Classical Mechanics...Force dynamics 2.notebook 1 June 09, 2014 Classical Mechanics In physics,...

Forcedynamics 2.notebook

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June 09, 2014

Classical MechanicsIn physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering, and technology.Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other major subfield of mechanics, quantum mechanics.


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June 09, 2014

Classical Mechanics

The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself. There are several branches of mechanis, on we already studied.

Kinematics

Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion. The study of kinematics is often referred to as the geometry of motion. Kinematics answer the question "HOW" something moved.

To describe motion, kinematics studies the trajectories of points, lines and other geometric objects and their differential properties such as velocity and acceleration. We focussed on displacement, velocity, and acceleration with the use of the "BIG" 3.


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June 09, 2014

What causes something to move (or not move) the way it does?To answer this question, we now look to another branch of classical mechanics. Dynamics.

Dynamics

Dynamics is a branch of physics (specifically classical mechanics) concerned with the study of forces and torques and their effect on motion, as opposed to kinematics, which studies the motion of objects without reference to its causes.

Generally speaking, researchers involved in dynamics study how a physical system might develop or alter over time and study the causes of those changes. In addition, Isaac Newton established the physical laws which govern dynamics in physics. By studying his system of mechanics, dynamics can be understood. In particular, dynamics is mostly related to Newton's second law of motion. However, all three laws of motion are taken into consideration, because these are interrelated in any given observation or experiment. Dynamics answers the question "WHY" something moves the way it does, it tells what causes that behavior.


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June 09, 2014

ForceIn physics, a force is any influence that causes an object to undergo a certain change, either concerning its movement, direction, or geometrical construction. It is measured with the SI unit of newtons (N) and represented by the symbol F. In other words, a force is that which can cause an object with mass to change its velocity (which includes speeding up, changing its direction, or slowing down), i.e., to accelerate. Also, a force can cause a flexible object to deform. Forces are an interaction between TWO objects. It is not possible to have a force on an object and not have another object involved. Force can also be described by intuitive concepts such as a push or pull. A force has magnitude, unit, and direction, making it a vector quantity. However, all forces may not cause a change in velocity or a deformation. When analyzing force, the cumulative effect of ALL the forces acting on an object must be accounted for, and will determine if the object accelerates, deforms, or if the objects state remains unchanged.

Understanding the definition of Force is another example when the familiarity of the topic is necessary to understand it.


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June 09, 2014

The Fundamental Forces of Nature

There are 4 fundamental forces that have been identified. In our present Universe, and they have rather different properties.

1. Strong Nuclear Force

2. Electromagnetic Force

3. Weak Nuclear Force

4. Gravitational Force


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June 09, 2014

Strong Nuclear Force

The strong nuclear force or strong interaction is very strong, but very shortranged. It acts only over ranges of order 1013 centimeters and is responsible for holding the nuclei of atoms together. It is basically attractive, but can be effectively repulsive in some circumstances.

A force which can hold a nucleus together against the enormous forces of repulsion of the protons is strong indeed. However, it is not an inverse square force like the electromagnetic force and it has a very short range. Yukawa modeled the strong force as an exchange force in which the exchange particles are pions and other heavier particles. The range of a particle exchange force is limited by the uncertainty principle. It is the strongest of the four fundamental forces.


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June 09, 2014

Electromagnetic ForceThe electromagnetic force causes electric and magnetic effects such as the repulsion between like electrical charges or the interaction of bar magnets. It is longranged, but much weaker than the strong force. It can be attractive or repulsive, and acts only between pieces of matter carrying electrical charge.The electromagnetic force manifests itself through the forces between charges (Coulomb's Law) and the magnetic force, both of which are summarized in the Lorentz force law. Fundamentally, both magnetic and electric forces are manifestations of an exchange force involving the exchange of photons. The electromagnetic force is a force of infinite range which obeys the inverse square law, and is of the same form as the gravity force.

Like charges repelOpposite charges attract

F = Kq1 q2r2


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June 09, 2014

Weak Nuclear ForceThe weak nuclear force or weak force is responsible for radioactive decay and neutrino interactions. It has a very short range and, as its name indicates, it is very weak.the weak interaction involves the exchange of the intermediate vector bosons, the W and the Z. This force influences the formation of the subatomic particles. The weak interaction changes one flavor of quark into another. It is crucial to the structure of the universe in that:

1. The sun would not undergo nuclear reactions without it. The weak interaction causes the transmutation protons into neutrons so that deuterium can form and deuterium fusion can take place.

2. It is necessary for the buildup of heavy nuclei. (elements with large atomic numbers)The weak force is responsible for radioactive decay and neutrino interactions.


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June 09, 2014

Gravitational ForcesThe gravitational force is the weakest fundamental force, but very long ranged. Furthermore, it is always attractive, and acts between any two pieces of matter in the Universe since mass is its source.Gravity is the weakest of the four fundamental forces, yet it is the dominant force in the universe for shaping the large scale structure of galaxies, stars, etc. Gravitational forces are usually the predominant forces acting on large objects. The gravitational force between two masses m1 and m2 is given by the relationship:

This is often called the "universal law of gravitation" and G the universal gravitation constant. It is an example of an inverse square law force. The force is always attractive and acts along the line joining the centers of mass of the two masses. The forces on the two masses are equal in size but opposite in direction, obeying Newton's third law. Viewed as an exchange force, the massless exchange particle is called the graviton.

F = Gm1 m2r2


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June 09, 2014

Common Forces In MechanicsIn mechanics, we encounter several kinds of forces. The gravitational force is, of course, all pervasive. Every object on the earth experiences the force of gravity due to the earth. Gravity also governs the motion of celestial bodies. The gravitational force can act at a distance without the need of any intervening medium.All the other forces common in mechanics are contact forces. As the name suggests, a contact force on an object arises due to contact with some other object: solid or fluid. When bodies are in contact (e.g. a book resting on a table, a system of rigid bodies connected by rods, hinges and other types of supports), there are mutual contact forces (for each pair of bodies) satisfying the third law. The component of contact force normal to the surfaces in contact is called normal reaction. The component parallel to the surfaces in contact is called friction. Contact forces arise also when solids are in contact with fluids. For example, for a solid immersed in a fluid, there is an upward buoyant force equal to the weight of the fluid displaced. The viscous force, air resistance, etc are also examples of contact forces


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June 09, 2014

Common Forces In Mechanics

The following is a list of the forces that are commonly used in mechanics. Along with their symbols. The symbols we will use are in red.

