According to Newton’s third law of motion, for every
action there is an equal and opposite reaction. However, consider
the case of a horse hitched to a cart. According to Newton’s
third law, when the horse exerts a force on the cart to cause forward
motion, the cart exerts a backward force of equal magnitude on the
horse (Figure 12-4). Considering the horse and the cart
as a single mechanical system, if the two forces are equal in magnitude
and opposite in direction, their vector sum is zero. How does the
horse-and-cart system achieve forward motion? The answer relates
to the presence of another force that acts with a different magnitude
on the cart than on the horse: the force of friction.
Friction is a force that acts at
the interface of surfaces in contact in the direction opposite the
direction of motion or impending motion. Because friction is a force,
it is quantified in units of force (N). The magnitude of the generated
friction force determines the relative ease or difficulty of motion
for two objects in contact.
Consider the example of a box sitting on a level tabletop (Figure
12-5). The two forces acting on the undisturbed box are its
own weight and a reaction force (R) applied by the table. In this
situation, the reaction force is equal in magnitude and opposite
in direction to the box’s weight.
The magnitude of the frictionforce changes with increasing
amounts of applied force.
When an extremely small horizontal force is applied to this box,
it remains motionless. The box can maintain its static position
because the applied force causes the generation of a friction force at
the box/table interface that is equal in magnitude and
opposite in direction to the small applied force. As the magnitude
of the applied force becomes greater and greater, the magnitude
of the opposing friction force also increases to a certain critical
point. At that point, the friction force present is termed maximum static friction (Fm).
If the magnitude of the applied force is increased beyond this value,
motion will occur (the box will slide).
Once the box is in motion, an opposing frictionforce continues
to act. The friction force present during motion is referred to
as kinetic friction(Fk).
Unlike static friction, the magnitude of kinetic friction remains
at a constant value that is less than the
magnitude of maximum static friction. Regardless of the amount of
the applied force or the speed of the occurring motion, the kinetic
friction force remains the same. Figure 12-6 illustrates
the relationship between friction and an applied external force.
As long as a body is static, the magnitude of the friction force developed is equal to that of an applied external force. Once
motion is initiated, the magnitude of the friction force remains
at a constant level below that of maximum static friction.
What factors determine the amount of applied force needed to
move an object? More force is required to move a refrigerator than
to move the empty box in which the refrigerator was delivered. More
force is also needed to slide the refrigerator across a carpeted
floor than to do so across a smooth linoleum floor. Two factors
govern the magnitude of the force of maximum static friction or
kinetic friction in any situation: the coefficient
of friction, represented by the lowercase Greek letter mu (μ),
and the normal (perpendicular) reaction force (R):
The coefficient of friction is a unitless number indicating the
relative ease of sliding, or the amount of mechanical and molecular
interaction between two surfaces in contact. Factors influencing
the value of μ are the relative roughness and hardness
of the surfaces in contact and the type of molecular interaction
between the surfaces. The greater the mechanical and molecular interaction,
the greater the value of μ. For example, the coefficient
of friction between two blocks covered with rough sandpaper is larger
than the coefficient of friction between a skate and a smooth surface
of ice. The coefficient of friction describes the interaction between
two surfaces in contact and is not descriptive of either surface
alone. The coefficient of friction for the blade of an ice skate
in contact with ice is different from that for the blade of the
same skate in contact with concrete or wood.
The coefficient of friction between two surfaces assumes one
or two different values, depending on whether the bodies in contact
are motionless (static) or in motion (kinetic). The two coefficients are
known as the coefficient of static friction (μs)
and the coefficient of kinetic friction (μk).
The magnitude of maximum static friction is based on the coefficient
of static friction:
The magnitude of the kinetic frictionforce is based on the coefficient
of kinetic friction:
For any two bodies in contact, μk is
always smaller than μs. Kinetic friction
coefficients as low as 0.003 have been reported between the blade
of a racing skate and a properly treated ice rink under optimal
conditions (47). Use of the coefficients of static and
kinetic friction is illustrated in Sample Problem 12.2.
The other factor affecting the magnitude of the frictionforce
generated is the normal reaction force. If weight is the only vertical
force acting on a body sitting on a horizontal surface, R is equal
in magnitude to the weight. If the object is a football blocking
sled with a 100 kg coach standing on it, R is equal to the weight
of the sled plus the weight of the coach. Other vertically directed
forces such as pushes or pulls can also affect the magnitude of
R, which is always equal to the vector sum of all forces or force
components acting normal to the surfaces in contact (Figure
As weight increases, the normal reaction force increases.
