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Physics For Scientists And Engineers 6E - part 31

S E C T I O N 5 . 6 • Newton’s Third Law

121

Another example of Newton’s third law is shown in Figure 5.5b. The force

ex-

erted by the hammer on the nail (the action) is equal in magnitude and opposite the
force

exerted by the nail on the hammer (the reaction). This latter force stops the

forward motion of the hammer when it strikes the nail.

You experience the third law directly if you slam your fist against a wall or kick a

football with your bare foot. You can feel the force back on your fist or your foot. You
should be able to identify the action and reaction forces in these cases.

The Earth exerts a gravitational force

on any object. If the object is a computer

monitor at rest on a table, as in Figure 5.6a, the reaction force to

is the force

exerted by the monitor on the Earth

" #

. The monitor does not accelerate

because it is held up by the table. The table exerts on the monitor an upward force
n "

, called the

normal force.

This is the force that prevents the monitor from

falling through the table; it can have any value needed, up to the point of breaking the
table. From Newton’s second law, we see that, because the monitor has zero accelera-
tion, it follows that

#F " n # mg " 0, or n " mg. The normal force balances the gravi-

tational force on the monitor, so that the net force on the monitor is zero. The reaction
to

n is the force exerted by the monitor downward on the table, F

" #

Note that the forces acting on the monitor are

and

n, as shown in Figure 5.6b. The

two reaction forces

and

are exerted on objects other than the monitor. Remem-

ber,

the two forces in an action–reaction pair always act on two different objects.

Figure 5.6 illustrates an extremely important step in solving problems involving

forces. Figure 5.6a shows many of the forces in the situation—those acting on the mon-
itor, one acting on the table, and one acting on the Earth. Figure 5.6b, by contrast,
shows only the forces acting on one object, the monitor. This is a critical drawing called a
free-body diagram. When analyzing an object subject to forces, we are interested in
the net force acting on one object, which we will model as a particle. Thus, a free-body
diagram helps us to isolate only those forces on the object and eliminate the other
forces from our analysis. The free-body diagram can be simplified further by represent-
ing the object (such as the monitor) as a particle, by simply drawing a dot.

= F

n = F

= F

(a)

(b)

n = F

Figure 5.6 (a) When a computer monitor is at rest on a table, the forces acting on the

monitor are the normal force n and the gravitational force F

. The reaction to n is the

force F

exerted by the monitor on the table. The reaction to F

is the force F

exerted by the monitor on the Earth. (b) The free-body diagram for the monitor.

Definition of normal force

▲

PITFALL PREVENTION

5.7 Newton’s Third Law

This is such an important and of-
ten  misunderstood  concept  that
it  will  be  repeated  here  in  a  Pit-
fall  Prevention.  Newton’s  third
law action and reaction forces act
on  different objects.  Two  forces
acting on the same object, even if
they are equal in magnitude and
opposite  in  direction,  cannot be
an action–reaction pair.

▲

PITFALL PREVENTION

5.6 n Does Not Always

Equal mg

In  the  situation  shown  in  Figure
5.6  and  in  many  others,  we  find
that  n " mg (the  normal  force
has  the  same  magnitude  as  the
gravitational  force).  However,
this is not generally true. If an ob-
ject  is  on  an  incline,  if  there  are
applied  forces  with  vertical  com-
ponents,  or  if  there  is  a  vertical
acceleration  of  the  system,  then
n ! mg.  Always apply  Newton’s
second  law  to  find  the  relation-
ship between n and mg.

Normal in this context means perpendicular.

122

C H A P T E R 5 • The Laws of Motion

5.7 Some Applications of Newton’s Laws

In this section we apply Newton’s laws to objects that are either in equilibrium (

a " 0)

or accelerating along a straight line under the action of constant external forces. Re-
member that

when we apply Newton’s laws to an object, we are interested only in

external forces that act on the object. We assume that the objects can be modeled as
particles so that we need not worry about rotational motion. For now, we also neglect
the effects of friction in those problems involving motion; this is equivalent to stating
that the surfaces are frictionless. (We will incorporate the friction force in problems in
Section 5.8.)