1. Gravitational Force Fg or W (for weight) (sometimes mg is used because Fg = mg)

2. Normal Force FN or N

3. Frictional force f, Ffrict, or Ff4. Tension FT or T

5. compression FC, FA, or Fapp (for applied force)

6. spring force FS7. Air Resistance or Drag Force FD or Fair

These are by NO means the only symbols you might find to represent these forces.


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June 09, 2014

Gravitational ForceThe force of gravity (Fg) is the force at which the earth, moon, or other massively large object attracts another object towards itself. By definition, this is the weight of the object. All objects upon earth experience a force of gravity which is directed "downward" towards the center of the earth. It is defined as the force of attraction acting on all objects on or near the surface of earth that acts towards the center of earth.

Note the direction is towards the center of Earth NOT down!!!


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June 09, 2014

Normal Force

The normal force (FN) is the support force exerted upon an object which is in contact with another stable object. For example, if a book is resting upon a surface, then the surface is exerting an upward force upon the book in order to support the weight of the book. On occasions, a normal force is exerted horizontally between two objects which are in contact with each other. It is defined as a force applied by a surface to an object that acts perpendicular to the surface.

FN

Fg

Note: The normal force does NOT always balance gravity, but will always act perpendicular to the surface.


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Friction ForceThe friction force (f) is the force exerted by a surface as an object moves across it or makes an effort to move across it. The friction force opposes the motion of the object. For example, if a book moves across the surface of a desk, then the desk exerts a friction force in the opposite direction of its motion. Friction results from the two surfaces being pressed together closely, causing intermolecular attractive forces between molecules of different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. It is defined as an electrical force that resists the sliding of an object on a surface that acts parallel (tangential) to the surface and opposite to the direction that the object is sliding or would tend to slide.

FrictionMo

tion

Note: the object does NOT have to be moving for there to be a frictional force. Friction acts on objects that are at rest to resist their moving.


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June 09, 2014

Tension ForceThe tension (Ft) is the force which is transmitted through a string, rope, or wire when it is pulled tight by forces acting from each end. The tensional force is directed along the wire and pulls equally on the objects on either end of the wire. It is defined as a pulling force that acts in the direction that the object is being pulled.

Motion

Note: The tension force does NOT always act in the direction that the object is moving.


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Compression ForceA compression force (Fc) is a force which is applied to an object by a person or another object. If a person is pushing a desk across the room, then there is a compression force acting upon the object. The compression force is the force exerted on the desk by the person. It is defined as a pushing force that acts in the direction that an object is being pushed.

compression

motion

Note: The compression force does NOT always act in the direction that the object is moving.


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Spring ForceThe spring force is the force exerted by a compressed or stretched spring upon any object which is attached to it. An object which compresses or stretches a spring is always acted upon by a force which restores the object to its original (un stretched) position. For most springs (specifically, for those which are said to obey "Hooke's Law"), the magnitude of the force is directly proportional to the amount of stretch or compression. It can be defined as a force applied by a spring to an object in a direction that restores the spring to its original shape.

originalshape

Red vectors are the Spring ForceBlue vectors are the Displacement

originalshape

compression

tension

Note: the spring force ALWAYS acts in the opposite direction of the displacement.


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June 09, 2014

Air resistance or DragAir resistance is a special type of frictional force which acts upon objects as they travel through the air. Like all frictional forces, the force of air resistance always opposes the motion of the object. This force will frequently be neglected due to its negligible magnitude. It is most noticeable for objects which travel at high speeds (e.g., a skydiver or a downhill skier), small mass, or for objects with large surface areas. Air resistance is a function of the velocity, crosssectional area, and the surface texture of the object. Air resistance is a force that is caused by air. The force acts in the opposite direction to an object moving through the air. A truck with a flat front will experience high air resistance while a sports car with a streamlined shape will experience lower air resistance, allowing the car to go faster.

FD

Fg

Motion

FC FC

FNFN


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June 09, 2014

Definitions

Gravitational Force the force of attraction acting on all objects on or near the surface of earth that acts towards the center of earth.

Normal Force a force applied by a surface to an object that acts perpendicular to the surface.

Friction Force an electrical force that resists the sliding of an object on a surface that acts parallel (tangential) to the surface and opposite to the direction that the object is sliding or would tend to slide.

Tension a pulling force that acts in the direction that the object is being pulled.

Compression a pushing force that acts in the direction that an object is being pushed.


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Definitions Quiz

1) Gravitational Force

2) Normal Force

3) Friction Force

4) Tension

5) Compression


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Applied ForceAn applied force is a force that is applied to an object by a person or another object. If a person is pushing a desk across the room, then there is an applied force acting upon the desk. The applied force is the force exerted on the desk by the person (compression). If a child pulls a wagon, then there is an applied force on the wagon. The applied force is the force exerted on the wagon by the child (tension). Some external forces (most commonly, tension and compression) are sometimes called applied forces.

Some other examples of applied forces are the thrust of a jet engine or rocket, the force a cars engine gives the wheels, and many more.


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Units of ForceThe basic unit of force in terms of physical dimensions is MLT2, or

FPS (footpoundsecond) Unit of ForceWhen a mass of 1 pound is subjected to an acceleration of 1 ft/sec2, the force applied is called 1 poundal. The unit is abbreviated as pdl. One poundal is 1lbmft/sec2

A pound force is defined as the gravitational force exerted on 1 pound mass. It is denoted as lbf. When we say that the weight of the person is 150 pounds, we actually mean 150 lbf. Since the 18th century, the unit has been used in lowprecision measurements, for which small changes in Earth's gravity (which varies from place to place by up to half a percent) can safely be neglected. (The pound mass or the slug is a unit of mass associated with Imperial units. It is a mass that accelerates by 1 ft/s2 when a force of one poundforce (lbF) is exerted on it.)

MLT2


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Metric Unit of Force In metric system 1 kilogram force is the gravitational force exerted on 1 kilogram mass. It is similar to pound force in fps system. It is denoted as kgf. The force that produces 1 m/sec2 acceleration on 1 kilogram mass is called as 1 Newton. This is named in the honor of Sir Isaac Newton. In short it is abbreviated as N. The metric unit of force has also smaller measures and that system is called cgs system. In cgs system a force is measured in terms of dynes. A force of 1 dyne produces an acceleration of 1cm/sec2 on a mass of 1 gram. The gravitational force exerted on 1 gram weight is called gram force and denoted as gf.

N = Kg mS2

Internationally, the engineers and scientists had jointly decided to use a single system of unit which is called as SI units.(Standard International). In SI units, Newton is the unit recognized for measurement of force.


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Important Properties of Forces:

• Forces are an interaction between TWO objects. It is not possible to have a force on an object and not have another object involved.

