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corresponds to the direction of motion of the body. If the body is moving, the direction ofthe applied force may change both the direction of motion of the body and how fast it is

moving. Newton defined the force acting on an object as proportional to the accelerationof the object. In order to define the magnitude of the force, he introduced a constant of

proportionality, the inertial mass, which Newton called quantity of matter.

Definition 1: Quantity of Matter

The quantity of matter is the measure of the same, arising from its density and bulkconjointly.

Thus air of double density, in a double space, is quadruple in quantity; in a triple

space, sextuple in quantity. The same thing is to be understood of snow, and finedust or powders, that are condensed by compression or liquefaction, and of all

bodies that are by any causes whatever differently condensed. I have no regard inthis place to a medium, if any such there is, that freely pervades the interstices

between the parts of bodies. It is this quantity that I mean hereafter everywhereunder the name of body or mass. And the same is known by the weight of each body,

for it is proportional to the weight, as I have found by experiment on pendulums,very accurately made, which shall be shown hereafter.3

1.2 International System of System of Units

The basic system of units used throughout science and technology today is theinternationally accepted Systme International(SI). It consists of seven base quantities

and their corresponding base units:

Base Quantity Base UnitLength meter (m)

Mass kilogram (kg)

Time second (s)

Electric Current ampere (A)

Temperature kelvin (K)

Amount of Substance mole (mol)

Luminous Intensity candela (cd)

We shall refer to the dimension of the base quantity by the quantity itself, for example

dim length ! length ! L, dim mass ! mass ! M, dim time ! time ! T. (2.1)

3 Isaac Newton.Mathematical Principles of Natural Philosophy. Translated by Andrew

Motte (1729). Revised by Florian Cajori. Berkeley: University of California Press, 1934.p. 1.

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Mechanics is based on just the first three of these quantities, the MKS or meter-kilogram-second system. An alternative metric system to this, still widely used, is the so-

called CGS system (centimeter-gram-second).

1.3 Standard Mass

The unit of mass, the kilogram (kg), remains the only base unit in the

International System of Units (SI) that is still defined in terms of a physical artifact,known as the International Prototype of the Standard Kilogram. George Matthey (of

Johnson Matthey) made the prototype in 1879 in the form of a cylinder, 39 mm high and39 mm in diameter, consisting of an alloy of 90 % platinum and 10 % iridium. The

international prototype is kept at the Bureau International des Poids et Mesures (BIPM) atSevres, France under conditions specified by the 1st Confrence Gnrale des Poids et

Msures (CGPM) in 1889 when it sanctioned the prototype and declared This prototypeshall henceforth be considered to be the unit of mass. It is stored at atmospheric pressure

in a specially designed triple bell-jar. The prototype is kept in a vault with six official

copies.

Figure 1.1 International prototype of the standard kilogram.

The 3rd CGPM (1901), in a declaration intended to end the ambiguity in popular usage

concerning the word weight confirmed that:

The kilogram is the unit of mass; it is equal to the mass of the internationalprototype of the kilogram.

There is a stainless steel one-kilogram standard that can travel for comparisons. In

practice it is more common to quote a conventional mass value (or weight-in-air, as

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measured with the effect of buoyancy), than the standard mass. Standard mass isnormally only used in specialized measurements wherever suitable copies of the

prototype are stored.

Example 1: The International Prototype Kilogram Determine the type of shape and

dimensions of the platinum-iridium prototype kilogram such that it has the smallestsurface area for a given volume. The standard kilogram is an alloy of 90 % platinum and

10 % iridium. The density of the alloy is!= 21.56 g "cm#3 . You may want to consider

the following questions:

1) Is there any reason that the surface area of the standard could be important?2) What is the appropriate density to use?3) What shape (that is, sphere, cube, right cylinder, parallelepiped, etc.) has the

smallest surface area for a given volume?

4) Why was a right-circular cylinder chosen?Solution: The standard kilogram is an alloy of 90 % platinum and 10 % iridium. The

density of platinum is 21.45 g !cm"3 and the density of iridium is 22.55g !cm"3 . Thus

the density of the standard kilogram, != 21.56 g "cm#3 , and its volume is

V= m / !" 1000 g / 22 g #cm$3 " 46.38 cm3 . (3.1)

Corrosion would affect the mass through chemical reaction; platinum and iridium werechosen for the standards composition as they resist corrosion.

