PROBLEM 13.178 (Continued)
4
4
0.8 0.8 (2.4)
(11.321) 0 ( ) ( )
11.321 ( ) 3
BC
BC
vv
g gg
vv
′′ ′
+= +
′′ ′
= +
PROBLEM 13.178 (Continued)
PROBLEM 13.179
A 5 kg sphere is dropped from a height y=2 m to test newly designed spring
floors used in gymnastics. The mass of the floor section is 10 kg, and the
effective stiffness of the floor is k= 120 kN/m. Knowing that the coefficient of
restitution between the ball and the platform is 0.6, determine (a) the height h
reached by the sphere after rebound, (b) the maximum force in the springs.
SOLUTION
PROBLEM 13.179 (Continued)
NC
22
11
max
PROBLEM 13.180
A 5 kg sphere is dropped from a height y=3 m to test a new spring floor used in
gymnastics. The mass of the floor section B is 12 kg, and the sphere bounces
back upwards a distance of 44 mm. Knowing that the maximum deflection of the
floor section is 33 mm from its equilibrium position, determine (a) the
coefficient of restitution between the sphere and the floor, (b) the effective
spring constant k of the floor section.
SOLUTION
PROBLEM 13.180 (Continued)
Apply Work-Energy to determine the stiffness of the springs (when the plate reaches its lowest point
and the speed is zero). Take the equilibrium position as the datum.
NC
22
11
PROBLEM 13.181
The three blocks shown are identical. Blocks B and C are at rest
when block B is hit by block A, which is moving with a
velocity
A
v
of 3 ft/s. After the impact, which is assumed to be
perfectly plastic
( 0),e=
the velocity of blocks A and B
decreases due to friction, while block C picks up speed, until all
three blocks are moving with the same velocity
.v
Knowing
that the coefficient of kinetic friction between all surfaces is
0.20,
k
m
=
determine (a) the time required for the three blocks
to reach the same velocity, (b) the total distance traveled by
each block during that time.
SOLUTION
PROBLEM 13.181 (Continued)
Blocks A and B: Impulse and momentum
A
A
PROBLEM 13.182
Block A is released from rest and slides down the
frictionless surface of B until it hits a bumper on
the right end of B. Block A has a mass of 10 kg
and object B has a mass of 30 kg and B can roll
freely on the ground. Determine the velocities of A
and B immediately after impact when (a)
0,e=
(b)
0.7.e=
A Ax A
ββ
++
PROBLEM 13.182 (Continued)
2
gh
B
PROBLEM 13.183
A 340–g ball B is hanging from an inextensible cord attached to a
support C. A 170-g ball A strikes B with a velocity
0
v
of magnitude
1.5 m/s at an angle of
60°
with the vertical. Assuming perfectly
elastic impact (
1e=
) and no friction, determine the height h reached
by ball B.
SOLUTION
60°
Ball
A alone
Momentum in
t-direction conserved
( ) ( )
AA AA
tt
mv mv
′
=
( ) ( )
0
vv
′
= =
12
2
g
PROBLEM 13.183 (C
ontinued)
12
0
B
V V Wh= =
( )
2
11 2 2
1
; 0.9446 0 ;
2
B
B
W
T V T V Wh
g
+=+ =+
( )
( )( )
2
0.9446 0.0455 m
2 9.81
h= =
45.5 mmh=
PROBLEM 13.184
A test machine that kicks soccer balls has a 5–lb simulated foot
attached to the end of a 6–ft long pendulum arm of negligible
mass. Knowing that the arm is released from the horizontal
position and that the coefficient of restitution between the foot
and the 1 lb ball is 0.8, determine the exit velocity of the ball
(a) if the ball is stationary, (b) if the ball is struck when it is
rolling towards the foot with a velocity of 10 ft/s.
SOLUTION
PROBLEM 13.184 (Continued)
PROBLEM 13.185
Ball B is hanging from an inextensible cord. An identical ball A is released
from rest when it is just touching the cord and drops through the vertical
distance
8 in.
A
h=
before striking ball B. Assuming
0.9e=
and no friction,
determine the resulting maximum vertical displacement hB of ball B.
SOLUTION
Ball A falls
PROBLEM 13.185 (Continued)
Ball A alone:
Momentum in t–direction:
B
PROBLEM 13.186
A 70 g ball B dropped from a height
0
1.5 mh=
reaches a
height
2
0.25h=
m after bouncing twice from identical 210-g
plates. Plate A rests directly on hard ground, while plate C rests
on a foam-rubber mat. Determine (a) the coefficient of
restitution between the ball and the plates, (b) the height
1
h
of
the ball’s first bounce.
03
70
BP
B
m′′
PROBLEM 13.186 (Continued)
Relative velocities:
33
2
[( ) ( )] ( )
BP P B
v ve v v
′′
− − =−−
11.278 mh=
PROBLEM 13.187
A 2–kg sphere moving to the right with a velocity of 5 m/s strikes at
A the surface of a 9–kg quarter cylinder which is initially at rest and
in contact with a spring of constant 20 kN/m. The spring is held by
cables so that it is initially compressed 50 mm. Neglecting friction
and knowing that the coefficient of restitution is 0.6, determine
(a) the velocity of the sphere immediately after impact, (b) the
maximum compressive force in the spring.
SOLUTION
Momentum:
1 12
0 (0.7071)
nn
mv mv Mv
′′
+= +
2
2
m
PROBLEM 13.188
When the rope is at an angle of
30
α
= °
the 1–lb sphere A has
a speed
0
4 ft/s.v=
The coefficient of restitution between A
and the 2–lb wedge B is 0.7 and the length of rope
2.6 ft.l=
The spring constant has a value of
2 lb/in.
and
20 .
θ
= °
Determine (a) the velocities of A and B immediately after the
impact, (b) the maximum deflection of the spring assuming A
does not strike B again before this point.
SOLUTION
22