


Newton’s Laws
Isaac Newton first published his three laws of motion
in 1687 in his monumental Mathematical Principles of Natural
Philosophy. In these three simple laws, Newton sums up everything
there is to know about dynamics. This achievement is just one of
the many reasons why he is considered one of the greatest physicists
in history.
While a multiplechoice exam can’t ask you to write down
each law in turn, there is a good chance you will encounter a problem
where you are asked to choose which of Newton’s laws best explains
a given physical process. You will also be expected to make simple calculations
based on your knowledge of these laws. But by far the most important
reason for mastering Newton’s laws is that, without them, thinking
about dynamics is impossible. For that reason, we will dwell at
some length on describing how these laws work qualitatively.
Newton’s First Law
Newton’s First Law describes how forces relate
to motion:
An object at rest remains at rest, unless acted
upon by a net force. An object in motion remains in motion, unless
acted upon by a net force.
A soccer ball standing still on the grass does not move
until someone kicks it. An ice hockey puck will continue to move
with the same velocity until it hits the boards, or someone else
hits it. Any change in the velocity of an object is evidence of
a net force acting on that object. A world without forces would
be much like the images we see of the insides of spaceships, where
astronauts, pens, and food float eerily about.
Remember, since velocity is a vector quantity, a change
in velocity can be a change either in the magnitude or the direction
of the velocity vector. A moving object upon which no net force
is acting doesn’t just maintain a constant speed—it also moves in
a straight line.
But what does Newton mean by a net force?
The net force is the sum of the forces acting on a body. Newton
is careful to use the phrase “net force,” because an object at rest
will stay at rest if acted upon by forces with a sum of zero. Likewise,
an object in motion will retain a constant velocity if acted upon
by forces with a sum of zero.
Consider our previous example of you and your evil roommate
pushing with equal but opposite forces on a box. Clearly, force
is being applied to the box, but the two forces on the box cancel
each other out exactly: F +
–F = 0. Thus the net
force on the box is zero, and the box does not move.
Yet if your other, good roommate comes along and pushes
alongside you with a force R,
then the tie will be broken and the box will move. The net force
is equal to:
Note that the acceleration, a,
and the velocity of the box, v,
is in the same direction as the net force.
Inertia
The First Law is sometimes called the law of inertia.
We define inertia as the tendency of an object to remain at a constant
velocity, or its resistance to being accelerated. Inertia is a fundamental
property of all matter and is important to the definition of mass.
Newton’s Second Law
To understand Newton’s Second Law, you must
understand the concept of mass. Mass is an intrinsic scalar quantity:
it has no direction and is a property of an object, not of the object’s
location. Mass is a measurement of a body’s inertia, or its resistance
to being accelerated. The words mass and matter are
related: a handy way of thinking about mass is as a measure of how
much matter there is in an object, how much “stuff” it’s made out
of. Although in everyday language we use the words mass and weight interchangeably,
they refer to two different, but related, quantities in physics.
We will expand upon the relation between mass and weight later in
this chapter, after we have finished our discussion of Newton’s
laws.
We already have some intuition from everyday experience
as to how mass, force, and acceleration relate. For example, we
know that the more force we exert on a bowling ball, the faster
it will roll. We also know that if the same force were exerted on
a basketball, the basketball would move faster than the bowling
ball because the basketball has less mass. This intuition is quantified
in Newton’s Second Law:
Stated verbally, Newton’s Second Law says that the net
force, F, acting on an
object causes the object to accelerate, a.
Since F = ma can
be rewritten as a = F/m,
you can see that the magnitude of the acceleration is directly proportional
to the net force and inversely proportional to the mass, m. Both
force and acceleration are vector quantities, and the acceleration
of an object will always be in the same direction
as the net force.
The unit of force is defined, quite appropriately, as
a newton (N). Because acceleration is given in units
of m/s^{2} and mass is given in units of
kg, Newton’s Second Law implies that 1 N = 1 kg
· m/s^{2}. In other words, one newton is
the force required to accelerate a onekilogram body, by one meter
per second, each second.
Newton’s Second Law in Two Dimensions
With a problem that deals with forces acting in two dimensions,
the best thing to do is to break each force vector into its x
and ycomponents. This will give you two equations instead
of one:
The component form of Newton’s Second Law tells us that
the component of the net force in the direction is directly proportional to the
resulting component of the acceleration in the direction, and likewise for the ycomponent.
Newton’s Third Law
Newton’s Third Law has become a cliché. The
Third Law tells us that:
To every action, there is an equal and opposite
reaction.
What this tells us in physics is that every push or pull
produces not one, but two forces. In any exertion of force, there
will always be two objects: the object exerting the force and the object
on which the force is exerted. Newton’s Third Law tells us that
when object A exerts a force F on
object B, object B will
exert a force –F on object A.
When you push a box forward, you also feel the box pushing back
on your hand. If Newton’s Third Law did not exist, your hand would
feel nothing as it pushed on the box, because there would be no reaction
force acting on it.
Anyone who has ever played around on skates knows that
when you push forward on the wall of a skating rink, you recoil
backward.
Newton’s Third Law tells us that the force that the skater
exerts on the wall, , is exactly equal in
magnitude and opposite in direction to the force that the wall exerts
on the skater, . The harder the skater
pushes on the wall, the harder the wall will push back, sending
the skater sliding backward.
Newton’s Third Law at Work
Here are three other examples of Newton’s Third Law at
work, variations of which often pop up on SAT II Physics:
You push down with your hand on a desk, and the desk pushes upward with a force equal in magnitude to your push. A brick is in free fall. The brick pulls the Earth upward with the same force that the Earth pulls the brick downward. When you walk, your feet push the Earth backward. In response, the Earth pushes your feet forward, which is the force that moves you on your way. 
The second example may seem odd: the Earth doesn’t move
upward when you drop a brick. But recall Newton’s Second Law: the
acceleration of an object is inversely proportional to its mass
(a = F/m).
The Earth is about 10^{24} times
as massive as a brick, so the brick’s downward acceleration of –9.8 m/s^{2} is
about 10^{24 }times as great
as the Earth’s upward acceleration. The brick exerts a force on
the Earth, but the effect of that force is insignificant.