The magnitude of R can be intentionally altered to increase or
decrease the amount of friction present in a particular situation.
When a football coach stands on the back of a blocking sled, the normal
reaction force exerted by the ground on the sled is increased, with
a concurrent increase in the amount of friction generated, making
it more difficult for a player to move the sled. Alternatively,
if the magnitude of R is decreased, friction is decreased and it
is easier to initiate motion.
How can the normal reaction force be decreased? Suppose you need
to rearrange the furniture in a room. Is it easier to push or pull
an object such as a desk to move it? When a desk is pushed, the force
exerted is typically directed diagonally downward. In contrast,
force is usually directed diagonally upward when a desk is pulled.
The vertical component of the push or pull either adds to or subtracts
from the magnitude of the normal reaction force, thus influencing
the magnitude of the friction force generated and the relative ease
of moving the desk (Figure 12-8).
From a mechanical perspective, it is easier to pull than
to push an object such as a desk, since pulling tends to decrease
the magnitude of R and F, whereas pushing tends to increase R and
The amount of friction present between two surfaces can also
be changed by altering the coefficient of friction between the surfaces.
For example, the use of gloves in sports such as golf and racquetball
increases the coefficient of friction between the hand and the grip
of the club or racquet. Similarly, lumps of wax applied to a surfboard
increase the roughness of the board’s surface, thereby increasing
the coefficient of friction between the board and the surfer’s
feet. The application of a thin, smooth coat of wax to the bottom
of cross-country skis is designed to decrease the coefficient of
friction between the skis and the snow, with different waxes used
for various snow conditions.
A widespread misconception about friction is that greater contact
surface area generates more friction. Advertisements often imply
that widetrack automobile tires provide better traction (friction) against
the road than tires of normal width. However, the only factors known
to affect friction are the coefficient of friction and the normal
reaction force. Because wide-track tires typically weigh more than
normal tires, they do increase friction to the extent that they
increase R. However, the same effect can be achieved by carrying
bricks or cinder blocks in the trunk of the car, a practice often
followed by people who regularly drive on icy roads. Wide-track
tires do tend to provide the advantages of increased lateral stability
and increased wear, since larger surface area reduces the stress
on a properly inflated tire.
Friction exerts an important influence during many daily activities.
Walking depends on a proper coefficient of friction between a person’s
shoes and the supporting surface. If the coefficient of friction
is too low, as when a person with smooth-soled shoes walks on a
patch of ice, slip-page will occur. The bottom of a wet bathtub
or shower stall should provide a coefficient of friction with the
soles of bare feet that is sufficiently large to prevent slippage.
The amount of friction present between ballet shoes and the dance
studio floor must be controlled so that movements involving some
amount of sliding or pivoting—such as glissades,
assembles, and pirouettes—can
be executed smoothly but without slippage. Rosin is often applied
to dance floors because it provides a large coefficient of static
friction but a significantly smaller coefficient of dynamic friction
(26). This helps to prevent slippage in static situations
and allows desired movements to occur freely.
The amount of friction present during sport situations has engendered
heated controversies. The National Football League Players Association
has attempted to have artificial turf declared a “hazardous
substance” partly because the high coefficient of friction
between artificial turf and a football shoe often does not allow
rotation of a planted foot. Many knee injuries have been attributed to
the immobility of a foot planted on artificial turf when a player
is tackled. This possibility is heightened with increased turf temperature,
since the coefficient of friction between football shoes and artificial
turf increases with temperature (46). Football shoes that
are not designed for use on artificial turf also tend to generate
more friction on artificial turf than those shoes that are designed
for such use (22). However, playing on artificial turf
continues, because the Consumer Products Safety Commission has concluded
that there is insufficient evidence supporting the NFL players’ claim
Another controversial disagreement occurred between Glenn Allison,
a retired professional bowler and member of the American Bowling
Congress Hall of Fame, and the American Bowling Congress. The dispute
arose over the amount of friction present between Allison’s
ball and the lanes on which he bowled a perfect score of 300 in
three consecutive games. According to the congress, his scores could
not be recognized because the lanes he used did not conform to congress
standards for the amount of conditioning oil present (25).
