Summary of the unit on force, motion, and energy
Mass and Weight
Mass is a property of an object. It indicates how much matter is present within the object. The
unit of mass is the kilogram. At the moment this is defined in terms of a certain lump of metal
kept in Paris, France, but in the foreseeable future it will be redefined as the mass of some
definite number (about 5.2 x 1025) carbon atoms.
The weight of an object is a measure of how hard the earth’s gravity is pulling on the object. Its
weight is proportional to its mass. The proportionality varies so little from place to place on Earth
that it can almost be represented by a constant (1 kilogram “equals” 9.8 newtons). This is a source
of confusion, it suggests that the force of gravity on an object and the mass of the object are the
same thing. Mass, however, is a property of the object itself, while its weight is a property of the
interacting system consisting of Earth and the object. Mass is independent of anything around the
object, while weight is dictated by the gravitational environment around the object. When you
visit the moon you will weigh much less, allowing you to climb stairs without losing your breath,
or lift a load that on earth would be too heavy for you, but your mass will be the same and the
clothes you wore in high school still will not fit.
Weight and mass are different. Weight is a force; mass is not.
Motion
Motion is what occurs when something changes its position. Measuring and describing distance,
position and time are where we begin our study of motion. Where did an object start? Where did
it end up? How far did it go? How long did it take? These are all familiar concepts, as is speed:
how fast did it go?
Speed is defined as:
Average speed = distance traveled/time elapsed.
We call it “average speed” because it is a property
of the whole trip, both the parts when you were
speeding down the highway and the parts while
you were stopped at the gas station. The car’s
speedometer tells us our speed at the instant we
look at it. When we measure distance in meters
and time in seconds, the speed is given in meters
per second (m/s).
Velocity is a related concept. It is also measured in
meters per second. It is the combination of the
speed and the direction of motion. For example,
you can go around a racetrack at constant speed,
but you cannot go around at constant velocity,
because you have to change direction as you go
around. We can use arrows (as in the figure) to
indicate direction of motion.
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Constant velocity turns out to be not very exciting from the point of view of the laws of motion
because moving in a straight line at constant speed is what things are supposed to do. Only when
we want the velocity to change do we have to do something. The relevant and interesting part of
motion is the acceleration, which is the rate of change of velocity:
Acceleration = (final velocity – initial velocity)/time elapsed
Acceleration occurs when there is a change in speed or direction or both. A large acceleration
means that velocity is changing a lot, or in a short time interval. Like velocity, acceleration also
has a direction. The more familiar case of acceleration is when motion is in just one direction, so
that all that is changing is the speed. Here are some examples of acceleration:
 As a car moves away from a stoplight, its forward speed is increasing. Its initial velocity
is zero, and so the car is accelerating forward. If it changes speed in a short time, the
acceleration is large. In the figure below, A is a good student driver, and B is an impatient
hot-rodder both starting to move after the stoplight turns green. Each car is shown in
three positions; 0 seconds shows the starting position, and the other positions show each
car after 1 second of acceleration and then after 2 seconds.
The acceleration can be small (A) or large (B), as in the example of these two cars:

Throwing a ball and catching it involves the same change in speed, but the throw takes
longer than the catch; the acceleration is much larger in catching the ball, which is why
baseball players throw a ball with bare hands but catch it with a glove.

The diagram below shows a car is moving from left to right. As it approaches a stop sign,
it slows down, making the direction of its acceleration to the left even though it is still
moving to the right. In this case, acceleration is negative; or we could say that the car is
accelerating backward.

When a
ball
tennis
bounces against a wall, its speed does not change much, but its
direction of motion does. The change in velocity is therefore
large and is directed away from the wall. Because the bounce
doesn’t take very long, the acceleration is huge. In the diagram,
the direction of the acceleration is to the left, away from the
wall.
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To calculate the acceleration of the ball, we would divide the change in velocity by the (short)
time it takes for the bounce—acceleration is huge! When the initial and final velocities are in
different directions, the directions of the velocities have to be included in the calculations. If a
moving object veers to the right, it has accelerated to the right, whether its speed has changed or
not. If it speeds up, it has a forward (or positive) acceleration. If it slows down, it has a backward
(or negative) acceleration.
Changing direction of motion changes the velocity, and so it is again an acceleration. When
moving in a circle, the acceleration is towards the center of the circle.
Force
A force is a push or pull. We can’t see forces, but we can see and feel their effects. They come in
all sizes, and we can measure them. We encounter several kinds of forces every day:
*The earth exerts forces on everything, due to gravity. The size of the earth’s gravitational force
on an object is called the weight of the object.
*Magnets exert forces on each other while they are still several centimeters apart.
*Running a comb through your hair charges it, creating an electrostatic force that enables it to
pick up small pieces of paper.
