Problem 5.101
A
1:34 lb
ball is dropped on a
10 lb
incline with
˛D33ı
. The ball’s
release height is
h1D5ft
, and the height of the impact point relative to
the ground is
h2D0:3 ft
. Assume that the contact between the ball and
the incline is frictionless, and let the COR for the impact be eD0:88.
Compute the distance
d
at which the ball will hit ground for the first
time if the incline cannot move relative to the floor.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1011
Observe that the argument of the square root in the result of Eq. (9) is larger than
vC
Ay
. Hence, the only
physically acceptable root is
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permission of McGraw-Hill, is prohibited.
Problem 5.102
A
1:34 lb
ball is dropped on a
10 lb
incline with
˛D33ı
. The ball’s
release height is
h1D5ft
, and the height of the impact point relative to
the ground is
h2D0:3 ft
. Assume that the contact between the ball and
the incline is frictionless, and let the COR for the impact be eD0:88.
Compute the distance
d
at which the ball will hit ground for the first
time if the incline can slide without friction relative to the floor.
v
Ax D0and v
Ay D
p2gh1:(4)
As far as
B
is concerned, we know it is at rest before impact and it can only move in the
x
direction after
impact. Hence, we have
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1013
Computation. Using the kinematic relations in Eqs. (4)–(8), we can rewrite Eqs. (1)–(3) as follows
dDp2.1 Ce/h1mBsin 2˛
pgh1.mAC2mBmAcos 2˛/(p2gh1ŒmACmBem
mA2mBCmAcos 2˛
B.mACmBCem
Bcos 2˛/2ç
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permission of McGraw-Hill, is prohibited.
Problem 5.103
A
1:34 lb
ball is dropped on a
10 lb
incline with
˛D33ı
. The ball’s
release height is
h1D5ft
, and the height of the impact point relative to
the ground is
h2D0:3 ft
. Assume that the contact between the ball and
the incline is frictionless, and let the COR for the impact be
eD0:88
.
Compute the distance
d
at which the ball will hit ground for the first
time if the combined stiffness of the supporting springs is
kD50 lb=in:
Assume that the incline can move only vertically.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1015
Computation. Using Eqs. (5)–(9), Eqs. (1)–(4) can be rewritten as
Ay Drgh1
2
mACmB
Now that we have the components of the postimpact velocity of
A
, we proceed to solve a projectile problem
to determine the required distance
d
. We begin by finding the time
t
f
that
A
takes to reach the floor via the
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permission of McGraw-Hill, is prohibited.
1016 Solutions Manual
Consider two balls
A
and
B
that are stacked one on top of the other and dropped
from rest from a height
h
. Let
eAG D1
be the COR for the collision of ball
A
with
the ground, and let
eAB D1
be the COR for the collision between balls
A
and
B
.
Finally, assume that the balls can move only vertically and that
mAmB
, that
is, that
mB=m
A⇡0
. Model the combined collision as a sequence of two-body
impacts, and predict the rebound speed of ball
B
as a function of
h
and
g
, the
acceleration due to gravity. Hint: Even though
A
and
B
are shown in contact,
assume that a small gap is present so that the impact between
A
and the ground
precedes that between Aand B.
Solution
We denote the Earth as “ball
G
” and we treat it as being initially stationary and
as having a mass far larger than the mass of ball
A
. Next, we observe that both
Dynamics 2e 1017
Kinematic Equations.
In this problem, as it seemed natural, we have analyzed the free fall portion of the
motion, which resulted in the determination the preimpact velocity. The kinematic equation relevant to the
study of the impact part of the problem is therefore the last of Eqs. (1).
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permission of McGraw-Hill, is prohibited.
Problem 5.105
Consider a stack of
N
balls dropped from rest from a height
h
. Let all impacts be
perfectly elastic, and assume that
mimiC1
, that is, that
miC1=mi⇡0
, with
iD1; : : : ; N 1
and
mi
being the mass of the
i
th ball. Model the combined
collision as a sequence of two-body impacts, and predict the rebound speed of the
topmost ball. Assume that the balls can move only vertically. Hint: Even though
the balls are shown in contact, assume a small gap is present between each pair so
that the impact between
B1
and the ground precedes that between
B1
and
B2
, etc.
2
traveling with a downward speed equal to
. Once ball
2
rebounds off of ball
1
, it will collide
with ball 3, and so on.
We can determine the outcome of the entire sequence of impacts by studying the impact between
ball
i
and ball
i1
(with
i>0
). The LOI for all impacts coincides with the
y
axis. Each impact
is an unconstrained perfectly elastic impact. Therefore the linear momentum of the system
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1019
Force Laws.
In addition to the conservation of momentum, each impact is also governed by the COR
equation, i.e,
.vC
i/y.vC
i1/yD.v
i1/y.v
i/y:(5)
. . . and so on.
By carefully reviewing the sequence of solutions just generated, we conclude that the velocity of ball
i
right
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permission of McGraw-Hill, is prohibited.