The magnitude of the rolling friction present between a rolling
object, such as a bowling ball or an automobile tire, and a flat
surface is approximately one-hundredth to one-thousandth of that present
between sliding surfaces. Rolling friction occurs because both the
curved and the flat surfaces are slightly deformed during contact.
The coefficient of friction between the surfaces in contact, the
normal reaction force, and the size of the radius of curvature of
the rolling body all influence the magnitude of rolling friction.
For bicycle tires, rolling friction is inversely proportional to
the wheel diameter (49). It decreases with bicycle tire
width and increases with reduced tire pressure (35).
The amount of friction present in a sliding or rolling situation
is dramatically reduced when a layer of fluid, such as oil or water,
intervenes between two surfaces in contact. The presence of synovial fluid
serves to reduce the friction, and subsequently the mechanical wear,
on the diarthrodial joints of the human body. The coefficient of
friction in a total hip prosthesis is approximately 0.01 (40).
Researchers attribute the extremely low coefficients of friction
between speed skates and the ice to a liquid-like film layer on
the surface of the ice (10). The amount of friction between
a bowling ball and a properly oiled lane is also extremely small, and
according to the American Bowling Congress, an insufficient amount
of oil on the lanes gave Allison the unfair advantage of added ball
Revisiting the question presented earlier about the horse and
cart, the force of friction is the determining factor for movement.
The system moves forward if the magnitude of the friction force
generated by the horse’s hooves against the ground exceeds
that produced by the wheels of the cart against the ground (Figure
12-9). Because most horses are shod to increase the amount
of friction between their hooves and the ground, and most cart wheels
are round and smooth to minimize the amount of friction they generate,
the horse is usually at an advantage. However, if the horse stands
on a slippery surface or if the cart rests in deep sand or is heavily
loaded, motion may not be possible.
A horse can pull a cart if the horse’s hooves
generate more friction than the wheels of the cart.
The coefficient of friction between a
dancer’s shoes and the floor must be small enough to allow freedom
of motion but large enough to prevent slippage.
The coefficient of static friction between a sled and the snow
is 0.18, with a coefficient of kinetic friction of 0.15. A 250 N
boy sits on the 200 N sled. How much force directed parallel to
the horizontal surface is required to start the sled in motion?
How much force is required to keep the sled in motion?
To start the sled in motion, the applied force must exceed the
force of maximum static friction:
To maintain motion, the applied force must equal the force of
Another factor that affects the outcome of interactions between
two bodies is momentum, a mechanical quantity that is particularly
important in situations involving collisions. Momentum may be defined
generally as the quantity of motion that an object possesses. More
specifically, linear momentum is the
product of an object’s mass and its velocity:
A static object (with zero velocity) has no momentum; that is,
its momentum equals zero. A change in a body’s momentum
may be caused by either a change in the body’s mass or
a change in its velocity. In most human movement situations, changes
in momentum result from changes in velocity. Units of momentum are
units of mass multiplied by units of velocity, expressed in terms
of kg ˙ m/s. Because velocity is a vector
quantity, momentum is also a vector quantity and is subject to the
rules of vector composition and resolution.
When a head-on collision between two objects occurs, there is
a tendency for both objects to continue moving in the direction
of motion originally possessed by the object with the greatest momentum.
If a 90 kg hockey player traveling at 6 m/s to the right
collides head-on with an 80 kg player traveling at 7 m/s
to the left, the momentum of the first player is the following:
The momentum of the second player is expressed as follows:
Since the second player’s momentum is greater, both
players would tend to continue moving in the direction of the second
player’s original velocity after the collision. Actual
collisions are also affected by the extent to which the players
become entangled, by whether one or both players remain on their
feet, and by the elasticity of the collision.
Neglecting these other factors that may influence the outcome
of the collision, it is possible to calculate the magnitude of the
combined velocity of the two hockey players after the collision
using a modified statement of Newton’s first law of motion
(see Sample Problem 12.3). Newton’s first law
may be restated as the principle of conservation
- In the absence of external forces, the total momentum
of a given system remains constant.
The principle is expressed in equation format as the following:
Subscript 1 designates an initial point in time and subscript
2 represents a later time.
