The Manga Guide to Physics

92 Chapter 2 Force and Motion

An Object Does Not Have Its Own Force

Those who have not studied physics tend to think, “An object in motion has a force.” This is a

common but incorrect notion. As we learned in Chapter 1, force is generated between paired

elements whose movement affects each other. An object in motion does not have an inter-

nal force that causes it to stay in motion—it’s simply the result of the first law of motion.

Let’s look at the example of a ball being thrown up in the air. The ball receives a force

from the hand until the moment it leaves the hand. (In response, due to the law of action

and reaction, the hand receives a force from the ball—but this force has nothing to do with

the ball’s motion.) Once the ball leaves the hand, it only receives the force of gravity from the

earth. The force on the ball from the hand does not remain after the ball leaves the hand.

The Unit for Force

Newton’s second law gives us the unit for force:

force = mass × acceleration

In this equation, the unit for mass is kilograms (kg), while the unit for acceleration is

meters per second squared (m/s

2

). Therefore, the unit for force is equal to kg × m/s

2

. To rep-

resent this more easily, we can use a unit called a newton (N):

1 newton = 1 (kg × m/s

2

)

You can use newtons to represent forces. As you can probably guess, this unit is named

after the great Isaac Newton, who established the foundations of physics. A force of 1N is

equivalent to the force required for accelerating an object with a mass of 1 kg by 1 m/s

2

.

The velocity of the ball

Notice how the horizontal component

of this vector does not change!

The orientation of

the force of gravity

(which is also the

orientation of

acceleration)

Path of the ball

t = 0

t = 0.2

t = 0.4

t = 0.6

t = 0.8

t = 0

t = 0.2

t = 0.4

t = 0.6

t = 0.8

Measuring Ma and Force 93

Measuring Mass and Force

How can we determine the mass of an object?

Mass can be measured with a scale, which takes

into account the fact that the force of gravity work-

ing on an object (that is, its weight) is proportional

to its mass. Mass that is measured based on

gravity is referred to as gravitational mass.

However, mass that is calculated using

Newton’s second law represents a measurement

of the resistance of an object against acceleration;

this mass has no direct relation to gravity. Mass as

calculated by Newton’s second law (mass = force /

acceleration) is referred to as inertial mass.

Inertial mass can be measured by combining

Newton’s second law and the law of action and

reaction. First, we need an object with a known

mass (we’ll call it the reference object and label

it m

1

in our diagram). Then, we’ll arrange the

object whose mass we want to measure (we’ll call

it the measurement object and label it m

2

in our

diagram) and the reference object so that their

forces work on each other through a collision. In

this collision, there are no external forces working

on the objects.

At this time, the forces of the reference object and the measurement object working on

each other are subject to the law of action and reaction. That is, they must be equal:

If F

1

= m

1

a

1

and F

2

= m

2

a

2

, we know that F

1

= F

2

, due to the law of action and reac-

tion. Therefore, we can express that relationship like so:

m

1

a

1

= m

2

a

2

Since we’re trying to solve for m

2

, our measurement object, we’ll rearrange that equa-

tion as follows:

m

2

=

m

1

a

1

—

a

2

Of course, these accelerations are actually in opposite directions, so we’ll consider their

magnitude alone.

The acceleration of an object can be found by measuring the distance the object travels

and the time it takes to travel that distance. If you have these measurements, you can find

the inertial mass of the measurement object.

Although experiments have shown that gravitational mass is the same as inertial mass,

Newton’s Laws don’t say that this has to be the case. Our understanding of this relationship

comes from Einstein, who founded general relativity on the equivalence principle—the idea

that inertial and gravitational mass are the same. This is still an active area of research.

Gravitational mass

Weight

m

2

Inertial mass

1

a

1

2

a

2

m

1

m

2

m

1

= m

2

a

1

a

2