To further minimize corrosion, the shape should be chosen to have the least surface area.

Ideally, this would be a sphere, but as spheres roll easily they become impractical,whereas cylinders have flat surfaces that prevent this. The volume for a cylinder or radius

r and height h is a constant and given by

V= !r2h . (3.2)

The surface area can be expressed in terms of the radius r as

A = 2!r2+ 2!rh = 2!r

2+2V

r. (3.3)

To find the smallest surface area, minimize the area with respect to the radius

dA

dr= 4!r"

2V

r2= 0 . (3.4)

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Solve for the radius

r3=

V

2!=

!r2h

2!. (3.5)

Thus the radius is one half the height,

r=h

2. (3.6)

For the standard mass, the radius is

r=V

2!

"#$

%&'

1 3

=

46.38 cm3

2!

"

#$%

&'

1 3

( 1.95 cm . (3.7)

Twice this radius is the diameter of the standard kilogram.

Alternative Definition of Mass

Since the prototype kilogram is an artifact, there are some intrinsic problems associated

with its use as a standard. It may be damaged, or destroyed. The prototype gains atomsdue to environment wear and cleaning, at a rate of change of mass corresponding to

approximately 1g / year (1g ! 1microgram ! 1"10-6 g ).

Several new approaches to defining the SI unit of mass (kg) are currently beingexplored. One possibility is to define the kilogram as a fixed number of atoms of aparticular substance, thus relating the kilogram to an atomic mass. Silicon is a good

candidate for this approach because it can be grown as a large single crystal, in a verypure form.

Example 2: Mass of a Silicon Crystal

A given standard unit cell of silicon has a volume V0

and contains N0

atoms. The

number of molecules in a given mole of substance is given by Avogadros constant

NA= 6.0221415 !10

23mole

-1 . The molar mass of silicon is given by Mmolar

. Find the

mass m of a volume V in terms of V0, N

0, V , M

molar, and N

A.

Solution: The mass m0

of the unit cell is the density ! of silicon cell multiplied by the

volume of the cell V0,

m0= !V

0. (3.8)

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The number of moles in the unit cell is the total mass, m0

, of the cell, divided by the

molar mass Mmolar

,

n0

= m0/ M

molar

=

!V

0/ M

molar. (3.9)

The number of atoms in the unit cell is the number of moles n0

times the Avogadro

constant, NA

,

N0= n

0N

A=

!V0N

A

Mmolar

(3.10)

The density of the crystal is related to the mass m of the crystal divided by the volume V of the crystal,

!= m /V (3.11)

So the number of atoms in the unit cell can be expressed as

N0=

mV0N

A

VMmolar

(3.12)

So the mass of the crystal is

m =M

molar

NA

V

V0

N0

(3.13)

The molar mass, unit cell volume and volume of the crystal can all be measured directly.

Notice that Mmolar

/NA

is the mass of a single atom, and (V /V0)N

0is the number of

atoms in the volume. This approach is therefore reduced to the problem of measuring the

Avogadro constant, NA

, with a relative uncertainty of 1 part in 108, which is equivalent

to the uncertainty in the present definition of the kilogram.

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1.4 Force

Definition: Force

An impressed force is an action exerted upon a body, in order to change its state,

either of rest, or of uniform motion in a right line.4

We shall define force first in terms of its effect on the standard body we introduced in

Section 1.3, which by definition has a mass ms= 1kg . We apply an action to the standard

body that will induce the body to accelerate with a magnitude!

a that can be measured by

an accelerometer (any device that measures acceleration). The magnitude of the force!

F

acting on the object is the product of the mass ms

with the magnitude of the

acceleration!a . Force is a vector quantity. The direction of the force on the standard bodyis defined to be the direction of the acceleration of the body. Thus

!

F ! ms

!

a (4.1)

The SI units for force are [kg!m! s"2] . This unit has been named the newton [N] and

1N = 1 kg!m! s"2 .

In order to justify the statement that force is a vector quantity, we need to apply two

forces!

F1

and!

F2

simultaneously to our standard body and show that the resultant force!

FT

is the vector sum of the two forces when they are applied one at a time.

Figure 1.2 Force adds as vectors.

We apply each force separately and measure the accelerations!

a1

and!

a2.

, noting that



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!

F1= m

s

!

a1

(4.2)

!

F2= m

s

!

a2

. (4.3)

When we apply the two forces simultaneously, we measure the acceleration

!

a

. The forceby definition is now

!