• Forces are vectors. They have magnitude, unit, and direction• The unit for force is the Newton. (1 N = 1 kg m / s2)• Forces are NOT properties of an object like mass, speed, or color. They

are properties of an interaction between two objects.


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June 09, 2014

Free Body DiagramsA free body diagram (FBD), sometimes called a force diagram is a pictorial device, often a rough working sketch, used by engineers and physicists to analyze the forces and moments acting on a body. Freebody diagrams are used to show the relative magnitude and direction of all forces acting upon an object in a given situation. A freebody diagram is a special example of the vector diagrams which were used when solving parallelogram and numerical vector addition problems; these diagrams are used throughout the study of physics. The size of the arrow in a freebody diagram is reflective of the magnitude of the force. The direction of the arrow reveals the direction which the force is acting. Each force arrow in the diagram is labeled to indicate the exact type of force. It is generally customary in a freebody diagram to represent the object by a dot and to draw the force arrow from the center of the dot outward in the direction that the force is acting. The dot represents the center of mass of the object.


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Free Body DiagramsA free body possesses three fundamental properties that make the concept useful in analyzing static and dynamic equilibrium:1. The internal forces of the free body are, by definition, assumed to

completely offset one another so that the body as a whole is in static equilibrium.

2. The body can be treated as a single entity with its total mass located entirely at the center of mass and subject to Newton's Laws.

3. A complete free body diagram shows all forces acting on an object with their directions.


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Constructing a freebody force diagram1. Select an object or group of objects to focus on as the "body", i.e. the

system.2. Sketch the body by itself, "free" of its surroundings. The body could be

represented by a single point located at the body's center of mass.3. Draw only those forces that are acting directly on the body. Include

both the magnitude and the direction of these forces.4. Except for rotational problems, you can normally sketch the forces as

though they were acting through a single point at the center of mass of the body. It is useful to draw the forcevectors with their tails at the center of mass.

5. Do not include any forces that the body exerts on its surroundings, they do not act on the body. However, there is always an equal reaction force acting on the body.

6. For a compound body you do not need to include any internal forces acting between the body's subparts, since these internal forces come in actionreaction pairs which cancel each other out because of Newton's Third Law.

7. Choose a coordinate system and sketch it on the freebody diagram. If you choose one of the axes to be parallel to the object's acceleration, it can sometimes simplify the equations you have to solve. In physics I, the freebody diagrams are shifted so that the X and Y axes are in their standard position.

8. All Forces need a direction. If a force is on an axis, then the angle does not have to be written down. However, if a force is in a quadrant, then an angle between the force vector and the closest X axis must be specified.


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Drawing Free Body DiagramsThe best way to learn how to draw free body diagrams is through repetition and practice. Knowledge of the force definitions will be of paramount importance. Consider the diagram below:

30o20oM

Vconst

First, replace the object with a dot that represents the center of mass of the object. If the object is supported by a surface, it may be helpful to draw a dotted line representing the surface through the dot and at the same angle that the surface has in the diagram (Note this line is NOT part of the free body diagram). It is also helpful to put a small piece of the X axis going through the dot to remind you to add angles for each vector relative to the X axis. See the diagram below:


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Fg

30o20oM

Vconst

Picture FBD

Drawing Free Body DiagramsSecondly, start adding forces into the diagram. (Remember, only those forces acting on the object are part of its FBD (Free Body Diagram). Never place forces that the object applies to other things in the diagram. The easiest force to add is the gravitational force. It is defined as the force of attraction acting on all objects on or near the surface of earth that acts towards the center of earth. The definition states that the force needs to be directed towards the center of Earth, which means that the force will be perpendicular to the X axis and act towards the negative Y domain. Placing this force on the diagram will look like this:

It is not necessary to write angles for vectors that are on an axis. It is understood that Fg acts at 270o.


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June 09, 2014

30o20oM

Vconst

Picture FBD

Drawing Free Body DiagramsNow add a second force, such as the normal force. The normal force (FN) is the support force exerted upon an object which is in contact with another stable object or surface. If there is no support from an object or surface, then there is no normal force. In this example there is a normal force present. By definition the normal force is a force applied by a surface to an object that acts perpendicular to the surface. (The force acts perpendicular or normal to the surface.) When drawing the force on the diagram it must be drawn at a 90o angle with the dotted line we drew initially. (The normal force will NOT always be on the + Y, DO NOT think of it as the force that is opposite gravity!) The free body diagram will now look like this:

Fg

FN


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June 09, 2014

30o20oM

Vconst

Picture FBD

Drawing Free Body DiagramsNext add a third force to the diagram, like friction. Since the object is sliding on a surface, there will be a frictional force acting against the motion. Friction is a resistive force that acts against the sliding of an object on a surface. The force acts parallel (tangential) to the surface and opposite to the direction that the object is sliding or would tend to slide. The definition states that friction acts parallel to the surface which means it will lie on the dotted line on our FBD. The only question is what the direction of the force will be. The definition has this covered also. The direction is opposite to the direction that the object is sliding or would tend to slide. Since the object is sliding to the right, the friction will act to the left. The result is the FBD below:

Fg

FN

f

Note: It is not necessary to write angles for vectors that are on an axis. These angles are understood.


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June 09, 2014

30o20oM

Vconst

Picture FBD

Drawing Free Body DiagramsNow, it is time to add the applied forces like tension and compression. In our example there is both a compression and a tension force acting on the object. A compression force can be recognized due to the fact that the arrow head is touching the block in the diagram, which means the force is being directed into the block. A tension force can be recognized due to the fact that the arrow head is not touching the block in the diagram, which means this force is being directed away from the block. Let's draw in the tension force. Since the tension force is acting in the first quadrant, it will be drawn in the first quadrant. The angle that is formed between the force and the X axis also must be specified. The FBD will now look like the diagram below:

Fg

FN

f

FT

30o

Note: the tension force will act in the direction as seen on the picture, but be careful all pictures may not be drawn to scale.


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Drawing Free Body DiagramsFinally, lets draw the compression force. This force is one of the more complicated forces to place on the free body diagram. Because the force is being directed into the object, one might wish to place the head of the vector on the dot in the diagram. Doing that, however, is incorrect! Think of it this way, a push is the same as a pull, only on the opposite side. If one pushes the object to the right with a force of 10 N it is same as pulling the object to the left with a 10 N force.

FC = 10 N FT = 10 NEquivolent

Imagine that the compression force is pushed directly through the object to the other side.