We usually neglect the mass of any ropes, strings, or cables involved. In this ap-

proximation, the magnitude of the force exerted at any point along a rope is the
same at all points along the rope. In problem statements, the synonymous terms light
and of negligible mass are used to indicate that a mass is to be ignored when you work
the problems. When a rope attached to an object is pulling on the object, the rope
exerts a force

T on the object, and the magnitude T of that force is called the

tension in the rope. Because it is the magnitude of a vector quantity, tension is a
scalar quantity.

▲

PITFALL PREVENTION

5.8 Free-body Diagrams

The most important step in solving
a problem using Newton’s laws is
to  draw  a  proper  sketch—the
free-body  diagram.  Be  sure  to
draw only those forces that act on
the  object  that  you  are  isolating.
Be  sure  to  draw  all forces  acting
on the object, including any field
forces,  such  as  the  gravitational
force.

Quick Quiz 5.7

If a fly collides with the windshield of a fast-moving bus,

which object experiences an impact force with a larger magnitude? (a) the fly (b) the
bus (c) the same force is experienced by both.

Quick Quiz 5.8

If a fly collides with the windshield of a fast-moving bus,

which object experiences the greater acceleration: (a) the fly (b) the bus (c) the same
acceleration is experienced by both.

Quick Quiz 5.9

Which of the following is the reaction force to the gravita-

tional force acting on your body as you sit in your desk chair? (a) The normal force ex-
erted by the chair (b) The force you exert downward on the seat of the chair (c) Nei-
ther of these forces.

Quick Quiz 5.10

In a free-body diagram for a single object, you draw

(a) the forces acting on the object and the forces the object exerts on other objects, or
(b) only the forces acting on the object.

Conceptual Example 5.3 You Push Me and I’ll Push You

less of which way it faced.) Therefore, the boy, having the
smaller  mass,  experiences  the  greater  acceleration.  Both
individuals accelerate for the same amount of time, but the
greater  acceleration  of  the  boy  over  this  time  interval  re-
sults  in  his  moving  away  from  the  interaction  with  the
higher speed.

(B)

Who moves farther while their hands are in contact?

Solution Because the boy has the greater acceleration and,
therefore, the greater average velocity, he moves farther dur-
ing the time interval in which the hands are in contact.

A large man and a small boy stand facing each other on fric-
tionless ice. They put their hands together and push against
each other so that they move apart.

(A)

Who moves away with the higher speed?

Solution This situation is similar to what we saw in Quick
Quizzes 5.7 and 5.8. According to Newton’s third law, the
force exerted by the man on the boy and the force exerted
by the boy on the man are an action–reaction pair, and so
they must be equal in magnitude. (A bathroom scale
placed between their hands would read the same, regard-

Rock climbers at rest are in equilib-

rium and depend on the tension

forces in ropes for their safety.

© John EIk III/Stock, Boston Inc./PictureQuest

S E C T I O N 5 . 7 • Some Applications of Newton’s Law

123

Objects in Equilibrium

If the acceleration of an object that can be modeled as a particle is zero, the particle is
in

equilibrium. Consider a lamp suspended from a light chain fastened to the ceiling,

as in Figure 5.7a. The free-body diagram for the lamp (Figure 5.7b) shows that the
forces acting on the lamp are the downward gravitational force

and the upward

force

T exerted by the chain. If we apply the second law to the lamp, noting that a " 0,

we see that because there are no forces in the x direction,

0 provides no helpful

information. The condition

0 gives

Again, note that

T and F

are not an action–reaction pair because they act on the

same object—the lamp. The reaction force to

T is T(, the downward force exerted by

the lamp on the chain, as shown in Figure 5.7c. The ceiling exerts on the chain a
force

T) that is equal in magnitude to the magnitude of T( and points in the opposite

direction.

Objects Experiencing a Net Force

If an object that can be modeled as a particle experiences an acceleration, then there
must be a nonzero net force acting on the object. Consider a crate being pulled to the
right on a frictionless, horizontal surface, as in Figure 5.8a. Suppose you are asked to
find the acceleration of the crate and the force the floor exerts on it. First, note that
the horizontal force

T being applied to the crate acts through the rope. The magni-

tude of

T is equal to the tension in the rope. The forces acting on the crate are illus-

trated in the free-body diagram in Figure 5.8b. In addition to the force

T, the free-

body diagram for the crate includes the gravitational force

and the normal force

exerted by the floor on the crate.

