What caused the baseball’s motion to change? It was the bat’s push. The term force describes a push or a pull. And the essence of dynamics is simply this:
Force causes change in motion.
We’ll soon quantify this idea, writing equations and solving numerical problems. But the essential point is in the simple sentence above. If you want to change an object’s motion, you need to apply a force. If you see an object’s motion change, you know there’s a force acting.
Contrary to Aristotle, and probably to your own intuitive sense, it does not take a force to keep something in unchanging motion; force is needed only to change an object’s motion.
The Net Force
You can push a ball left or right, up or down. Your car’s tires can push the car forward or backward, or make it round a curve. Force has direction and is a vector quantity. Further- more, more than one force can act on an object. We call the individual forces on an object interaction forces because they always involve other objects interacting with the object in question. In Fig. 4.2a, for example, the interaction forces are exerted by the people push- ing the car. In Fig. 4.2b, the interaction forces include the force of air on the plane, the engine force from the hot exhaust gases, and Earth’s gravitational force.
We now explore in more detail the relation between force and change in motion.
Experiment shows that what matters is the net force, meaning the vector sum of all individual interaction forces acting on an object. If the net force on an object isn’t zero, then the object’s motion must be changing—in direction or speed or both (Fig. 4.2a). If the net force on an object is zero—no matter what individual interaction forces contribute to the net force—then the object’s motion is unchanging (Fig. 4.2b).
Newton’s First Law
The basic idea that force causes change in motion is the essence of Newton’s first law:
Newton’s first law of motion: A body in uniform motion remains in uniform motion, and a body at rest remains at rest, unless acted on by a nonzero net force.
Figure 4.1 Galileo considered balls rolling on inclines and concluded that a ball on a horizontal surface should roll forever.
cit always rises to its starting height c If a ball is
released here c
cso if the surface is made horizontal, the ball should roll forever.
Figure 4.2 The net force determines the change in an object’s motion.
Here there’s a nonzero net force acting on the car, so the car’s motion is changing.
The three forces sum to zero, so the plane moves in a straight line with constant speed.
(a)
(b) FSnet
F1
S
F2
S
Fair
S FSnet = S0 Fengine S
FSg
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4.2 Newton’s First and Second Laws 53 The word “uniform” here is essential; uniform motion means unchanging motion—that
is, motion in a straight line at constant speed. The phrase “a body at rest” isn’t really necessary because rest is just the special case of uniform motion with zero speed, but we include it for consistency with Newton’s original statement.
The first law says that uniform motion is a perfectly natural state, requiring no explana- tion. Again, the word “uniform” is crucial. The first law does not say that an object moving in a circle will continue to do so without a nonzero net force; in fact, it says that an object moving in a circle—or in any other curved path—must be subject to a nonzero net force because its motion is changing.
GoT IT? 4.1 A curved barrier lies on a horizontal tabletop, as shown. A ball rolls along the barrier, and the barrier exerts a force that guides the ball in its curved path. After the ball leaves the barrier, which of the dashed paths shown does it follow?
Newton’s first law is simplicity itself, but it’s counter to our Aristotelian preconcep- tions; after all, your car soon stops when you take your foot off the gas. But because the motion changes, that just means—as the first law says—that there must be a nonzero net force acting. That force is often a “hidden” one, like friction, that isn’t as obvious as the push or pull of muscle. Watch an ice show or hockey game, where frictional forces are minimal, and the first law becomes a lot clearer.
Newton’s Second Law
Newton’s second law quantifies the relation between force and change in motion. Newton reasoned that the product of mass and velocity was the best measure of an object’s “quan- tity of motion.” The modern term is momentum, and we write
p! = mv! 1momentum2 (4.1)
for the momentum of an object with mass m and velocity v!. As the product of a scalar (mass) and a vector (velocity), momentum is itself a vector quantity. Newton’s second law relates the rate of change of an object’s momentum to the net force acting on that object:
Newton’s second law of motion: The rate at which a body’s momentum changes is equal to the net force acting on the body:
FSnet = d p!
dt 1Newton>s 2nd law2 (4.2) When a body’s mass remains constant, we can use the definition of momentum, p! = mv!, to write
FS
net = dp!
dt = d1mv!2
dt = m dv! But dv!>dt is the acceleration a!, so dt
FS
net = ma! 1Newton>s 2nd law, constant mass2 (4.3) We’ll be using the form given in Equation 4.3 almost exclusively in the next few chapters.