FC = 10 N

FT = 10 N

Equivolent30o

30o30o

FC

FC


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June 09, 2014

30o20oM

Vconst

Picture FBD

Drawing Free Body DiagramsNow, it is time to add the the compression force. On the original diagram, it is helpful to draw the compression force so that it looks like its tension analogue. Since the equivalent compression force is acting in the fourth quadrant, it will be drawn in the fourth quadrant. Notice, the angle that is formed between the original force and the surface; and the equivalent force and the surface, are equal. This happens because the two forces are collinear and forms vertical angles with the surface. Adding the compression force to the FBD results in the following:

Fg

FN

f

FT

30o

Note: the compression force will NOT act in the quadrant it is seen in on the picture.

20o

20o

FC


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June 09, 2014

FBD for Objects on an Incline1) Draw the dot for the center of mass, a dotted line representing the surface, and a small section of the X axis.

20 o

Vconst 40o40o

2) Add the gravitational force.

40oFg

20 o

Vconst 40o


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20 o

Vconst 40o 40oFg

3) Add the normal force. (Notice that it is not opposite the gravitational force.)

FN

?

In order to calculate the angle for the normal force, we will use geometry.

40oFg

FN

?

We know by definition that the normal force acts perpendicular to the surface. Therefore, it will form a 90o angle with the dotted line.


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Alternate Interior Angles Theorem and Vertical AnglesWhen two lines are crossed by another line (which is called the Transversal), the pairs of angles on opposite sides of the transversal but inside the two lines are called Alternate Interior Angles. The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal, the resulting alternate interior angles are congruent.

Vertical anglesA = DG = F

A B

C D

E F

G H

A B

C D

E F

G H

Alternate interiorAnglesE = D


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Using the fact that alternate interior angles are congruent. We can find the missing angle.

40oFg

FN

? We know that the alternate interior angle with the surface angle is congruent.

40oFg

FN

40o

We, also, know that the unknown angle for FN and the 40o angle must add up to 90o. They are complimentary angles. So the missing angle is 50o. As it turns out, in an unshifted FBD, the angle for the normal force is always the complimentary angle to the surface!

50o


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June 09, 2014

40oFg

FN

40o50o

5) Add the frictional force: By definition it is parallel to the surface, which means it will be drawn on the dotted line. But in what direction? Again by definition it is drawn in the opposite direction of the motion of the object. Since the object is moving up the incline, friction will act down the incline or in the fourth quadrant. Which yields the following FBD. (note the angle for friction had previously been calculated.)

20 o

Vconst 40of

Remember friction will ALWAYS be drawn on the dotted line.


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6) Finally add the tension force.

40oFg

FN

40o50o

20 o

Vconst 40of

Note: the dotted line represents the surface, and the tension force is acting 20o above the surface. So the tension force must be drawn 20o above the dotted line. But the angle that is needed is the angle between the tension force and the negative X axis.

FT

20 o?


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40oFg

FN

40o50o

f

FT

20 o

We know that the 40o angle for friction is equal to the angle between the dotted line and the negative X axis because they are vertical angles.

The angle we need to find is the sum of the 2 angles 20o and 40o which is 60o.

Note: Finding the angle for tension or compression is complicated, you will NOT always be adding, and the vectors may end up in a different quadrent than was seen in the picture. (pictures ARE NOT drawn to scale)

40o60o


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The final FBD looks like this:

40oFg

FN

40o50o

f

FT

60o20 o

Vconst 40o

Picture FBD


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Draw an FBD for the following. Be sure to label each force and show the correct directions. (remember forces that are on an axis do not need an angle)

1)

20o

Picture FBD

2)

20o

Vconst

Vconst


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3) 35o

Picture FBD

4)

15 oVcons

t

Vconst

20o 20o

35o 35o


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5)

33o

Picture FBD

6)

35 oVcons

t

Vconst

33o 33o

27o 27o


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June 09, 2014

How do you deal with the situation when 2 or more forces are acting in the same direction?

20o

Vconst

Fg

FN

f

FT

20o

Fc is acting in the same

direction as Fg

Fc

Draw the vectors acting in the same direction by offsetting the vectors and then drawing them parallel to each other.


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30o50o

Fg

30o50o

FT1FT2

When you have the same type of force acting multiple times on the object each force has to be specified with a different subscript. Ex. FT1 and FT2.


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7)

Picture FBD

8)

35 o

Vconst

27o27o

20o

Vconst

63 o

Fg

Fg

FN

FN

63o

f

f

27o

20o

FT1FT2

63 o

Fc1

8o

Fc2

35 o


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June 09, 2014

We now know something about force and how to draw FBD's. So it's time to answer the question "WHY".1. Why doesn't a boulder move? Why is it at rest?2. Why does a car move with a constant velocity on the highway?3. Why does a cannon ball dropped from a building speed up?4. Why does a truck slow down while braking?5. Why does a bicyclist change direction?6. Why can a plane fly?What are the causes of all these things?


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The answer to the question "Why"

Isaac Newton put forth a variety of laws that explain why objects move (or don't move) as they do. These laws are so complete that they can explain everything from the falling of an apple to the motion of the planets. Newtons Laws are the basis from which all physics has started, and is still used today to analyze objects that are relatively large (bigger than an atom) and relatively slow (less than the speed of light).

Newton's laws are universal; they apply to objects on Earth and to objects in outer space. Newton's laws are also mathematical; they permit the computation of the positions and velocities of objects. Newton's laws of motion and law of gravity, for instance, were used to compute the trajectories of interplanetary spacecraft like the Voyager spacecraft that visited many of the outer planets in our solar system.


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Newton's First Law

Newton's First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. It may be seen as a statement about inertia, that objects will remain in their state of motion unless a force acts to change the motion. Any change in motion involves an acceleration, and then Newtons Second Law applies; in fact, the First Law is just a special case of the Second Law for which the net external force is zero.

Newton's First Law contains implications about the fundamental symmetry of the universe in that a state of motion in a straight line must be just as "natural" as being at rest. If an object is at rest in one frame of reference, it will appear to be moving in a straight line to an observer in a reference frame which is moving by the object. There is no way to say which reference frame is "special", so all constant velocity reference frames must be equivalent.


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Newton's First Law

The first law says that an object at rest tends to stay at rest, and an object in motion tends to stay in motion, with the same direction and speed. Motion (or lack of motion) cannot change without an unbalanced force acting. If nothing is happening to you, and nothing does happen, you will never go anywhere. If you're going in a specific direction, unless something happens to you, you will always go in that direction. Forever.

You can see good examples of this idea when you see video footage of astronauts. Have you ever noticed that their tools float? They can just place them in space and they stay in one place. There is no interfering force to cause this situation to change. The same is true when they throw objects for the camera. Those objects move in a straight line. If they threw something when doing a spacewalk, that object would continue moving in the same direction and with the same speed unless interfered with; for example, if a planet's gravity pulled on it.


