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Problem 7.110
A bowling ball is thrown onto a lane with a forward spin
!0
and forward
velocity
v0
. The mass of the ball is
m
, its radius is
r
, its radius of gyration is
given by
kG
, and the coefficient of kinetic friction between the ball and the
lane is k. Assume the mass center Gis at the geometric center.
Assuming that
v0> r!0
, determine the acceleration of the center of the
ball and the angular acceleration of the ball until it starts rolling without slip.
Solution
We must begin this problem with a little bit of kinematics because we need to determine whether or not the
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 1637
Problem 7.111
A bowling ball is thrown onto a lane with a forward spin
!0
and forward
velocity
v0
. The mass of the ball is
m
, its radius is
r
, its radius of gyration is
given by
kG
, and the coefficient of kinetic friction between the ball and the
lane is k. Assume the mass center Gis at the geometric center.
Assuming that
v0< r!0
, determine the acceleration of the center of the
ball and the angular acceleration of the ball until it starts rolling without slip.
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 7.112
The car, as seen from the front, is traveling at a constant speed
vc
on
a turn of constant radius
R
that is banked at an angle
with respect
to the horizontal. The coefficient of static friction between the tires
and the road is s. The car’s center of mass is at G.
Determine the bank angle
so that there is no tendency to slip or
tip, i.e., so that no friction is required to keep the car on the road.
Solution
Since no friction is required to keep the car on the road, the FBD is as
shown on the right, where we have chosen to use a normal-tangential
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 1639
Problem 7.113
The car, as seen from the front, is traveling at a constant speed
vc
on
a turn of constant radius
R
that is banked at an angle
with respect
to the horizontal. The coefficient of static friction between the tires
and the road is s. The car’s center of mass is at G.
For a given bank angle
, and assuming that the car does not
tip, find the maximum speed
vm
that the car can achieve without
slipping.
Solution
The FBD shown at the right reflects the assumption that the car does
not tip.
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 7.114
An overhead fold-up door, with height
h
and mass
m
, consists of
two identical sections hinged at
C
. The roller at
A
moves along
a horizontal guide, whereas the rollers at
B
and
D
, which are the
midpoints of sections
AC
and
CE
, move along a vertical guide.
The door’s operation is assisted by two identical springs attached
to the horizontally moving rollers (only one of the two springs is
shown). The springs are stretched an amount
ı0
when the door is
fully open.
Let
hD10
m and
mD380 kg
. In addition, let
kD
2400 N=m
and
ı0D0:15
m. Assuming that the door is released
from rest when
D10ı
and that all sources of friction are negligi-
ble, determine the angular acceleration of each section of the door
right after release.
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 1641
Force Laws. The spring force can be expressed as
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.
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 1643
Problem 7.115
An overhead fold-up door, with height
h
and mass
m
, consists of
two identical sections hinged at
C
. The roller at
A
moves along
a horizontal guide, whereas the rollers at
B
and
D
, which are the
midpoints of sections
AC
and
CE
, move along a vertical guide.
The door’s operation is assisted by two identical springs attached
to the horizontally moving rollers (only one of the two springs is
shown). The springs are stretched an amount
ı0
when the door is
fully open.
Assuming that friction between the rollers and the guide can
be neglected, determine the equation(s) of motion of the system.
Solution
The FBDs of the two segments of the door for an arbitrary angle
are shown
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.
Force Laws. The spring force can be expressed as
FsDk.ıCı0/;
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 1645
Problem 7.116
An overhead fold-up door, with height
h
and mass
m
, consists of
two identical sections hinged at
C
. The roller at
A
moves along
a horizontal guide, whereas the rollers at
B
and
D
, which are the
midpoints of sections
AC
and
CE
, move along a vertical guide.
The door’s operation is assisted by two identical springs attached
to the horizontally moving rollers (only one of the two springs is
shown). The springs are stretched an amount
ı0
when the door is
fully open.
Let
hD10
m and
mD320 kg
. In addition, let
kD
2400 N=m
and
ı0D0:15
m. Assuming that the door is released
from rest when
D5ı
, determine the time the door will take to
close and the speed of Eat closing.
Solution
The FBDs of the two segments of the door for an arbitrary angle
are shown
on the right.
Balance Principles.
Based on the FBD of section
AC
, its Newton-Euler
equations are
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permission of McGraw-Hill, is prohibited.