But keep in mind that Equation 4.2 is Newton’s original expression of the second law, that it’s more general than Equation 4.3, and that it embodies the fundamental concept of momentum. We’ll return to Newton’s law in the form of Equation 4.3, and elaborate on momentum, when we consider many-particle systems in Chapter 9.
(a)
(c) (b)
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54 Chapter 4 Force and Motion
Newton’s second law includes the first law as the special case FSnet = S0. In this case Equation 4.3 gives a! = S0, so an object’s velocity doesn’t change.
✓TIp Understanding Newton
To apply Newton’s law successfully, you have to understand the terms, summarized in Fig. 4.3. On the left is the net force FSnet—the vector sum of all real, physical interac- tion forces acting on an object. On the right is ma!—not a force but the product of the object’s mass and acceleration. The equal sign says that they have the same value, not that they’re the same thing. So don’t go adding an extra force ma! when you’re applying Newton’s second law.
Mass, Inertia, and Force
Because it takes force to change an object’s motion, the first law implies that objects nat- urally resist changes in motion. The term inertia describes this resistance, and for that reason the first law is also called the law of inertia. Just as we describe a sluggish person as having a lot of inertia, so an object that is hard to start moving—or hard to stop once started—has a lot of inertia. If we solve the second law for the acceleration a!, we find that a!
= FS/m—showing that a given force is less effective in changing the motion of a more massive object (Fig. 4.4). The mass m that appears in Newton’s laws is thus a measure of an object’s inertia and determines the object’s response to a given force.
By comparing the acceleration of a known and an unknown mass in response to the same force, we can determine the unknown mass. From Newton’s second law for a force of magnitude F,
F = mknownaknown and F = munknownaunknown
where we’re interested only in magnitudes so we don’t use vectors. Equating these two expressions for the same force, we get
munknown
mknown = aknown
aunknown (4.4)
Equation 4.4 is an operational definition of mass; it shows how, given a known mass and force, we can determine other masses.
The force required to accelerate a 1-kg mass at the rate of 1 m/s2 is defined to be 1 newton (N). Equation 4.3 shows that 1 N is equivalent to 1 kg#m/s2. Other common force units are the English pound (lb, equal to 4.448 N) and the dyne, a metric unit equal to 10-5 N. A 1-N force is rather small; you can readily exert forces measuring hundreds of newtons with your own body.
Figure 4.3 Meaning of the terms in Newton’s second law.
Product of object’s mass and its acceleration;
not a force
Net force: the vector sum of all real, physical forces acting on an object
F S net = ma u
Equal sign indicates that the two sides are mathematically equal — but that doesn’t mean they’re the same physically. Only Fnet
involves physical forces.
S
Figure 4.4 The loaded truck has greater mass—more inertia—so its acceleration is smaller when the same force is applied.
au
au FS
FS
ExAMpLE 4.1 Force from Newton: A Car Accelerates
A 1200-kg car accelerates from rest to 20 m/s in 7.8 s, moving in a straight line with constant acceleration. (a) Find the net force acting on the car. (b) If the car then rounds a bend 85 m in radius at a steady 20 m/s, what net force acts on it?
Interpret In this problem we’re asked to evaluate the net force on a car (a) when it undergoes constant acceleration and (b) when it rounds a turn. In both cases the net force is entirely horizontal, so we need to consider only the horizontal component of Newton’s law.
Develop Figure 4.5 shows the horizontal force acting on the car in each case; since this is the net force, it’s equal to the car’s mass mul- tiplied by its acceleration. We aren’t actually given the acceleration in this problem, but for (a) we know the change in speed and the time
involved, so we can write a = ∆v/∆t. For (b) we’re given the speed and the radius of the turn; since the car is in uniform circular motion, Equation 3.16 applies, and we have a = v2/r.
Figure 4.5 Our sketch of the net force on the car in Example 4.1.
(a) (b)
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