Applying this principle to the hypothetical example of the colliding
hockey players, the vector sum of the two players’ momenta
before the collision is equal to their single, combined momentum
following the collision (see Sample Problem 12.3). In
reality, friction and air resistance are external forces that typically
act to reduce the total amount of momentum present.
A 90 kg hockey player traveling with a velocity of 6 m/s
collides head-on with an 80 kg player traveling at 7 m/s.
If the two players entangle and continue traveling together as a
unit following the collision, what is their combined velocity?
The law of conservation of momentum may be used to solve the
problem, with the two players considered as the total system.
When external forces do act, they change the momentum present
in a system predictably. Changes in momentum depend not only on
the magnitude of the acting external forces but also on the length of
time over which each force acts. The product of force and time is
known as impulse:
When an impulse acts on a system, the result is a change in the
system’s total momentum. The relationship between impulse
and momentum is derived from Newton’s second law:
Subscript 1 designates an initial time and subscript 2 represents
a later time. An application of this relationship is presented in Sample
Significant changes in an object’s momentum may result
from a small force acting over a large time interval or from a large
force acting over a small time interval. A golf ball rolling across
a green gradually loses momentum because its motion is constantly
opposed by the force of rolling friction. The momentum of a baseball
struck vigorously by a bat also changes because of the large force
exerted by the bat during the fraction of a second it is in contact
with the ball.
The amount of impulse generated by the human body is often intentionally
manipulated. When a vertical jump is performed on a force platform,
a graphical display of the vertical GRF across time can be generated
(Figure 12-10). Since impulse is the product of force and
time, the impulse is the area under the force–time curve. The
larger the impulse generated against the floor, the greater the
change in the performer’s momentum, and the higher the
resulting jump. Theoretically, impulse can be increased by increasing either
the magnitude of applied force or the time interval over which the
force acts. Practically, however, when time of force application
against the ground is prolonged during vertical jump execution,
the magnitude of the force that can be generated is dramatically
reduced, with the ultimate result being a smaller impulse. For performing
a maximal vertical jump, the performer must maximize impulse by
optimizing the trade-off between applied force magnitude and force
duration. The motion of the arms during a counter-movement jump
contributes 12–13% of the total upward momentum,
indicating the importance of arm motion when jumping for maximum
Force–time histories for (A)
high, and (B) low vertical jumps by
the same performer. The shaded area represents the impulse generated against
the floor during the jump.
Impulse can also be intentionally manipulated during a landing
from a jump (Figure 12-11). A performer who lands rigidly
will experience a relatively large GRF sustained over a relatively short
time interval. Alternatively, allowing the hip, knee, and ankle
joints to undergo flexion during the landing increases the time
interval over which the landing force is absorbed, thereby reducing the
magnitude of the force sustained. Research has shown that females
tend to land in a more erect posture than males, with greater shock
absorption occurring in the knees and ankles, and a concomitant
greater likelihood of lower-extremity injury (11). One-foot
landings also tend to generate higher impactforces and faster loading
rates than two-foot landings (45).
Representations of ground reaction forces during vertical
jump performances: (A) a rigid landing,
(B) a landing with hip, knee, and
ankle flexion occurring. Note the differences in the magnitudes
and times of the landing impulses.
It is also useful to manipulate impulse when catching a hard-thrown
ball. “Giving” with the ball after it initially
contacts the hands or the glove before bringing the ball to a complete
stop will prevent the force of the ball from causing the hands to
sting. The greater the period is between making initial hand contact
with the ball and bringing the ball to a complete stop, the smaller
is the magnitude of the force exerted by the ball against the hand,
and the smaller is the likelihood of experiencing a sting.
A toboggan race begins with the two crew members pushing the
toboggan to get it moving as quickly as possible before they climb
in. If crew members apply an averageforce of 100 N in the direction
of motion of the 90 kg toboggan for a period of 7 s before jumping
in, what is the toboggan’s speed (neglecting friction)
at that point?
The crew members are applying an impulse to the toboggan to change
the toboggan’s momentum from zero to a maximum amount.
The impulse–momentum relationship may be used to solve
The type of collision that occurs between a struck baseball and
a bat is known as an impact. An impact
involves the collision of two bodies over an extremely small time
interval during which the two bodies exert relatively large forces
on each other. The behavior of two objects following an impact depends
not only on their collective momentum but also on the nature of
For the hypothetical case of a perfectly
elastic impact, the relative velocities of the two bodies after
impact are the same as their relative velocities before impact.
