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which can be rearranged to obtain the following equation of motion
G
Computer Solution
This equation of motion can now be integrated using
mD200 kg
,
rD0:8
m,
kGD0:65
m,
D38ı
,
FindRoot!x'!t"!0#.Motion,$t, 5.%"
which yields the following result for the period of oscillation
Period of Oscillation D5:12 s:
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 1587
Problem 7.88
The uniform bar
AB
of mass
m
and length
L
is leaning against the corner with
0
when end
B
is given a slight nudge so that end
A
starts sliding down the wall as
B
slides along the floor. Assuming that friction is negligible between the bar and the two
surfaces against which it is sliding, determine the angle
at which end
A
will lose
contact with the vertical wall.
Solution
The FBD of the bar as it is sliding down the wall is shown at the right.
Balance Principles.
The Newton-Euler equations corresponding to this FBD
are:
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.
Expanding cross products and equating coefficients, we obtain
2˛AB sin L
2!2
Computation.
Substituting the expressions for the mass moment of inertia,
aGx
, and
aGy
into the Newton-
Euler equations, we obtain the following system of three equations in the three unknowns
NA
,
NB
, and
˛AB
Solving the first equation for
NA
, the second for
NB
, and then substituting those results into the third equation,
The value
s
for which the bar separates from the wall is the value of
at which the force
NA
becomes
equal to zero. We can see from Eq. (1) that we need to know
˛AB
and
!AB
as functions of
before we can
find this angle. Having just found ˛AB , we can integrate it using the chain rule to find !AB as follows
d
d!AB
3g
)!2
AB D3g
L.1 cos /:
Substituting the expressions for !2
AB and ˛AB into Eq. (1) and setting the result equal to zero, we obtain
The values of
for which
sin D0
do not make physical sense in this context and the only value of
for
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 1589
Problem 7.89
The uniform thin bar, which is leaning on the incline, is released
from rest in the position shown and slides in the vertical plane. The
contacts between the bar and the surface at ends
A
and
B
have
negligible friction. Determine the angular acceleration of the bar
immediately after it is released. Evaluate your answer for
mD3kg
,
LD0:75 m, D45ı, and D30ı.
Solution
The FBD of the bar immediately after it is released is shown on the
right, where
NA
is perpendicular to the inclined surface and
NB
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.
which, when equating components, gives
cos
cos
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 1591
Problem 7.90
The uniform ball of radius
and mass
m
is gently placed in the
bowl
B
with inner radius
R
and is released. The angle
measures
the position of the center of the ball at
G
with respect to a vertical
line through
O
. Assume that the system lies in the vertical plane.
Hint: In working the following problems, we recommend using
the r coordinate system shown.
Assuming that the ball rolls without slip, determine the acceler-
ation of the center of the ball at
G
, the angular acceleration of the
ball, and the force on the ball due to the bowl immediately after
the ball is released.
IGD2
5m2:
Force Laws. All forces have been accounted for on the FBD.
Kinematic Equations.
Because the center of the ball is in circular motion about point
O
, the acceleration
of point Gcan be written as
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permission of McGraw-Hill, is prohibited.
Computation.
Substituting the kinematic equations into the Newton-Euler equations yields the following
system of three equations in the three unknowns N,F, and R
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1593
Problem 7.91
The uniform ball of radius
and mass
m
is gently placed in the
bowl
B
with inner radius
R
and is released. The angle
measures
the position of the center of the ball at
G
with respect to a vertical
line through
O
. Assume that the system lies in the vertical plane.
Hint: In working the following problems, we recommend using
the r coordinate system shown.
Assuming that the ball rolls without slip, that it weighs
3lb
,
is at the position
D40ı
, and is moving clockwise at
10 ft=s
,
determine the acceleration of the center of the ball at
G
and the
normal and friction force between the ball and the bowl. Use
RD4ft and D1:2 ft.
Solution
The FBD of the sphere as it rolls without slipping in the bowl is shown
on the right.
Balance Principles.
The Newton-Euler equations corresponding to
this FBD are:
XFrWmg cos NDmaGr ;
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.
Since the ball rolls without slip on a stationary surface, aQ D0, and so
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 1595
Problem 7.92
The uniform ball of radius
and mass
m
is gently placed in the
bowl
B
with inner radius
R
and is released. The angle
measures
the position of the center of the ball at
G
with respect to a vertical
line through
O
. Assume that the system lies in the vertical plane.
Hint: In working the following problems, we recommend using
the r coordinate system shown.
Assuming that friction is sufficient to prevent slipping, derive
the equation(s) of motion of the ball in terms of the angle .
IGD2
5m2:
Force Laws. All forces have been accounted for on the FBD.
Kinematic Equations.
Because the center of the ball is in circular motion about point
O
, the acceleration
R
Computation. Substituting the kinematic equations into the Newton-Euler equations gives
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.