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June 09, 2014

Newton's First Law

Newton's First Law is commonly stated as an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. But this presupposes too much prior knowledge of the subject. For example, what is meant by:

1. uniform motion

2. external force

Uniform motion relates to a constant velocity, and the external force is an unbalanced net force.

So the first law we will use is:

An object at rest remains at rest, and an object moving with a constant velocity in a straight line will continue moving with a constant velocity in a straight line, unless acted on by an unbalanced force. All forces acting on the object ARE balanced, net force on the object equals zero, the velocity of the object is constant and equal to anything, and the acceleration of the object equals zero. Rx and Ry are both equal to zero.


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June 09, 2014

An object at rest remains at rest, and an object moving with a constant velocity in a straight line will continue moving with a constant velocity in a straight line, unless acted on by an unbalanced force. All forces acting on the object ARE balanced, net force on the object equals zero, the velocity of the object is constant and equal to anything, and the acceleration of the object equals zero. Rx and Ry are both equal to zero.


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June 09, 2014

Fc = 60 N 12 Kg

Vconst

Find ALL the missing forces acting on the block.


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June 09, 2014

15o

FT = 50 N

8 Kg

Vconst



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June 09, 2014

24om = 8 kgVconst = 0



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June 09, 2014

Friction

Frictional Force: (f or Ff) An electrical force that resists the sliding of an object on a surface that acts parallel (tangential) to the surface and opposite to the direction that the object is sliding or would tend to slide.

There are 2 main types of friction, sliding/static (rubbing) friction and rolling friction.

1. Sliding/static friction occurs when two surfaces are in contact with each other, and move (or not) relative to each other without rotation.

2. Rolling friction is caused when an object rolls on a surface.

• Friction acts against the motion of an object. It will cause a moving object to slow down, and act in opposition to any force trying to move an object that is at rest. Friction tries to keep objects from moving.

• Friction does negative work on a system.

• Friction converts mechanical energy into nonmechanical energy. It converts Kinetic energy to Thermal energy.


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June 09, 2014

The force between surfaces in contact that resists their relative tangential motion (slipping).

1. Static: The object is at rest.

2. Kinetic: The object is in motion. (The type of motion does NOT matter) (also called sliding friction)

• Static and Kinetic friction is mainly caused by surface roughness.

• The classic law of friction states that friction is the product of a coefficient and a force.

• The main two types of frictionstatic and kineticcreate different amounts of resistance for the same objects.

The 3 Main causes of Sliding Friction

1) The roughness of the materials rubbing against each other2) The molecular attraction between materials3) The deformations in the materials

Static & Kinetic


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June 09, 2014

Static friction

• Objects are AT REST.• designated as fs• proportional to Fn (surface normal force) • independent of area in contact • variable to a maximum value (which depends on the surfaces)

Kinetic (sliding) friction

• Objects ARE MOVING.• designated as fk • proportional to Fn • independent of area in contact • independent of speed • always less than static friction


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June 09, 2014

Surface Roughness

Most friction results because the surfaces of materials being rubbed together are not completely smooth. If you looked at what seems to be a smooth surface under a microscope, you would see bumps, ridges, hills and valleys that would interfere with the sliding motion. Of course, the rougher the surface, the more the friction.• The bumps and ridges of the two surfaces interlock. The greater the

number of bumps and ridges, increases the number of interlocks.• To move through the interlocks, either the material breaks or one surface

has to ride up the ridge of the other surface.


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June 09, 2014


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June 09, 2014

The figure below illustrates one macroscopic characteristic of friction that is explained by microscopic (smallscale) research. We have noted that friction is proportional to the normal force, but not to the area in contact, a somewhat counterintuitive notion. When two rough surfaces are in contact, the actual contact area is a tiny fraction of the total area since only high spots touch. When a greater normal force is exerted, the actual contact area increases, and it is found that the friction is proportional to this area.


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June 09, 2014

Molecular Adhesion

If both surfaces become ultrasmooth and flat, the friction from surface roughness becomes negligible, but then friction from molecular attraction comes into play, often becoming greater than the normal friction. This is called stickiness.Soft rubber is an example of a material that can have this type of friction. This factor is usually seen in rolling friction. The stickiness will create a resistance to any motion. Although this force is the smallest, it still can be a factor when the other causes of friction are low.


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June 09, 2014

An atomicscale view can explain far more than the simpler features of friction. The mechanism for how thermal energy is generated is now being determined. In other words, why do surfaces get warmer when rubbed? Essentially, atoms are linked with one another to form lattices. When surfaces rub, the surface atoms adhere and cause atomic lattices to vibrate essentially creating sound waves that penetrate the material. The sound waves diminish with distance and their energy is converted into thermal energy. Chemical reactions that are related to frictional wear can also occur between atoms and molecules on the surfaces. The figure below shows how the tip of a probe drawn across another material is deformed by atomicscale friction.


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June 09, 2014

Surface Deformation

Soft materials will deform when under pressure. This also increases the resistance to motion. For example, when you stand on a rug, you sink in slightly, which causes resistance when you try to drag your feet along the rug's surface. Another example is how rubber tires flatten out at the area on contact with the road. When materials deform, you must "plow" through to move, thus creating a resistive force.The soft material may also deform causing it to fill into the cracks and crevices of the other substrate.


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June 09, 2014

Friction

While this general description of friction has practical utility, it is by no means a precise description of friction. Friction is in fact a very complex phenomenon which cannot be represented by a simple model. Almost every simple statement you make about friction can be countered with specific examples to the contrary. Saying that rougher surfaces experience more friction sounds safe enough two pieces of coarse sandpaper will obviously be harder to move relative to each other than two pieces of fine sandpaper. But if two pieces of flat metal are made progressively smoother, you will reach a point where the resistance to relative movement increases. If you make them very flat and smooth, and remove all surface contaminants in a vacuum, the smooth flat surfaces will actually adhere to each other, making what is called a "cold weld".


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June 09, 2014

Friction lawThe classic Friction Law applies to trying to slide one object along the surface of another. It is the most common form of friction. This law states that the force required to overcome friction for two objects or materials pushing against each other equals the coefficient of friction for the two materials times the perpendicular or normal force pushing the objects together. The equation is:

f = μ x FnWhere f = the resistive force of friction, μ = the coefficient of friction, and Fn = the normal or perpendicular force pushing the two objects together. f and Fn are measured in units of force, which are newtons in the mks system. The coefficient of friction is a decimal number that is mainly determined by the comparative roughness of the two materials. For example, if the coefficient of friction (μ) for automobile disc brakes is 1.5 and the force pushing the pad and the disc together (Fn) is 500 newtons, the braking force f = 750 newtons.The most important thing about the friction law or equation is that friction is independent of the surface area in contact. In other words, it is just as difficult to move a 1 cm square object as a 1 meter square object, if they both are pressed to the surface with the same amount of force.