FT! m

s

!

a . (4.4)

We then compare the accelerations. The results of these three measurements, and for thatmatter any similar experiment, confirms that the accelerations add as vectors

!

a =!

a1+!

a2. (4.5)

Therefore the forces add as vectors as well,

!

FT=

!

F1+

!

F2. (4.6)

This last statement is not a definition but a consequence of the experimental resultdescribed by Equation (4.5) and our definition of force.

Example 3: Vector Decomposition Solution

Two horizontal ropes are attached to a post that is stuck in the ground. The ropes pull the

post producing the vector forces!

F1= 70 N i + 20 N j and

!

F2= !30 N i + 40 N j as

shown in the figure. Find the direction and magnitude of the horizontal component of a

third force on the post that will make the vector sum of forces on the post equal to zero.

Solution: Since the ropes are pulling the post horizontally, the third force must also havea horizontal component that is equal to the negative of the sum of the two horizontal

forces exerted by the rope on the post. Since there are additional vertical forces acting onthe post due to its contact with the ground and the gravitational force exerted on the post

by the earth, we will restrict our attention to the horizontal component of the third force.

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Let!

F3

denote the sum of the forces due to the ropes. Then we can write the vector!

F3

as

!

F3

= (F1x

+F2x

) i + (F1y

+F2y

)j = (70 N + ! 30 N) i + (20 N + 40 N)j

= (40 N) i + (60 N)j

Therefore the horizontal component of the third force of the post must be equal to

!

Fhor

= !

"

F3= !(

"

F1+

"

F2) = (!40 N) i + (!60 N)j .

The magnitude is!

Fhor

= (!40 N)2+ (!60 N)

2= 72 N . The horizontal component of

the force makes an angle

!= tan"160 N

40 N

#

$%

&

'( = 56.3

as shown in the figure above.

Mass Calibration

So far, we have only used the standard body to measure force. Instead of performingexperiments on the standard body, we can calibrate the masses of all other bodies in

terms of the standard mass by the following experimental procedure. We shall refer to the

mass measured in this way as the inertial mass and denote it by min .

We apply a force of magnitude F to the standard body and measure the

magnitude of the acceleration as. Then we apply the same force to a second body of

unknown mass min

and measure the magnitude of the acceleration ain

. Since the same

force is applied to both bodies,

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F= minain= m

sa

s, (4.7)

Therefore the ratio of the inertial mass to the standard mass is equal to the inverse ratio ofthe magnitudes of the accelerations,

min

ms

=

as

ain

. (4.8)

Therefore the second body has inertial mass equal to

min! m

s

as

ain

. (4.9)

This method is justified by the fact that we can repeat the experiment using a different

force and still find that the ratios of the acceleration are the same. For simplicity we shalldenote the inertial mass by m .

1.5 Newtons First Law

The First Law of Motion, commonly called the Principle of Inertia, was first realized

by Galileo. (Newton did not acknowledge Galileos contribution.) Newton wasparticularly concerned with how to phrase the First Law in Latin, but after many rewrites

Newton perfected the following expression for the First Law (in English translation):

Law 1: Every body continues in its state of rest, or of uniform motion in a right

line, unless it is compelled to change that state by forces impressed upon it.

Projectiles continue in their motions, so far as they are not retarded by the

resistance of air, or impelled downwards by the force of gravity. A top, whoseparts by their cohesion are continually drawn aside from rectilinear motions,

does not cease its rotation, otherwise than as it is retarded by air. The greaterbodies of planets and comets, meeting with less resistance in freer spaces,

preserve their motions both progressive and circular for a much longer time.

The first law is an experimental statement about the motions of bodies. When abody moves with constant velocity, there are either no forces present or there are forces

acting in opposite directions that cancel out. If the body changes its velocity, then theremust be an acceleration, and hence a total non-zero force must be present. We note that

velocity can change in two ways. The first way is to change the magnitude of thevelocity; the second way is to change its direction.

After a bus or train starts, the acceleration is often so small we can barely perceiveit. We are often startled because it seems as if the station is moving in the opposite

direction while we seem to be still. Newtons First Law states that there is no physical

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way to distinguish between whether we are moving or the station is, because there isessentially no total force present to change the state of motion. Once we reach a constant

velocity, our minds dismiss the idea that the ground is moving backwards because wethink it is impossible, but there is no actual way for us to distinguish whether the train is

moving or the ground is moving.