We can now apply Newton’s second law in component form to the crate. The only

force acting in the x direction is

T. Applying #F

to the horizontal motion

gives

No acceleration occurs in the y direction. Applying

with a

0 yields

That is, the normal force has the same magnitude as the gravitational force but acts in
the opposite direction.

T is a constant force, then the acceleration a

T/m also is constant. Hence, the

constant-acceleration equations of kinematics from Chapter 2 can be used to obtain
the crate’s position x and velocity v

as functions of time. Because a

T/m " con-

stant, Equations 2.9 and 2.12 can be written as

In the situation just described, the magnitude of the normal force

n is equal to the

magnitude of

, but this is not always the case. For example, suppose a book is lying

on a table and you push down on the book with a force

F, as in Figure 5.9. Because the

book is at rest and therefore not accelerating,

0, which gives n # F

F " 0, or

n " F

F. In this situation, the normal force is greater than the force of gravity. Other

examples in which n ! F

are presented later.

t $

n $ (# F

) " 0

n " F

T " ma

T # F

T " F

(b)

(c)

′

′′ = T

(a)

Figure 5.7 (a) A lamp suspended

from a ceiling by a chain of negligi-

ble mass. (b) The forces acting on

the lamp are the gravitational force

and the force T exerted by the

chain. (c) The forces acting on the

chain are the force T! exerted by

the lamp and the force T" exerted

by the ceiling.

(a)

(b)

Figure 5.8 (a) A crate being

pulled to the right on a frictionless

surface. (b) The free-body diagram

representing the external forces

acting on the crate.

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C H A P T E R 5 • The Laws of Motion

Figure 5.9 When one object

pushes downward on another

object with a force F, the normal

force n is greater than the

gravitational force: n " F

P R O B L E M - S O LV I N G H I N T S

Applying Newton’s Laws

The following procedure is recommended when dealing with problems involving
Newton’s laws:

•

Draw a simple, neat diagram of the system to help conceptualize the problem.

•

Categorize the problem: if any acceleration component is zero, the particle is
in equilibrium in this direction and

#F " 0. If not, the particle is undergoing

an acceleration, the problem is one of nonequilibrium in this direction, and
#F " ma.

•

Analyze the problem by isolating the object whose motion is being
analyzed. Draw a free-body diagram for this object. For systems containing
more than one object, draw separate free-body diagrams for each object.
Do not include in the free-body diagram forces exerted by the object on its
surroundings.

•

Establish convenient coordinate axes for each object and find the
components of the forces along these axes. Apply Newton’s second law,
#F " ma, in component form. Check your dimensions to make sure that all
terms have units of force.

•

Solve the component equations for the unknowns. Remember that you must
have as many independent equations as you have unknowns to obtain a
complete solution.

•

Finalize by making sure your results are consistent with the free-body diagram.
Also check the predictions of your solutions for extreme values of the
variables. By doing so, you can often detect errors in your results.

Example 5.4

A Traffic Light at Rest

A traffic light weighing 122 N hangs from a cable tied to two
other cables fastened to a support, as in Figure 5.10a. The
upper cables make angles of 37.0° and 53.0° with the hori-
zontal. These upper cables are not as strong as the vertical
cable, and will break if the tension in them exceeds 100 N.
Will the traffic light remain hanging in this situation, or will
one of the cables break?

Solution We  conceptualize the  problem  by  inspecting  the
drawing  in  Figure  5.10a.  Let  us  assume  that  the  cables  do
not break so that there is no acceleration of any sort in this
problem  in  any  direction.  This  allows  us  to  categorize the
problem as one of equilibrium. Because the acceleration of
the system is zero, we know that the net force on the light
and the net force on the knot are both zero. To analyze the

53.0

37.0

(a)

53.0

37.0

(b)

(c)

Figure 5.10 (Example 5.4) (a) A traffic light suspended by cables. (b) Free-body diagram

for the traffic light. (c) Free-body diagram for the knot where the three cables are joined.

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