows of the table and make sure that you are not referencingempty cells. Delete invalid equations if needed.

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Test: Plot acceleration versus time. Look for obvious outliers and delete if needed.

12. PLOT POSITION COMPONENTS VERSUS TIME: Create one properly formattedplot that contains all three position components, ps, px, pz versus time t. An exampleof what your plot should look like is shown in Figure 3.17.

Test: Is position changing in the correct direction with the correct sign? Does theinitial position correspond to the distance measured from the bottom to the top of theramp? Is the final position occurring at s=0? Why or why not? Make corrections asneeded.

Figure 3.17: Example plot of block position along the ramp

13. RESULTS - ESTIMATE FRICTION: Create a results table in your spreadsheet, asshown in Figure 3.18. Enter equations as needed to calculate the required information.Notice that these friction values are primarily calculated from the mean block acceler-ation, that was determined by applying central di↵erence method first to the positionmeasured in lab and then to the calculated velocity. You should be able to derive theseequations using a FBD of the block and Newton’s 2nd Law.

Test: Compare your acceleration to a neighbor’s. If your angle was larger is your meanacceleration also larger? Make corrections as needed.

Test: Compare the friction force applied to block calculated two ways. Do they agreewithin a reasonable amount and can you justify the di↵erences? Make corrections ifneeded.

Test: Compare the friction force to the tangential component of the weight. Which

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force is larger? Can you justify the di↵erence, particularly if the result does not makephysical sense? Make corrections if needed.

Figure 3.18: Template for recording results

14. CALCULATE THE UNCERTAINTY DUE TO MEASUREMENT PRECISION: Thisweek we have compound error in the position of the block due to the uncertainty inL and the uncertainty in pB. In addition, the angle measurement has an uncertaintythat will create an uncertainty in the trigonometric function calculations. Create theuncertainties table shown in the upper table of Figure 3.19. Enter values (in the shadedcells) and equations (in the white cells) as needed to obtain the required information.

Test: These equations are complicated and it is easy to mistype. Double check your ✏values with a neighbor. If you don’t have the same values for epsilon✓s (cell B29) andepsilon✓ (cell B31) due to di↵erent experimental conditions, temporarily change thesevalues so that you can make an exact comparison. Make corrections to your equationsas needed.

15. DETERMINE THE UNCERTAINTY IN INSTANTANEOUS VELOCITY AND AC-CELERATION: Create the middle table shown in Figure 3.19. Enter equations tocalculate the uncertainties in the estimate for instantaneous velocity, ✏v and the in-stantaneous acceleration, ✏a. Note that since we used the central di↵erence method,rather than the forward di↵erence method, the instantaneous error equation will bedi↵erent.

Test: Has the uncertainty reduced compared to last week? Why or why not? Hint:Since the derivative is now taken across two time steps, the error should be reduced by1/2 compared to last week, assuming the same time step was used in the experiment.

16. DETERMINE THE UNCERTAINTY IN THE CALCULATED RESULTS: Create anuncertainty table for the calculated results and a new type of table called RelativeUncertainties as shown in the lower and right side tables of Figure 3.19. The relativeuncertainties should be determined first by dividing the uncertainty by the magnitudeof the variable to get a fractional uncertainty. These values can be used to simplifythe analysis process for finding compound uncertainties. For example, the relative

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Figure 3.19: Template for calculating the uncertainties for this week’s studio.

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uncertainty of the mass will likely be on the order of 10�4 or 0.001 percent, whereasthe relative uncertainty of the average acceleration will be of oder 1 or 0.1. Therefore,it would be reasonable to treat the mass of the system as a constant in consideringthe compound uncertainty analysis. Similar inspection of the the relative uncertaintyof other values may lead to a similar simplifications. A document will be available inmyCourses content for this week that describes the compound uncertainty analysis inmore detail.

Note: You may want to refer to Section 3.2.9 and the Engineering Mechanics ReferenceTable, currently available on myCourses under Week 1 content. You have the equationfor uncertainty in the mean acceleration from last week.

Test: The error in the mean acceleration should be smaller than the error in theinstantaneous acceleration. Make corrections to your equations as needed.

Test: These equations are complicated and it is easy to mistype. Double check yourvalues with a neighbor. Temporarily change constants if needed so that an exactcomparison can be made. Make corrections to your equations as needed.

17. UPDATE YOUR ENGINEERING LOGBOOK: Please print out your graphs, resultstable and uncertainty tables and paste them in your logbook. You may also want toprint all of the tables, but this is left for the student to decide. Sign and date yourlogbook before you leave Studio.

