1858 Solutions Manual
Consider the impact-relevant
FBD
of a car involved in a collision. Assume
that, at the time of impact, the car was stationary. In addition, assume that
the impulsive force
F
, with line of action
`
, is the only impulsive force
acting on the car at the time of impact. The point
P
at the intersection
of
`
and the line perpendicular to
`
and passing through
G
, the center of
mass of the car, is sometimes referred to as the center of percussion (for an
alternative definition of center of percussion see Example 7.5 on p. 542). Is
it true that, at the time of impact, the instantaneous center of rotation of the
car lies on the same line as Pand G?
Solution
Yes, it is true that the instantaneous center of rotation will lie on the line passing through points
P
and
G
.
Dynamics 2e 1859
Problem 8.118
A basketball with mass
mD0:6 kg
is rolling without slipping as shown
when it hits a small step with
`D7cm
. Letting the ball’s diameter
be
rD12:0 cm
, modeling the ball as a thin spherical shell (the mass
moment of inertia of a spherical shell about its mass center is
2
3mr2
),
and assuming that the ball does not rebound off the step or slip relative
to it, determine v0such that the ball barely makes it over the step.
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permission of McGraw-Hill, is prohibited.
Balance Principles.
Applying the work-energy principle as a statement of conservation of energy, we have
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Dynamics 2e 1861
Problem 8.119
A bullet
B
of mass
mb
is fired with a speed
v0
as shown against a uniform thin rod
A
of length
`
and mass
mr
that is pinned at
O
. Determine the distance
d
, such that
no horizontal reaction is felt at the pin when the bullet strikes the rod.
Solution
Under the assumption that no horizontal reaction is felt at the pin during the impact,
1862 Solutions Manual
A bullet
B
weighing
147 gr
(
1lb D7000 gr
) is fired with a speed
v0
as shown
and becomes embedded in the center of a rubber block of dimensions
hD4:5 in:
and
wD6in:
weighing
Wrb D2lb
. The rubber block is attached to the end of a
uniform thin rod
A
of length
LD1:5 ft
and weight
WrD5lb
that is pinned at
O
. After the impact, the rod (with the block and the bullet embedded in it) swings
upward to an angle of 60ı. Determine the speed of the bullet right before impact.
Solution
The solution of this problem is organized in two parts. First, we determine the postimpact angular velocity of
Balance Principles.
Applying the work-energy principle as a statement of conservation of energy, we have
T1CV1DT2CV2;(1)
where
V
is the potential energy of the system, and where, denoting by
IO
the mass moment of inertia of the
Dynamics 2e 1863
Kinematic Equations. In ¡, the pendulum comes to a temporary stop. So,
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permission of McGraw-Hill, is prohibited.
1864 Solutions Manual
Solve the problem in Example 7.5 on p. 542 using momentum methods
and the concept of impulsive force. Specifically, consider a ball hitting
a bat at a distance
d
from the handle when the batter has “choked up” a
distance
ı
. Find the “sweet spot”
P
(more properly called the center of
percussion) of the bat
B
by determining the distance
d
at which the ball
should be hit so that the lateral force (i.e., perpendicular to the bat) at
O
is
zero. Assume that the bat is pinned at
O
, it has mass
m
, the mass center is
at G, and the mass moment of inertia is IG.
Solution
The impact-relevant FBD of the bat assuming no impulsive forces at
Dynamics 2e 1865
Problem 8.122
A batter is swinging a
34 in:
long bat with weight
WBD32 oz
, mass
center
G
, and mass moment of inertia
IGD0:0413 slugft2
. The
center of rotation of the bat is point
Q
. Compute the distance
d
identifying the position of point
P
, the bat’s “sweet spot” or center
of percussion, such that the batter will not feel any impulsive forces
at
O
where he is grasping the bat. In addition, knowing that the
ball, weighing
5oz
, is traveling at a speed
vbD90 mph
and that the
batter is swinging the bat with an angular velocity
!0D45 rad=s
,
determine the speed of the ball and the angular velocity of the bat
immediately after impact. To solve the problem, use the following
data: ıD6in:,⇢D14 in:,`D22:5 in:, and COR eD0:5.
Solution
Determination of d
The impact-relevant FBD of the bat assuming no impulsive forces
at the grip is shown on the right, where
N
is the impulsive force
1866 Solutions Manual
The impact-relevant FBD of the bat-ball system is shown to the right.
We observe that the LOI is parallel to the
y
direction and that the sys-
tem’s angular momentum about the fixed point
Q
must be conserved
since there are no external impulsive forces acting on the system when
Dynamics 2e 1867
Problem 8.123
A thin homogeneous bar
A
of length
`D1:75
m and mass
mD23 kg
is
translating as shown with a speed
v0D12 m=s
when it collides with the fixed
obstacle
B
. Modeling the contact between the bar and obstacle as frictionless,
letting
ˇD32ı
, and letting the distance
dD0:46
m, determine the angular
velocity of the bar immediately after the collision, knowing that the
COR
for
the impact is eD0:74.
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.