596 Solutions Manual
Problem 3.122
A simple elevator consists of a
15;000 kg
car
A
connected to a
12;000 kg
counterweight
B
. Suppose that a failure occurs when
the car is at rest and
50
m above its buffer, causing the elevator
car to fall. Model the car and the counterweight as particles and
the cord as massless and inextensible; and model the action of the
emergency brakes using a Coulomb friction model with kinetic
friction coefficient
kD0:5
and a normal force equal to 35% of
the car’s weight. Determine the speed with which the car impacts
the buffer.
Solution
Referring to the figure at the right, we model the car
A
and the coun-
terweight
B
as particles of mass
mA
and
mB
, respectively. We assume
Dynamics 2e 597
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598 Solutions Manual
Problem 3.123
The linear elastic spring with stiffness
k
and unstretched length
`u
is
attached to both the vertical wall and the metal block
A
of mass
mA
. The
metal block
B
is pushed into
A
so that the spring compresses a distance
d
.
The block Bis then released from rest.
Assuming that friction between the blocks and the horizontal surface
is negligible, determine the distance the blocks slide before they separate
and their speed at separation. Hint: The blocks will start to separate when
the normal force between them goes to zero.
Solution
Dynamics 2e 599
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permission of McGraw-Hill, is prohibited.
600 Solutions Manual
Problem 3.124
The linear elastic spring with stiffness
k
and unstretched length
`u
is
attached to both the vertical wall and the metal block
A
of mass
mA
. The
metal block
B
is pushed into
A
so that the spring compresses a distance
d
.
The block Bis then released from rest.
Assuming that friction between the blocks and the surface is non-
negligible and that the coefficient of static friction is insufficient to prevent
motion, determine the condition on the compression distance
d
for the
blocks to separate. The coefficient of kinetic friction between the blocks
and the surface is
k
.Hint: The blocks will start to separate when the
normal force between them goes to zero.
Solution
Dynamics 2e 601
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permission of McGraw-Hill, is prohibited.
602 Solutions Manual
Problem 3.125
Two particles
A
and
B
with masses
mA
and
mB
, respectively, are a distance
r0
apart, and both masses are initially at rest. Using Eq. (1.5) on p. 3, determine the
amount of time it takes for the two masses to come into contact if
mAD1kg
,
mBD2kg
, and
r0D1
m. Assume that the two masses are only influenced by
their mutual gravitational attraction. Hint: Rr0
0pr0r=.r0r/dr Dl
2⇡r3=2
0.
Solution
We model
A
and
B
as particles under the action of their mutual gravitational
attraction. We view the problem as one-dimensional. We treat the
r
axis shown in
Dynamics 2e 603
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604 Solutions Manual
Problem 3.126
Energy storage devices that use spinning flywheels to store en-
ergy are becoming available. To maximize energy storage, the
flywheel must spin as fast as possible. Unfortunately, if it spins
too fast, internal stresses in the flywheel cause it to come apart
catastrophically. Therefore, it is important to keep the speed at
the periphery of the flywheel below about
1000 m=s
. It is also
critical that the flywheel be well balanced to avoid the damaging
vibrations that would otherwise result. With this in mind, let the
flywheel
D
with diameter
0:3
m rotate at
!D60;000 rpm
. In ad-
dition, assume that the cart
B
is constrained to move rectilinearly
along the guide tracks. Given that the flywheel is not perfectly
balanced, that the unbalanced weight
A
has mass
mA
, and that
the total mass of the flywheel
D
, cart
B
, and electronics package
E
is
mB
, determine the constraint force between the wheels of
the cart and the guide tracks as a function of
✓
, the masses, the
diameter, and the angular speed of the flywheel. What is the
maximum constraint force between the wheels of the cart and
the guide tracks? Finally, evaluate your answers for
mAD1
g
(about the mass of a paper clip) and
mBD70 kg
. Assume that
the unbalanced mass is at the periphery of the flywheel.
Solution
Dynamics 2e 605
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permission of McGraw-Hill, is prohibited.