*A spring or a rubber band exerts a force when it is stretched. When you let go of the rubber
band, this force causes it to contract back to its original length.
*The sail of a ship deflects the wind, which causes a force on the sail that propels the ship.
*When an object slides on a surface, it encounters the friction force, which opposes the motion.
* A book sitting on a table experiences two forces: gravity and an upward force that prevents it
from moving downward. We don’t need to know much about this response force. We can explain
its existence because it prevents something from happening, and is as large as it needs to be to
accomplish this; it’s bigger for a concrete block than it is for a cell phone, because the block
weighs more. In some cases we can ignore both this response force and any other forces that
would cause the system being studied to do something that is being prevented. Thus a ball on a
level table in effect does not experience gravity, because it can’t move downward; a ball on an
inclined surface experiences a force only along the surface, which is due to gravity but less than
the ball’s weight.
One way to measure force is with a force scale. When we weigh an object with a
force scale, gravity pulls it downward, stretching the spring in the scale. The unit
of force is the newton, so force scales are calibrated in newtons. The weight of 1
kilogram is 9.8 newtons (on Earth); a newton is about 1/5 of a pound. Why don’t
we use the weight of a 1 kilogram mass as the unit of force? Because the
gravitational force on an object varies from place to place, even on the earth, and
we prefer to use a set of units that works all over the universe. The weight of the
standard kilogram mass in Paris, France is not exactly the weight of a kilogram in
your town, and pretty irrelevant to the weight of a kilogram on the moon.
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Like velocity and acceleration, force has both a size and a direction. The gravitational
force is downward (by definition) but we can use the force scale to measure forces in
any direction—for example, to determine how hard you have to pull to open a door.
In this explanation we use thick and bold arrows to show the direction of forces, as
opposed to the lighter and thinner arrows we use for other things, such as direction of
motion. In high school physics these bold arrows are called vectors, and their size and
direction become more important.
To be completely clear, we have to specify “who” is pulling on “what.” For example,
just weighing a rock allows us to define many different forces:
A. the force of gravity on the rock
B. the force of the scale on the rock
C. the force of the rock on the scale
D. the force exerted at the top end of the scale to hold it in place (if we don’t hold the
scale up, everything will fall down)
E. the force exerted by the scale on whatever is holding it up
Some of these forces are upward, and some are downward. They are all the same size,
in this particular system, for reasons to be explained below. For now, the point is that
conceptually these forces are all distinct. We should recognize that they are all
present, and when we talk about “the force in this system” we have to be clear about which one
we have in mind.
The Laws of Motion
We can understand the motion of a complicated system by breaking it up into interacting parts.
The motion of each part is described by Newton’s laws of motion.. Physicists usually refer to
Newton’s laws by number, using the order and even the exact words that Newton used to describe
the laws. We prefer to give the three laws descriptive names with clearer statements:
The Law of Inertia, Newton’s first law: An isolated object (that is, an object that is not
interacting with anything else) will maintain constant speed and direction of motion.
In the real world, the Law of Inertia may seem against our experience: moving objects always
come to a stop, it seems. But this happens because in all the examples we have observed, the
object we were studying was touching something else and interacting with it. We can come close
to realizing the condition described by the law (an isolated object) by rolling a ball on a smooth
track. Significantly, what we have done is to remove ways for the ball to interact with other
things. The essential point being made is that a moving object slows down because something
has interfered with it. Slowing down is not a property built into the object. We should search for
the cause outside the object itself.
Another way to state the Law of Inertia is: In the absence of external forces (or when the external
forces cancel), an object will maintain constant velocity. Velocity includes both speed and
direction. For example, hockey puck
coasts in a straight line until deflected
by an external force (like a hockey
stick). Objects that are not moving at
all are also included in this law; if it’s
not moving, it won’t start to move
unless a force makes it move.
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The Law of Force and Acceleration, Newton’s second law: When a net force acts on an object,
it accelerates in the direction of the force, according to the mathematical rule Force = mass x
acceleration. This law describes what happens when forces act on an object. It is the most
important and interesting of the three laws.
If there is more than one force acting on the object, we combine them (noting the directions as
well as the magnitudes of the forces) to get the net force on the object. Force = mass x
acceleration tells us it is easier to get a ping-pong ball moving (and easier to make it stop) than it
is a watermelon. Because the watermelon has larger mass, it will take a larger force to achieve the
same change in motion. Similarly, the Law of Force and Acceleration explains why it is possible
to kick a football and send it flying, but using the same force on a bowling ball makes it barely
budge. The applied force (F, your kick) is equal each time, but the mass of the bowling ball is
much larger than that of the football, so
Force = mass x acceleration
tells us that the acceleration of the bowling ball’s acceleration will be much smaller than the
football’s
acceleration.
