138
CHAPTE R 6 FORCE AN D M OTION—I I
write Newton’s second law for components along the y axis
(F net,y ϭ ma y ) as
F N cos u Ϫ mg ϭ m(0),
from which
F N cos u ϭ mg.
( 6 - 2 4 )
Combining results: Equation 6-24 also contains the
unknowns F N and m, but note that dividing Eq. 6-23 by
Eq. 6-24 neatly eliminates both those unknowns. Doing so,
replacing (sin u)/(cos u) with tan u, and solving for u then
yield
.
(Answer)
ϭ tan
Ϫ1
(20 m/s)
2
(9.8 m/s
2
)(190 m)
ϭ 12Њ
ϭ tan
Ϫ1
v
2
gR
Radial calculation: As Fig. 6-11b shows (and as you
should verify), the angle that force
makes with the vertical is equal to the bank angle u of the track. Thus, the radial component F Nr is equal to F N sin u. We can now write
Newton’s second law for components along the r axis
(F net,r ϭ ma r ) as
.
( 6 - 2 3 )
We cannot solve this equation for the value of u because it
also contains the unknowns F N and m.
Vertical calculations: We next consider the forces and acceleration along the y axis in Fig. 6-11b. The vertical component of the normal force is F Ny ϭ F N cos u, the gravitational force
on the car has the magnitude mg, and the
acceleration of the car along the y axis is zero. Thus we can
F
:
g
ϪF N sin u ϭ m Ϫ
v
2
R
F
:
N
Additional examples, video, and practice available at WileyPLUS
Friction When a force tends to slide a body along a surface, a
frictional force from the surface acts on the body. The frictional force
is parallel to the surface and directed so as to oppose the sliding. It is
due to bonding between the atoms on the body and the atoms on the
surface, an effect called cold-welding.
If the body does not slide, the frictional force is a static
frictional force . If there is sliding, the frictional force is a kinetic
frictional force .
1. If a body does not move, the static frictional force
and the
component of parallel to the surface are equal in magnitude,
and
is directed opposite that component. If the component
increases, f s also increases.
2. The magnitude of has a maximum value f s,max given by
f s,max ϭ m s F N ,
( 6 - 1 )
where m s is the coefficient of static friction and F N is the magnitude of the normal force. If the component of parallel to the
surface exceeds f s,max , the static friction is overwhelmed and the
body slides on the surface.
3. If the body begins to slide on the surface, the magnitude of the
frictional force rapidly decreases to a constant value f k given
by
f k ϭ m k F N ,
( 6 - 2 )
where m k is the coefficient of kinetic friction.
Drag Force When there is relative motion between air (or
some other fluid) and a body, the body experiences a drag force
that opposes the relative motion and points in the direction in
which the fluid flows relative to the body. The magnitude of is
D
:
D
:
F
:
f
:
s
f
:
s
F
:
f
:
s
f
:
k
f
:
s
F
:
Review & Summary
related to the relative speed v by an experimentally determined
drag coefficient C according to
(6-14)
where r is the fluid density (mass per unit volume) and A is the
effective cross-sectional area of the body (the area of a cross section taken perpendicular to the relative velocity ).
Terminal Speed When a blunt object has fallen far enough
through air, the magnitudes of the drag force
and the gravitational force on the body become equal. The body then falls at a
constant terminal speed v t given by
(6-16)
Uniform Circular Motion If a particle moves in a circle or a
circular arc of radius R at constant speed v, the particle is said to be
in uniform circular motion. It then has a centripetal acceleration
with magnitude given by
(6-17)
This acceleration is due to a net centripetal force on the particle,
with magnitude given by
(6-18)
where m is the particle’s mass. The vector quantities and are
directed toward the center of curvature of the particle’s path. A
particle can move in circular motion only if a net centripetal
force acts on it.
F
:
a
:
F ϭ
mv
2
R
,
a ϭ
v
2
R
.
a
:
v t ϭ A
2F g
CrA
.