18. SUBMIT YOUR FILES: Submit your excel spreadsheet to your individual Week 3dropbox on myCourses before leaving the Studio. If you have not completed all thesteps, upload what you have done and within 24 hours, upload your final completedversion. Remember to save your work to you USB drive, and take it with you whenyou leave the Studio.

19. OBSERVATIONS AND ANALYSIS: Write responses to the following questions in yourlogbook. Be sure to include a justification for your answer by referring to the data,plots, and derivations that are contained within your logbook. You may want to cross-reference equations from Sections 3.1, 3.3.1 and 3.2.9 in your work.

(a) How do your findings show agreement or disagreement with Newton’s First Law?

(b) How do your findings show agreement or disagreement with Newton’s SecondLaw?

(c) How do your findings show agreement or disagreement with Newton’s Third Law?

(d) How do your findings show agreement or disagreement with Newton’s Law ofgravity?

20. CONGRATULATIONS! You have just completed the Studio portion for week 3.

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21. WRITE THE REPORT: Please refer to section 3.3.3 Report on details for the reportsubmission. Before leaving Studio, decide on a date and time to meet up with yourteam mates to prepare the report. Reports are due Monday by 5 pm.

3.3.3 Report

Please use the same task distribution for writing the report that was outlined in Week 1.Refine your “Team Norms” to enhance your team’s ability to work e↵ectively with oneanother. Prepare a report to include only the following components:

• TITLE PAGE: Include the title of your experiment, “Single Body in ConstrainedLinear Motion”, Team Number, date, authors, with the scribe first, the teammember’s role for the week, and a photograph of each person beginning to initiatetheir trial, with a label below each photo providing team member’s name.

• PAGE 1: The heading on this page should read Experimental Set-up. Create adiagram of the experimental set-up. This week we will include only the diagram andits caption. Thus, is it important that your diagram clearly communicate the set-up,including each key component and where measurements were taken. The importantinformation to communicate are the variable names and datums that relate to yourmeasurements and results. It is a good practice to add a legend that defines anyvariables or components of the schematic that are not obvious. At the bottom of thefigure include a figure caption, for example Figure 1. A brief figure caption. Referto the text for examples.

Note: Figure captions are required for every plot and diagram in the report, exceptfor the title page. Figure captions are placed below the figures, and are numberedsequentially beginning with Figure 1 for the first figure in the report.

• PAGE 2: The heading on this page should read Results. Include the table shown inTable 3.2 summarizing each team member’s results. At the top of the table, includea table caption, for example Table 1. A brief table caption. Refer to the text forexamples.

This week we include only tables and plots with no accompanying text. Thus, it isimportant that your tables, graphs and captions clearly communicate to the readerwhat the data represents.

Note: Table captions are required for every table in the report, except for the title page.Unlike figure captions, table captions are placed above the tables, and are numberedsequentially (independent of figure caption numbering) beginning with Table 1 for thefirst table in the report.

• PAGE 3: No heading is needed on this page, since it is a continuation of the Resultssection. Include the plots for position components , ps, px and pz. The position values

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Table 3.2: Results from the Inclined Ramp Experiment

should be consistent with the datums shown in your schematic, which should agreewith the conventions provided in the text. Include one plot for each team member,with one figure caption for each plot. Each figure caption should be located below thegraph and include the name of the test engineer responsible for that series of trials. Thefirst caption in the Results section should be for example Figure 2. Block positionresults for Princess Leia.

Strive for uniformity among the graphs, including axis values and plotting styles. Thiswill enable the reader to easily compare the results between di↵erent team members.

• The final report should be collated into one document with page numbers and a con-sistent formatting style for sections, subsections and captions. Before uploading thefile, you must convert it to a pdf. Non-pdf version files may not appear the same indi↵erent viewers. Be sure to check the pdf file to make sure it appears as you intend.

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3.4 Recitation

Many students taking this class are enrolled in Calculus I along with this class. Limits anddi↵erentiation are among the earliest topics covered in Calculus I. Later in Calculus I andthroughout Calculus II you will also study the topic of integration. Let’s re-visit the balldrop problem from the perspective of integral calculus. This presentation will show theproblem solving method in its basic form, similar to how a student would solve the problem,with minimal expository commentary.

3.4.1 Formulate

Given a mass, m, released from rest at an initial elevation z(t = 0) = z0

and allowed to fallto the ground under the action of the local acceleration of gravity, g, predict the verticalposition of the object as a function of time.