1.6 Momentum, Newtons Second Law and Third Laws

Newton began his analysis of the cause of motion by introducing the quantity of motion:

Definition: Quantity of Motion:

The quantity of motion is the measure of the same, arising from the velocity and

quantity of matter conjointly.

The motion of the whole is the sum of the motion of all its parts; and therefore ina body double in quantity, with equal velocity, the motion is double, with twice the

velocity, it is quadruple.5

Our modern term for quantity of motion is momentum and it is a vector quantity

!

p = m!

v . (6.1)

where m is the inertial mass. Newtons Second Law is the most important experimental

statement about motion in physics.

Law II: The change of motion is proportional to the motive force impressed,and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, atriple force triple the motion, whether that force is impressed altogether and at

once or gradually and successively. And this motion (being always directed thesame way with the generating force), if the body moved before, is added or

subtracted from the former motion, according as they directly conspire with orare directly contrary to each other; or obliquely joined, when they are oblique, so

as to produce a new motion compounded from the determination of both.6

5 Isaac Newton.Mathematical Principles of Natural Philosophy. Translated by AndrewMotte (1729). Revised by Florian Cajori. Berkeley: University of California Press, 1934.

p. 13.6 Isaac Newton.Mathematical Principles of Natural Philosophy. Translated by Andrew


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Suppose that a force is applied to a body for a time interval !t. The impressed force orimpulse produces a change in the momentum of the body,

!

I =!

F!t= !!

p . (6.2)

From the commentary to the second law, Newton also considered forces that wereapplied continually to a body instead of impulsively. The instantaneous action of the totalforce acting on a body at a time t is defining by taking the mathematical limit as the time

interval !t becomes smaller and smaller,

!

Ftotal

= lim!t"0

!!

p

!t#

d!

p

dt. (6.3)

When the mass remains constant in time, for example point-like objects, the Second Lawcan be recast in its more familiar form,

!

Ftotal

= md!

v

dt. (6.4)

Because the derivative of velocity is the acceleration, the force is the product of mass and

acceleration,

!

Ftotal

= m

!

a . (6.5)

Since we defined force in terms of change in motion, the Second Law appears to bea restatement of this definition, and devoid of predictive power since force is only

determined by measuring acceleration. What transforms the Second Law from justmerely a definition is the additional input that comes from Force Laws that are based on

experimental observations on the interactions between bodies. Throughout the semesterwe shall investigate these force laws and learn to use them in our free-body force

diagrams in order to determine the left-hand side of Newtons Second Law.

The right-hand-side of Newtons Second Law is the product of mass with acceleration.Acceleration is a mathematical description of how the velocity of a body changes. We

shortly see that if we know the acceleration of a body we can in principle, predict thevelocity and position of that body at all future times.

Newtons Third Law: Action-Reaction Pairs

Newton realized that when two bodies interact via a force, then the force on one body is

equal in magnitude and opposite in direction to the force acting on the other body.

Law III: To every action there is always opposed an equal reaction: or, the

mutual action of two bodies upon each other are always equal, and directed tocontrary parts.

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Whatever draws or presses another is as much drawn or pressed by that other. If

you press on a stone with your finger, the finger is also pressed by the stone.7

The Third Law, commonly known as the action-reaction law, is the most

surprising of the three laws. Newtons great discovery was that when two objects interact,they each exert the same magnitude of force on each other. We shall refer to objects thatinteract as an interaction pair.

Consider two bodies engaged in a mutual interaction. Label the bodies 1 and 2

respectively. Let!

F1,2

be the force on body 2 due to the interaction with body 1, and!

F2,1

be the force on body 1 due to the interaction with body 2. These forces are depicted in

Figure 1.3.

Figure 1.3 Action-reaction pair of forces

These two vector forces are equal in magnitude and opposite in direction,

!

F1,2= !

!

F2,1

. (6.6)

Example 1: A large truck collides head-on with a small car. During the collision

a) the truck exerts a greater force on the car than the car exerts on the truck.

b) the car exerts a greater force on the truck than the truck exerts on the car.

c) the truck exerts the same force on the car as the car exerts on the truck.

d) the truck exerts a force on the car but the car does not exert a force on the truck.

Solution: c). Thecar and the truck form an interaction pair. By Newtons Third Law the

collision forces on each object have equal magnitude but point in opposite directions.



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