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June 09, 2014

Coefficients of FrictionFriction is typically characterized by a coefficient of friction which is the ratio of the frictional resistance force to the normal force which presses the surfaces together. In this case the normal force is the weight of the block. Typically there is a significant difference between the coefficients of static friction and kinetic friction.

μ = coefficient of friction

μk = coefficient of kinetic friction

μs = coefficient of static friction


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June 09, 2014

Coefficients of Friction

Static Coefficient of Friction: (μs) The dimensionless ratio between the critical frictional force and the normal force for an object that is at rest but about to slide. (A property of the two materials that are in contact with each other)

Kinetic Coefficient of Friction: (μk) The dimensionless ratio between the kinetic frictional force and the normal force for an object that is sliding. (A property of the two materials that are in contact with each other)

Critical Frictional Force: (fcrit) The maximum frictional force that an object at rest can exert to remain at rest.


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June 09, 2014

Static Kinetic

fs = μs FNfs is a variable forcefs varies between

0 and fcrit

fcrit = μs FN

The object is at rest but on the verge of moving. fcrit does not vary

but is the maximum static frictional force.

μs =fcritFN

fK = μK FNfK is a constant and does not depend on how the object is moving.

μK =fKFN

μs can ONLY be calculated using fcrit.


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June 09, 2014

FTFT

FN

f

Fg

friction (N)

critical friction(Max value ofstatic friction)

At Rest Moving

Kinetic FrictionfK

Static Frictionfs

Tension (N)

at rest but on the verge of movingfcrit


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June 09, 2014

A 15 kg block is resting on a horizontal surface. The coefficient of static friction is 0.45 and the coefficient of kinetic friction is 0.30. 1. What is the minimum amount of force required to get the block to

move?

2. What is the minimum amount of force required to keep the block moving?

The force is applied to the block as shown in the diagram below.

Fc 15 kg


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June 09, 2014

A 7 kg block is pulled with a constant velocity of 3 m/s by a 50 N tension force as shown below. Determine the objects coefficient of kinetic friction.

15o

FT = 50 N

7 Kg

Vconst


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June 09, 2014

A 20 kg block is pushed by a 70 N compression force as shown below. The block is at rest but on the verge of moving. Determine the objects coefficient of static friction.

25oFC = 70 N

20 Kg

V = 0But about to move


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June 09, 2014

A body of weight Fg = 25 N is placed on a rough inclined plane with angle of inclination 30o. Calculate the coefficient of static friction if the body is just at the point of slipping.

30o

25 N


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June 09, 2014

Derive an equation to determine the angle of motion.

θ


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June 09, 2014

A 9 kg block is sitting on the inclined plane as shown below. If the coefficient of static friction is 0.40 determine if the objects is at rest or moving for different values of θ.

θ

θ = 10o, 15o, 20o, and 25o


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June 09, 2014

A 9 kg block is sitting on the inclined plane as shown below. Find the coefficient of kinetic friction.

20o30

oFc = 70 N


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June 09, 2014

FC

m = 5 kgμk = 0.2Vconst


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June 09, 2014

FC


30o


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June 09, 2014

30o

20o

Ft



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June 09, 2014

55o


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June 09, 2014

What would happen if 3 different balls, a cue ball, a candle pin bowling ball and a ten pin bowling ball, are rolled down the same inclined plane?

Would they have the same velocity when they reached the floor?

If they all rolled on the same floor would they travel the same distance?

θ


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June 09, 2014

Inertia

Inertia is a property of matter. It is that property of matter which opposes changes in velocity. Simply stated, a common object will not change its velocity spontaneously. That is, if something is moving along at a constant speed in a straight line, it will continue to move along at the same constant speed in the same straight line. It will not, all on its own, speed up, slow down, or change direction. Something else must push or pull on the object to cause it to speed it up, slow it down, or change its direction. Also, if something is standing still, it will, if left to itself, continue to stand still. Something else must push or pull on the object to get it moving. Actually, an object standing still is just a special case of an object keeping its velocity constant. Its velocity is constantly 0 m/s.

Inertia is defined as a physical property of matter by which an object remains at rest or in uniform motion in the same straight line unless acted upon by some external force. The resistance an object has to changing its state or to accelerating. (ie. An object at rests’ resistance to moving, and an object moving with a constant velocity in a straight lines’ resistance to speeding up, slowing down, or changing direction.)

A descriptive term for that property of a body which resists change in its motion. Two kinds of changes of motion are recognized: changes in translational motion, and changes in rotational motion.


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June 09, 2014

An object's velocity will not change all on its own. Pushes, or pulls, are necessary to change an object's velocity. Therefore: pushes, or pulls, are not necessary to keep an object moving. An object will keep moving all on its own.

Galileo is traditionally credited with being the first scientist to formalize this concept. People before him, however, often believed that a push was necessary to keep something moving. It certainly seems that way at first thought. If you push a chair across a room, it seems that your push is necessary to sustain the velocity of the chair. If you stop pushing, the chair stops moving.

Galileo, though, believed that when the push on the chair is taken away, the chair should continue to move along without any assistance. And, as it turns out, it will if the chair is entirely left alone. But it is not left alone. Friction between the chair and the floor continues to apply a push to the chair after you take your hand away from it. It is this friction that prevents the chair from continuing its motion.


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June 09, 2014

Galileo figured this out by thinking of a ball rolling back and forth between two identical inclines. He imagined this motion:

We must think of a very smooth ball and very smooth inclines. So smooth, in fact, that we do not have to worry about friction slowing down the ball. Also, do not worry about the ball bouncing about when it abruptly changes direction at the bottom of each incline. Imagine this transaction to be gradual, much like a marble rolling in a large bowl.


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June 09, 2014

Galileo noticed that when the ball is started at a certain height (h) on the left incline, that It will roll over to the right incline so that it will roll up that incline to the same height from which it was released on the left incline.

h


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June 09, 2014

If the ball always rolls up to the same vertical height, then he reasoned what would happen if the right incline was not so steep, would the ball again rises to the same height from which it was released? The answer was yes, except now, the ball must roll a greater distance up the right incline before coming to a stop for an instant at the top of its journey. Therefore, it takes more time for the ball to roll to a stop on the right incline before it turns around.

Final Height = Initial Height

With a steep angle the ball will roll a small distance to attain the original

hieght.

As the angle of the opposing incline is reduced, the ball must roll a farther distance in order to attain the original height. What happens as the angle of the opposing incline approaches zero?