Given:

m = [kg] � Mass of Dead Weight (Constant) (3.96)

g = 9.81 [m/s2] � Standard value of gravitational acceleration (3.97)

W = mg [N ] � Gravitational Force Upon m (Constant) (3.98)

z(t = 0) = z0

[m] � Initial Elevation of m (Constant) (3.99)

Vz(t = 0) = Vz0

[m] � Initial Velocity of m (Constant) (3.100)

Find:

z(t) = ? [m] ↵ Vertical position of m vs time (3.101)

3.4.2 Assume

Assume:

f = 0 [N ] Neglect air friction (3.102)

z = 0 [m] Origin measured from floor (3.103)

Equation 3.102 is reasonable for aerodynamically shaped objects dropped from a small ele-vation. Equation 3.103 is consistent with our coordinate system.

3.4.3 Chart

The Schematic and FBD for the problem are shown in Figure 3.20.

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Figure 3.20: Schematic and FBD of a mass in the proximity of the Earth.

3.4.4 Execute

Recall Newton’s Second Law:

X�!F =

d(m�!V )

dt=

d�!pdt

Newton’s 2nd Law (3.104)

Using Eq. 3.96 we can move m outside of the di↵erential, and write Equation 3.104 as

X�!F = m

d�!V

dt(3.105)

Now, since we are only concerned with the change in vertical position, we can rewrite thevector form of Equation 3.105 for the vertical component only, and use the FBD to write:

�mgk = mdVz

dtk (3.106)

Since m is common to both sides of Equation 3.106, and it is a single component equation,we can drop the m and k notation. Then, we can separate variables as:

�gdt = dVz (3.107)

We take the indefinite integral of 3.107 and add a constant of integration, C0

:

�Z

gdt = �gt+ C0

=

ZdVz = Vz(t) (3.108)

We know from Equation 3.100, that the initial velocity is zero when t = 0, so we must haveC

0

= 0 for the constant of integration, and we can write the vertical velocity as simply:

Vz(t) = �gt (3.109)

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Now, the vertical velocity component is defined as the time rate of change of vertical position,Vz(t) ⌘ dz(t)/dt, which we can substitute into Equation 3.109 to get

�gt =dz(t)

dt(3.110)

Once again, we can separate the time and position terms:

(�gt)dt = dz(t) (3.111)

Now, we integrate both sides:Z

(�gt) dt =

Zdz(t) (3.112)

upon taking the indefinite integral, we add another constant:

�gt2

2+ C

1

= z(t) (3.113)

We know from Equation 3.99, that the initial elevation is z0

when t = 0, so we must haveC

1

= z0

for the constant of integration, and we can write our final answer for the verticalposition as a function of time:

z(t) = z0

� g

2t2 (3.114)

3.4.5 Test

Let’s validate that the equations for position and velocity are consistent with our intuition,and that they have appropriate units. The velocity only has a vertical component, and wasgiven by Equation 3.109. We know from the FBD that three should be no motion in the xor y directions, so we write the velocity vector as

��!Vz(t) = �gtk (3.115)

[m/s] = [m/s2][s] Units validation

At time t = 0, the velocity is zero, consistent with the given information. The velocity isnegative, which is consistent with a ball falling down toward the Earth, when the z coordinatepoints upward away from the Earth. Equation 3.115 is dimensionally correct. The positionequation is

��!z(t) = z

0

� g

2t2k (3.116)

[m] = [m]� [m/s2]

[�][s2] Units validation

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At time t = 0, the elevation is z0

, consistent with the given information. Equation 3.116 isdimensionally correct. The position starts at a maximum value, and then decreases parabol-ically with time. This is consistent with our previous observations. As an intuitive check,assume for the moment that z

0

= 1.5[m]. When the object impacts the ground, we wouldhave z(timpact) = 0. We can then evaluate Equation 3.116 to determine how long it wouldtake the object to drop 1.5[m] from rest under the action of gravity as:

0 = z0

� g

2t2impact (3.117)

z0

=g

2t2impact (3.118)

2z0

g= t2impact (3.119)

r2z

0

g= timpact (3.120)

r2(1.5)

9.81= 0.553 (3.121)

s[�][m]

[m/s2]= [s] Units validation

timpact ⇡ 0.553[s]

This result is consistent with our experimental observations in Lab last week, so we arehighly confident that the result is correct.

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3.5 Homework Problems

Complete all assigned homework problems in your logbook.

3.5.1 Do you expect the magnitude of the acceleration of the ball rolling down the ramp tobe greater than, equal to, or less than the magnitude of acceleration that was observedin the previous free fall experiment? Justify your answer.