F g
:
D
:
v
:
D ϭ
1
2 CAv
2
,
CHAPTE R 6 FORCE AN D M OTION—I I
write Newton’s second law for components along the y axis
(F net,y ϭ ma y ) as
F N cos u Ϫ mg ϭ m(0),
from which
F N cos u ϭ mg.
( 6 - 2 4 )
Combining results: Equation 6-24 also contains the
unknowns F N and m, but note that dividing Eq. 6-23 by
Eq. 6-24 neatly eliminates both those unknowns. Doing so,
replacing (sin u)/(cos u) with tan u, and solving for u then
yield
.
(Answer)
ϭ tan
Ϫ1
(20 m/s)
2
(9.8 m/s
2
)(190 m)
ϭ 12Њ
ϭ tan
Ϫ1
v
2
gR
Radial calculation: As Fig. 6-11b shows (and as you
should verify), the angle that force
makes with the vertical is equal to the bank angle u of the track. Thus, the radial component F Nr is equal to F N sin u. We can now write
Newton’s second law for components along the r axis
(F net,r ϭ ma r ) as
.
( 6 - 2 3 )
We cannot solve this equation for the value of u because it
also contains the unknowns F N and m.
Vertical calculations: We next consider the forces and acceleration along the y axis in Fig. 6-11b. The vertical component of the normal force is F Ny ϭ F N cos u, the gravitational force
on the car has the magnitude mg, and the
acceleration of the car along the y axis is zero. Thus we can
F
:
g
ϪF N sin u ϭ m Ϫ
v
2
R
F
:
N
Additional examples, video, and practice available at WileyPLUS
Friction When a force tends to slide a body along a surface, a
frictional force from the surface acts on the body. The frictional force
is parallel to the surface and directed so as to oppose the sliding. It is
due to bonding between the atoms on the body and the atoms on the
surface, an effect called cold-welding.
If the body does not slide, the frictional force is a static
frictional force . If there is sliding, the frictional force is a kinetic
frictional force .
1. If a body does not move, the static frictional force
and the
component of parallel to the surface are equal in magnitude,
and
is directed opposite that component. If the component
increases, f s also increases.
2. The magnitude of has a maximum value f s,max given by
f s,max ϭ m s F N ,
( 6 - 1 )
where m s is the coefficient of static friction and F N is the magnitude of the normal force. If the component of parallel to the
surface exceeds f s,max , the static friction is overwhelmed and the
body slides on the surface.
3. If the body begins to slide on the surface, the magnitude of the
frictional force rapidly decreases to a constant value f k given
by
f k ϭ m k F N ,
( 6 - 2 )
where m k is the coefficient of kinetic friction.
Drag Force When there is relative motion between air (or
some other fluid) and a body, the body experiences a drag force
that opposes the relative motion and points in the direction in
which the fluid flows relative to the body. The magnitude of is
D
:
D
:
F
:
f
:
s
f
:
s
F
:
f
:
s
f
:
k
f
:
s
F
:
Review & Summary
related to the relative speed v by an experimentally determined
drag coefficient C according to
(6-14)
where r is the fluid density (mass per unit volume) and A is the
effective cross-sectional area of the body (the area of a cross section taken perpendicular to the relative velocity ).
Terminal Speed When a blunt object has fallen far enough
through air, the magnitudes of the drag force
and the gravitational force on the body become equal. The body then falls at a
constant terminal speed v t given by
(6-16)
Uniform Circular Motion If a particle moves in a circle or a
circular arc of radius R at constant speed v, the particle is said to be
in uniform circular motion. It then has a centripetal acceleration
with magnitude given by
(6-17)
This acceleration is due to a net centripetal force on the particle,
with magnitude given by
(6-18)
where m is the particle’s mass. The vector quantities and are
directed toward the center of curvature of the particle’s path. A
particle can move in circular motion only if a net centripetal
force acts on it.
F
:
a
:
F ϭ
mv
2
R
,
a ϭ
v
2
R
.
a
:
v t ϭ A
2F g
CrA
.
F g
:
D
:
v
:
D ϭ
1
2 CAv
2
,