D 1 D2


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June 09, 2014


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June 09, 2014

If a ball stops when it reaches its original vertical level, then the ball (in the absence of friction) will never stop, because it can never reach that level!Galileo realized the ball would roll for an infinite amount of time and an infinite distance; the ball would not stop rolling. It would continue moving along with an unchanging velocity as long as nothing else affected it. That is, as long as it experienced no other pushes or pulls.The ball will, all on its own, continue in its state of motion, moving at a constant speed in a straight line. This property of matter is called inertia.Isaac Newton built on Galileo's thoughts about motion. Newton's first law of motion declares that a force is not needed to keep an object in motion.

If friction could be eliminated....


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June 09, 2014

Newtons 2nd LawNewton's first law of motion predicts the behavior of objects for which all existing forces are balanced. The first law sometimes referred to as the law of inertia states that if the forces acting upon an object are balanced, then the acceleration of that object will be 0 m/s/s. Objects at equilibrium (the condition in which all forces balance) will not accelerate. According to Newton, an object will only accelerate if there is a net or unbalanced force acting upon it. The presence of an unbalanced force will accelerate an object changing either its speed, its direction, or both its speed and direction.Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is decreased.


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June 09, 2014

Force

Balanced1st Law

Not Balanced2nd Law

Constantvelocity ≠ 0

At restV = 0

Acceleration = 0

NonConstantVelocity

ConstantVelocity

Acceleration ≠ 0

ConstantAcceleration ≠ 0

NonConstantAcceleration

1st Law or 2nd Law?


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June 09, 2014

Newton's second law of motion can be formally stated as follows:

The acceleration of an object is directly proportional to and in the same direction as the net unbalanced force acting on the object. The acceleration of the object is inversely proportional to the inertial mass of the object. All forces acting on the object are NOT balanced, the net force acting on the object does NOT equal zero, the velocity of the object is NOT constant, and the acceleration of the object is NOT equal to zero. Rx and Ry cannot equal zero at the same time.

This verbal statement can be expressed in equation form as follows:

The above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration.

Fnet = ma

mFneta =


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June 09, 2014

Second LawThe emphasis in the second law is on the net force. • The acceleration is directly proportional to the net force• The net force equals mass times acceleration• The acceleration is in the same direction as the net force• An acceleration is produced by a net force. All relate to NET FORCE. It is important to remember this distinction. Do not use the value of merely "any old force" in the FBD. It is the net force which is related to acceleration. Remember, the net force is the vector sum of all the forces (a resultant vector). If all the individual forces acting upon an object are known, then the net force can be determined. If the net force is known then an individual unknown force acting on an object might be able to be determined.


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June 09, 2014

The Big MisconceptionSo what's the big deal? Many people have known Newton's first law since

eighth grade (or earlier). And if prompted with the first few words, most people could probably recite the law word for word. And what is so terribly difficult about remembering that Fnet = ma? It seems to be a simple algebraic statement for solving word problems. The big deal however is not the ability to recite the first law nor to use the second law to solve problems; but rather the ability to understand their meaning and to believe their implications. While most people know what Newton's laws say, many people do not know what they mean (or simply do not believe what they mean). Some physics concepts are nonintuitive.This misconception sticks out its ugly head in a number of different ways and at a number of different times. While reading through the following discussion, give careful attention to your own belief systems. View physics as a system of thinking about the world rather than information that can be dumped into your brain without evaluating its consistency with your own belief systems. Newton's laws declare loudly that a net force (an unbalanced force) causes an acceleration; the acceleration is in the same direction as the net force. But an object moving with a constant velocity in a straight line requires no net force acting on it. In fact the net force must be zero!


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June 09, 2014

Consider a child sledding down a hill and across the level surface at the local park?

A

B

Imagine that there was no friction along the level surface from point B to point C and that there was no air resistance to impede your motion. How far would the sled travel? And what would its motion be like?Without friction or air resistance to slow it down, the sled would continue in motion with the same speed and in the same direction. The forces acting upon the sled from point B to point C would be the normal force and the gravitational force. These forces are balanced and since the sled is already in motion at point B it will continue in motion with the same speed and direction. So, an object can be moving to the right even if the only forces acting upon the object are vertical forces. Forces do not cause motion; but forces can cause accelerations, if they are unbalanced.

C


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June 09, 2014

Limitations to the Laws of MotionNewton arranged his laws in hierarchical order for good reason. Essentially,

the first law establishes frames of reference for which the other laws are applicable, such frames being called inertial frames. To understand why this is required, consider a ball at rest within an accelerating body: an airplane on a runway will suffice for this example. From the perspective of anyone within the airplane (that is, from the airplane's frame of reference when put in technical terms) the ball will appear to move backwards as the plane accelerates forwards (the same feeling of being pushed back into your seat as the plane accelerates). This appears to contradict Newton's second law as, from the point of view of the passengers, there appears to be no force acting on the ball which would cause it to move. The reason why there is in fact no contradiction is because Newton's second law (without modification) is not applicable in this situation because Newton's first law was never applicable in this situation (i.e. the stationary ball does not remain stationary). Thus, it is important to establish when the various laws are applicable or not since they are not applicable in all situations. On a more technical note, although Newton's laws are not applicable in noninertial frames of reference, such as the accelerating airplane, they can be made to do so with the introduction of a "fictitious force" acting on the entire system: basically, by introducing a force that quantifies the anomalous motion of objects within that system (such as the ball moving without an apparent influence in the example above).


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June 09, 2014

30o

Aconst = ?m = 8 kgμk = 0.25

Fc = 100 N

Find ALL the missing quantities acting on the block.


100

June 09, 2014

20o

Aconst = ?m = 8 kgμk = 0.4

Fc = 100 N



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June 09, 2014

30o20

oFc = 100 N m = 8 kgμk = 0.20Aconst = ?



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June 09, 2014

35o30 o

m =4 kgμk = 0.20Aconst = ?

Fc = 100 N



103

June 09, 2014

60om = 8 kgμk = 0.10Aconst = ?



104

June 09, 2014

θμk = 0Aconst = ?



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June 09, 2014

m = 25 kg

Ft2 = 100 N

60o

Ft1 = 60 N

33.56o

Aconst = ?



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June 09, 2014

20o

Aconst = 5 m/s2 at LOAm = 8 kgμk = ?

Fc = 100 N



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June 09, 2014

30o20

oFc = 100 N m = 8 kgμk = ?Aconst = 2 m/s2 at LOA



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June 09, 2014

60o

m = 5000 kgμk = 0Aconst = ?

a = g sin θ

Quiz


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June 09, 2014

Aconst = 4 m/s2 at LOAm = 8 kgμk = 0.2

Fc = ?