The force unit of a newton is defined so that a 1 newton force will cause 1 kilogram to accelerate
at 1 m/sec2
1 newton = 1 kg * 1 meter/second2
If we measure mass in kilograms and acceleration in meters per second per second, the Law of
Force and Acceleration lets us calculate the corresponding force, in newtons. If we were to use
the weight of a kilogram as the force unit, the Law of Force and Acceleration would have to be
more complicated.
A falling rock gains speed because only the force of gravity is present. Its acceleration is in the
direction of this force (down!). Then the Law of Force and Acceleration says
Net force = weight = mass x acceleration
We observed earlier that the weight of an object is proportional to its mass. Then different sized
objects will fall with the same acceleration: a larger object has more weight, but it also has more
mass, and the effect of the two increases cancels out. We also noted that the weight of a 1 Kg
object is 9.8 newtons (near earth’s surface). Then the acceleration of all falling objects is 9.8
m/sec2.
A satellite orbits the earth with constant speed but changing direction. It is accelerating
downward (towards the earth, in the direction that it is veering), again due to the unbalanced force
of gravity.
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The Law of Interaction, Newton’s third law: When object A exerts a force on object
B, object B exerts an equal and opposite force on A.
This is a statement about interacting systems. Pairs of forces that are related this way
are always equal and opposite, in any system, no matter what kinds of forces are
involved, and even if the system is moving or accelerating. It is a completely general
rule. The law implies that you can’t pull yourself up by your shoelaces because the
upward force you exert on the laces is exactly equal and opposite to the downward
force your shoelaces exert on your hand.
As another example of the law of interaction, imagine taking
your pet pig for a walk. When the pig balks and doesn’t want
to move, you feel the force of the pig pulling back on you.
At the same time, the pig feels exactly the same force in the
opposite direction.
Let’s return to the example of weighing a rock to see the interplay among these laws of motion.
When we weigh a rock, the scale tells us the force acting on it, where what we wanted to know is
the gravitational force on the rock. Earlier we identified the five forces in action:
A. the force of gravity on the rock
B. the force of the scale on the rock
C. the force of the rock on the scale
D. the force exerted at the top end of the scale to hold it in place (if we don’t hold the scale
up, everything will fall down)
E. the force exerted by the scale on whatever is holding it up
These are related to each other as follows:
*The forces B and C (and also D and E) form a natural pair, because they are “the force of
this on that” and “the force of that on this.” According to the Law of Interaction, these
forces are opposite in direction and equal in size.
*The forces A and B form a different kind of pair. These are forces acting on the same
thing: the rock. Since the rock is not accelerating, these forces must be equal according to
the Law of Force and Acceleration. So long as the system holds steady, forces A and B
must be equal. For the same reason, forces C and D are equal.
Thus the scale exerts a force on the object that balances the force of gravity, preventing the
object from accelerating. The scale actually reports the force C that the object exerts on the
scale. But C = B (Law of Interaction) and B = A (Law of Force and Acceleration), so the
scale does tell us the weight of the rock.
If you try to weigh a rock while waving it and the scale around, you will see that the scale reads
different values at different times; this means that the force of the scale on the rock is not the
same as the force of gravity on the rock. The scale reading is C (the force of the rock on the
scale), which is the same size as B (the force of the scale on the rock). This, however, is not equal
to force A (the force of gravity on the rock, otherwise known as the weight of the rock) when the
rock is accelerating, and now the scale reading is wrong.
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Misusing the law of force and acceleration
The laws of motion are all one needs to solve a wide range of problems. Unfortunately,
there are alternate viewpoints that work some of the time but which in the end are
inconsistent with the laws we have described. The purpose of this page is to point out
some popular ways of getting confused, or even getting the wrong answer.
Using the wrong name, or using the wrong concept
To begin with, we have to know what name goes with what concept, and understand the
concepts themselves. Acceleration is different from velocity, and velocity is not quite the
same thing as speed. "Energy," "power," "pressure," "momentum," and "velocity" are not
alternate names for force. Students will sometimes use one word when another is
appropriate, either because they are misnaming a concept or because they are applying
the wrong concept.
Mass and weight are distinct, too. This distinction is very hard to maintain, because all
weights have mass, and because there is no word like "weigh" that means "determine the
mass of this object." If you reread the instructions we gave for the simulation of the
Towed Truck you will see that the same object could be used both as a mass (on the
truck) or as a weight (on the hanger), and that we were forced to refer to "weights" that
were being used as masses.
Using yourself as the coordinate system (don’t do this!)
Riding around in a car gives us a lot of experience with velocity and acceleration. The car
even has an accelerator! So this could be a good source of relevant experience.