3.5.2 Describe the di↵erence between vector and a scalar.

3.5.3 Define “magnitude,” “direction,” “component, ” “resultant” and “unit vector.”

3.5.4 Give an example of the “associative rule” as it applies to scalars.

3.5.5 Give an example of the “commutative rule” as it applies to scalars.

3.5.6 Give an example of the “associative rule” as it applies to vectors.

3.5.7 Give an example of the “commutative rule” as it applies to vectors.

3.5.8 Given a first vector �!s1

= 3ı+3|�2k and a second vector �!s2

= �1ı+7|+4k, computethe following:

(a) �!r1

= �!s1

+�!s2

(b) �!r2

= �!s1

��!s2

(c) �!r3

= 2�!s1

� 5�!s2

(d) |�!r2

|(e) |�!r

3

|

3.5.9 Determine graphically and numerically the resultant vector �!r for the following prob-lems. Express your answer for each problem in three forms: “graphically” (tip-to-tail),in Cartesian form (�!r = xı+ zk), and in polar form (�!r = |r|\✓).

(a) �!r1

= �!s1

+�!s2

(b) �!r2

= �!s1

��!s2

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3.5.10 Determine graphically and numerically the resultant vector �!r for the following prob-lems. Express your answer for each problem in three forms: “graphically” (tip-to-tail),in Cartesian form (�!r = xı+ zk), and in polar form (�!r = |r|\✓).

(a) �!r1

= �!s1

+�!s2

+�!s3

(b) �!r2

= �!s1

��!s2

+�!s3

(c) �!r3

= �!s1

��!s2

��!s3

(d) �!r4

= �!s3

+�!s1

+�!s2

3.5.11 Given that a floor tile in the hallway measures 1[ft] ⇥ 1[ft], show all work as youdetermine the following

(a) The edge length expressed in units [in].

(b) The edge length expressed in units [m].

(c) The edge length expressed in units [cm].

(d) The surface area in SI units.

3.5.12 Given that the nominal acceleration of gravity near the Earth’s surface has a magnitude9.81[m/s2], show all work as you determine the magnitude of the acceleration of gravitywhen the length is measured in [miles] and the time is measured in [hours].

3.5.13 Given that a rocket leaves the launch pad with an acceleration of 5g[m/s2], and ne-glecting the change in the mass of the rocket and its fuel, what is the speed of therocket (in units [miles/hour]) at a time 60[s] after lift-o↵? What is the elevation ofthe rocket at that time, in units [ft]? If there is a significant reduction in the mass ofthe rocket and its fuel, will the rocket be moving faster or slower in reality? Justifyyour answer using Newton’s Laws.

3.5.14 Convert each of the following angular values from degrees into radians: 1�, 45�, 60�,90�, 180�, 360�. Show all work, and the unit conversion factors.

3.5.15 Convert each of the following angular values from radians into degrees: ⇡/6, ⇡/4, ⇡/3,⇡/2, 2⇡. Show all work, and the unit conversion factors.

3.5.16 Consider a block of massm placed on a test apparatus as shown. Determine the normaland tangential components of the weight due to the block acting upon the apparatusfor each of the following scenarios. Show all work.

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Scenario Mass, m Angle, ✓[�](a) 10 [kg] 10(b) 5 [slugs] 55(c) 25 [kg] 80(d) 10 [slugs] 35(e) 15 [kg] 45(f) 30 [slugs] 25(g) 20 [kg] 0(h) 20 [slugs] 90

3.5.17 Given a block of mass, m = 0.2[kg], released from rest on an inclined ramp makingan angle ✓ = 45[�] to the horizontal at an initial distance s(t = 0) = 0.5[m] fromthe top of the ramp and allowed to slide to the bottom of the ramp subject to thelocal acceleration of gravity, g = 9.81[m/s2], and a dynamic coe�cient of frictionµd = 0.3[�], derive an expression for the tangential position of the mass as a functionof time until it reaches the bottom of the ramp. How long will it take for the mass toreach the bottom of the ramp?

3.5.18 What force must you apply to push a box containing your new HD LED TV weighing120[lbs] up an incline of 20[deg] at a constant velocity? Assume the coe�cient of staticfriction is 0.35 and the coe�cient of dynamic friction is 0.25 between the box and thefloor. Clearly identify all forces acting on the box in your Free Body Diagram.

3.5.19 A block is being pulled up a ramp at a constant velocity by a force of 100[N ]. Calculatethe coe�cient of friction between the block and the inclined ramp if it is known thatthe mass of the block is 12[kg] and the ramp is inclined 35[degrees] from the horizontal.

3.5.20 A system with a mass on an inclined ramp similar to the lab experiment for this week(reference schematic below) has a known static coe�cient of friction of 0.55 and adynamic coe�cient of friction of 0.4. Calculate the maximum value the mass [kg]can have and still remain stationary for each angle of inclination indicated below.

Scenario Angle, ✓[�](a) 15(b) 25(c) 35

159