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June 09, 2014

Fnet = mA or A = Fnetm

fk = μk FN or μk = FNfk

Fg = mg

fcrit = μs FN or μs = FNfcrit


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June 09, 2014

Newton Third LawAll forces in the universe occur in equal but oppositely directed pairs. There are no isolated forces; for every external force that acts on an object there is a force of equal magnitude but opposite direction which acts back on the object which exerted that external force. In the case of internal forces, a force on one part of a system will be countered by a reaction force on another part of the system so that an isolated system cannot by any means exert a net force on the system as a whole. A system cannot "bootstrap" itself into motion with purely internal forces to achieve a net force and an acceleration, it must interact with an object external to itself.Without specifying the nature or origin of the forces on the two masses, Newton's 3rd law states that if they arise from the two masses themselves, they must be equal in magnitude but opposite in direction so that no net force arises from purely internal forces.Newton's third law is one of the fundamental symmetry principles of the universe. Since we have no examples of it being violated in nature, it is a useful tool for analyzing situations which are somewhat counterintuitive. For example, when a small truck collides headon with a large truck, your intuition might tell you that the force on the small truck is larger. Not so!


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June 09, 2014

Newtons Third Law

A force is a push or a pull upon an object which results from its interaction with another object. Forces result from interactions! Some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces are the result of actionatadistance interactions (gravitational, electrical, and magnetic forces). According to Newton, whenever objects A and B interact with each other, they exert forces upon each other. When you sit in your chair, your body exerts a downward force on the chair and the chair exerts an upward force on your body. There are two forces resulting from this interaction; a force on the chair and a force on your body. These two forces are called action and reaction forces and are the subject of Newton's third law of motion. Formally stated, Newton's third law is:

When two or more objects interact the force exerted on body 1 by body 2 will be equal in magnitude but opposite in direction to the force exerted on body 2 by body 1. All forces come in pairs.


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June 09, 2014

Note the important provision: two objects must be involved! There exists a whole set of situations where two equal and opposite forces act on the same object, sometimes canceling each other so that no acceleration (or even no motion) occurs, and sometimes causing accelerations. A variety of actionreaction force pairs are evident in nature. Some examples are: 1. The propulsion of a fish through the water. A fish uses its fins to push

water backwards. But a push on the water will only serve to accelerate the water. Since forces result from mutual interactions, the water must also be pushing the fish forwards, propelling the fish through the water. The size of the force on the water equals the size of the force on the fish; the direction of the force on the water (backwards) is opposite the direction of the force on the fish (forwards).

2. The motion of a car. A car is equipped with wheels that spin. As the wheels spin, they grip the road and push the road backwards. Since forces result from mutual interactions, the road must also be pushing the wheels forward. The size of the force on the road equals the size of the force on the wheels (or car); the direction of the force on the road (backwards) is opposite the direction of the force on the wheels (forwards).

3. Rockets operate by combustion. This combustion expels large quantities of gases from the back of the rocket at high speeds. By pushing the gas down and out of the back of the rocket, the reaction force pushes the rocket upwards.


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June 09, 2014

Tug of War


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June 09, 2014

What is the initial acceleration for both people?

m1 = 50 kgμ1 = 0.45a1 = ?

m2 = 80 kgμ2 = 0.10a2 = ?

800 N

Tug of War Problem


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June 09, 2014


m1 = 70 kgμ1 = 0.45a1 = ?

m2 = 80 kgμ2 = 0.20a2 = ?

800 N

Tug of War Problem


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June 09, 2014


m1 = 50 kgμ1 = 0.40a1 = ?

m2 = 80 kgμ2 = 0.25a2 = ?

800 N

Tug of War Problem

Adam Arnold S.


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June 09, 2014

A disco ball is suspended from an elevator by a cord. The disco ball has a mass of 5 kg. The cord has a maximum break strength of 75 N. Calculate the force in the cord when the elevator is:

A) at restB) Accelerating up at 3 m/s2

C) Accelerating down at 3 m/s2

D) moving up with a constant velocity of 10 m/s

Elevator Problem


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June 09, 2014

A disco ball is suspended from an elevator by a cord. The disco ball has a mass of 5 kg. The cord has a maximum break strength of 75 N. Calculate the force in the cord when the elevator is:

A) Accelerating up at 10 m/s2

B) Accelerating up at 20 m/s2

C) Accelerating down at 10 m/s2

D) Accelerating down at 20 m/s2

Elevator Problem


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June 09, 2014


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June 09, 2014

A cord passing over an ideal pulley (one that is both massless and frictionless) has a 8 kg (m) mass hanging from one end and a 12 kg (M) mass hanging from the other, as shown below. (this is also called an Atwood machine) Find the acceleration for both masses and the tension in the cord.

FT2

FT1


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June 09, 2014

A cord passing over an ideal pulley (one that is both massless and frictionless) has a 15 kg (m) mass hanging from one end and a 25 kg (M) mass hanging from the other, as shown below. (this is also called an Atwood machine) Find the acceleration for both masses and the tension in the cord.

FT2

FT1

(M + m)(M m)

gA =

(25 kg + 15 kg)(25 kg 15 kg) 10 m/s2A =

40 kg10 kg 10 m/s2A =

A = 2.5 m/s2


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June 09, 2014

A cord passing over an ideal pulley (one that is both massless and frictionless) has a small mass m hanging from one end and a large mass (M) hanging from the other, as shown below. (this is also called an Atwood machine) Derive an equation to find the acceleration for both masses.

FT2

FT1

FBD's

FT1 FT2

Fg1 = mg Fg2 = Mg

x = 0y = FT1

x = 0y = FT2

x = 0y = mg

x = 0y = Mg

Ry = "Fnety"FT1 mg = "Fnety1"FT1 mg = + mA1FT1 = + mA1 + mg

FT1 = m(g + A)

m

Rx = 00 + 0 = 0

M

Rx = 00 + 0 = 0

LOA90o

LOA270o

Ry = "Fnety"FT2 Mg = "Fnety2"FT2 Mg = MA2FT2 = Mg MA2

FT2 = M(g A)

Fnet1 = mA1at 90o

"Fnety1" = + mA1

Fnet2 = MA2at 270o

"Fnety2" = MA2

FT1 = FT2

A1 = A2 = A

mA + mg = Mg MA

MA + mA = Mg mg

(M + m)A = (M m)g

(M m) (M + m)

This formula is known as the Atwood acceleration equation, and is worth memorizing!

m M

gA =


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June 09, 2014

A 32 kg block (m1) is c

Classical Mechanics...Force dynamics 2.notebook 1 June 09, 2014 Classical Mechanics In physics,...

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