Unfortunately, there is an aspect of riding in a car that confuses more than it informs: it
becomes very easy to adopt a point of view that the car is standing still and the world is
rushing by. When you are moving in a straight line at constant speed, this is
unobjectionable; indeed, as you sit reading this you are moving east at about 1000 miles
per hour (due to the rotation of the earth), and in another direction at 65,000 miles per
hour as the earth orbits the sun, and still faster than that due to the rotation of the galaxy - and not only are you unaware of these motions, there is no measurement you can do that
would detect them.
However, a problem arises when the velocity of the car changes, for example, by going
around a curve. Most people believe there is a force pulling them towards the outside of
the curve. All they actually experience is that the seatbelt, car seat, or car door exerts a
force towards the inside of the curve, and they invent the outward force to explain it.
They are regarding themselves as not moving in the car, and so there must be an outward
force that the inward forces are balancing (an application of the Law of Inertia).
But this is not the right point of view. As a car goes around a curve, it is accelerating
towards the inside of the curve. If the passengers obeyed the Law of Inertia, they would
continue in a straight line and end up in the bushes on the outside of the curve. This is not
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the consequence of some force pulling out, but rather the lack of a force pulling in.
Fortunately, the car seat, the seat belt, and the car door exert forces on the passengers
towards the center of the curve to prevent this. The magnitude and the direction of the net
force on the passengers is given by the Law of Force and Acceleration.
We can avoid introducing unnecessary forces if we don't use a corrdinate system that is
being carrying around by the object that is being studied. Discuss the car from the point
of view of someone standing on the sidewalk.
The acceleration force (there isn’t one!) Some people try to
turn acceleration into a force, using the Law of Force and
Acceleration, and include this "acceleration force" in the set of
forces that are acting. For this to work at all, the "acceleration
force" has to be in the opposite direction as the acceleration.
The diagram at right shows how this point of view would
discuss a falling ball. Gravity acts downward, and there is an
upward force (labeled "reaction") that "balances" gravity; its
magnitude is given by mass x acceleration. One objection to the
introduction of the "reaction" force is that it doesn't obey the Law of Interaction: if this is
the force acceleration imposes on the object, there is no force that the object "imposes on
acceleration."
In the theory of motion we are trying to teach, the "reaction" force doesn't exist. Instead,
there is an unbalanced force (gravity) acting on the ball, and this causes the downward
acceleration. In general, the Net Force is the combination of all the forces that are acting
on the object, and doesn't appear separately; it is the net force that causes the acceleration
of the object.
The force of the hand (also doesn’t exist!)
Consider a ball that has been thrown upward. Some people try to discuss
this situation as a competition between the downward force of gravity
and an upward "force of the hand." There are three possible
interpretations. Sometimes, the discussion really is about how the ball
was thrown: there once was a force that made the ball start moving. But in fact, it doesn't
matter how the ball achieved its upward motion; the issue is what is happening after it has
been thrown. What people mean by "the force of the hand" is either the velocity itself,
or a closely related concept called momentum (this is the mass multiplied by the
velocity). These are not forces, and calling them a force is confusing, but the issue may
only be one of communication. If we change the words to say that the ball starts with an
upward velocity and that the force causes this to change, we are back to the Law of Force
and Acceleration
Net Force = mass x (rate of change of velocity)
Just please don't call the velocity (or the momentum) a force.
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Another version of the same communication problem comes in describing the car that
fails to stay on a curving road. Some people will say, "the force of the velocity pushed it
off the road." This gets the physics professors all upset, because falling off the road
doesn't need at force at all. A force would be needed to keep the car going around the
curve. But the statements "The velocity caused the car to go off the road" or "The
momentum carried the car off of the road" are fine.
Using the Law of Interaction
Here's a common kind of
physics question.
The car has suddenly stopped,
and there is a backward force
of the seat belt on the
passenger. The question is,
how big is this force?
Some people will try to use the Law of Inertia, explaining that the passenger would
continue moving if there was no external force. This is true, but the question is about the
external force.
Some people will try to use the Law of Interaction, and conclude that it is the same size
as the force of the passenger on the seat belt. This is also true, but not useful. There
wouldn't be a force on the seatbelt if there wasn't a passenger; the passenger is causing
the force, and to determine the size of the force we have to consider the passenger. Once
we have found out how large is the force of the seatbelt on the passenger, we can discuss
how large is the force of the passenger on the seatbelt (the manufacturer of the seatbelt
might want to know this, but the passenger doesn't care).
The point is that the passenger needs to stop; this implies a backwards acceleration (a decrease in
forwards velocity) and thus an unbalanced backwards force on the passenger. That’s what the
seat belt (and air bag) provide. The size of the force can be calculated using the Law of Force
and Acceleration.
Unbalanced force = mass x acceleration
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