The impact of a superball with a hard surface approaches perfect
elasticity, because the ball’s speed diminishes little
during its collision with the surface. At the other end of the range
is the perfectly plastic impact, during
which at least one of the bodies in contact deforms and does not
regain its original shape, and the bodies do not separate. This
occurs when modeling clay is dropped on a surface.
Most impacts are neither perfectly elastic nor perfectly plastic,
but somewhere between the two. The coefficient
of restitution describes the relative elasticity of an impact.
It is a unitless number between 0 and 1. The closer the coefficient
of restitution is to 1, the more elastic is the impact; and the
closer the coefficient is to 0, the more plastic is the impact.
The coefficient of restitution governs the relationship between
the relative velocities of two bodies before and after an impact.
This relationship, which was originally formulated by Newton, may
be stated as follows:
- When two bodies undergo a direct collision, the difference
in their velocities immediately after impact is proportional to
the difference in their velocities immediately before impact.
This relationship can also be expressed algebraically as the
In this formula, e is the coefficient of restitution, u1 and
u2 are the velocities of the bodies just before impact,
and v1 and v2 are the velocities of the bodies
immediately after impact (Figure 12-12).
The differences in two ball’s velocities before
impact is proportional to the difference in their velocities after
impact. The factor of proportionality is the coefficient of restitution.
In tennis, the nature of the game depends on the type of impacts
between ball and racket and between ball and court. All other conditions
being equal, a tighter grip on the racket increases the apparent
coefficient of restitution between ball and racket (21).
When a pressurized tennis ball is punctured, there is a reduction
in the coefficient of restitution between ball and surface of 20% (20).
Other factors of influence are racket size, shape, balance, flexibility,
string type and tension, and swing kinematics (18, 19, 51).
The nature of impact between the bat and the ball is also an
important factor in the sports of baseball and softball. The hitting
surface of the bat is convex, in contrast to the surface of the
tennis racquet, which deforms to a convex shape during ball contact.
Consequently, hitting a baseball or softball in a direct, rather
than a glancing, fashion is of paramount concern. Research has shown that
aluminum baseball bats produce significantly higher batted ball
speeds than do wood bats, which suggests that the coefficient of
restitution between an aluminum bat and baseball is higher than
that between a wood bat and baseball (17). The condition
of the ball is also significant. Tests have shown that rebound from
a surface is higher after 800 impacts than when a ball is new, because
the loss of nap increases the coefficient of restitution between
ball and surface, and decreases the ball’s aerodynamic
drag. However, the major factor affecting ball rebound is the amount
of time the ball has been outside a pressurized can. A loss of rebound
height for both used and unused balls occurs after five days out
of a can (38). The surface of the court also influences
ball rebound during play, with differences in the coefficients of
restitution and friction between ball and surface making some courts “fast” and
In the case of an impact between a moving body and a stationary
one, Newton’s law of impact can be simplified because the
velocity of the stationary body remains zero. The coefficient of
restitution between a ball and a flat, stationary surface onto which
the ball is dropped may be approximated using the following formula:
In this equation, e is the coefficient of restitution, hd is
the height from which the ball is dropped, and hb is the
height to which the ball bounces (see Sample Problem 12.5).
The coefficient of restitution describes the interaction between
two bodies during an impact; it is not descriptive
of any single object or surface. Dropping a basketball, a golf ball,
a racquetball, and a baseball onto several different surfaces demonstrates
that some balls bounce higher on certain types of surfaces (Figure
Bounce heights of a basketball, golf ball, racquetball, and baseball
all dropped onto the same surface from a height of 1 m.
The coefficient of restitution is increased by increases in both
impact velocity and temperature. In sports such as baseball and
tennis, increases in both incoming ball velocity and bat or racket
velocity increase the coefficient of restitution between bat or
racket and ball, and contribute to a livelier ball rebound from
the striking instrument. In racquetball and squash, where the ball
is constantly being deformed against the wall, the ball’s
thermal energy (temperature) is increased over the course of play.
As ball temperature increases, its rebound from both racquet and
wall becomes more lively.
“Giving” with the ball
during a catch serves to lessen the magnitude of the impactforce
sustained by the catcher.
A basketball is dropped from a height of 2 m onto a gymnasium
floor. If the coefficient of restitution between ball and floor
is 0.9, how high will the ball bounce?