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of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1527
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Problem 7.57
The sphere, cylinder, and thin ring each have mass
m
and radius
r
. Each is released from rest on identical
inclines. Assuming they all roll without slipping, which will have the largest initial angular acceleration?
In addition, which will reach the bottom of the incline first?
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1529
Problem 7.58
A bowling ball of radius
r
, mass
m
, and radius of gyration
kG
is released from rest
on a rough surface that is inclined at the angle
with respect to the horizontal. The
coefficients of static and kinetic friction between the ball and the incline are
s
and
k, respectively. Assume that the mass center Gis at the geometric center.
Assuming the ball rolls without slip, determine expressions for the angular
acceleration of the ball and the friction and normal force between the ball and the
incline. In addition, find the minimum value of
s
that is compatible with this
motion.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Problem 7.59
A bowling ball of radius
r
, mass
m
, and radius of gyration
kG
is released from rest
on a rough surface that is inclined at the angle
with respect to the horizontal. The
coefficients of static and kinetic friction between the ball and the incline are
s
and
k, respectively. Assume that the mass center Gis at the geometric center.
Let the weight of the ball be
14 lb
, the radius be
4:25 in:
, and the radius of
gyration be
kGD2:6 in
. If the incline is
10 ft
long, determine the time it takes the
ball to reach the bottom of the incline and the speed of
G
when it reaches the bottom.
Use D40ı,sD0:2, and kD0:15.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
k2
G
Since the acceleration of the center of mass of the ball
G
is constant, applying constant acceleration equations,
and recalling that the ball is released from rest, we have
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Problem 7.60
A bowling ball is thrown onto a lane with a backspin
!0
and forward velocity
v0
. The mass of the ball is
m
, its radius is
r
, its radius of gyration is
kG
, and
the coefficient of kinetic friction between the ball and the lane is
k
. Assume
the mass center Gis at the geometric center.
Find the acceleration of
G
and the ball’s angular acceleration while the
ball is slipping.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1533
Problem 7.61
A bowling ball is thrown onto a lane with a backspin
!0
and forward velocity
v0
. The mass of the ball is
m
, its radius is
r
, its radius of gyration is
kG
, and
the coefficient of kinetic friction between the ball and the lane is
k
. Assume
the mass center Gis at the geometric center.
For a
14 lb
ball with
rD4:25 in:
,
kGD2:6 in
,
!0D10 rad=s
, and
v0D17 mph
, determine the time it takes for the ball to start rolling without
slip and its speed when it does so. In addition, determine the distance it travels
before it starts rolling without slip. Use kD0:10.
k2
G
Now that we have the angular acceleration of the ball and the acceleration of its center, we can find the
time it takes for the ball to start rolling without slip by noting that the kinematic condition for no slip is
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
where
.vGx/iDv0
is the initial
x
component of the velocity of
G
and
!bi D!0
is the initial angular
velocity of the ball. Substituting these expressions and the acceleration solutions into the kinematic condition
for no slip, we obtain
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1535
Problem 7.62
Solve Example 7.3 on p. 532 by assuming that the crate slips
and tips. In doing so, show that this motion is not possible
for the given conditions since part of your solution will not be
physically admissible.
2Nh
2FPdh
2DIG˛c;
where ˛cis the angular acceleration of the crate and where, treating the crate as a uniform body,
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
