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PROBLEM 13.31
A 5-kg collar A is at rest on top of, but not attached to, a spring
with stiffness k1 = 400 N/m; when a constant 150-N force is
applied to the cable. Knowing A has a speed of 1 m/s when the
upper spring is compressed 75 mm, determine the spring
stiffness k2. Ignore friction and the mass of the pulley.
SOLUTION
1
400 N/m
f
PROBLEM 13.31 (Continued)
Work of the force exerted by the lower spring:
1 21 1 1
0
22 2 2
() ( )
1 11
f
x
U F k x dx
→= −
∫
PROBLEM 13.32
A piston of mass m and cross-sectional area A is equilibrium
under the pressure p at the center of a cylinder closed at both
ends. Assuming that the piston is moved to the left a distance
a/2 and released, and knowing that the pressure on each side of
the piston varies inversely with the volume, determine the
velocity of the piston as it again reaches the center of the
cylinder. Neglect friction between the piston and the cylinder
and express your answer in terms of m, a, p, and A.
SOLUTION
PROBLEM 13.33
An uncontrolled automobile traveling at 65 mph strikes squarely a highway crash cushion of the type shown
in which the automobile is brought to rest by successively crushing steel barrels. The magnitude F of the force
required to crush the barrels is shown as a function of the distance x the automobile has moved into the
cushion. Knowing that the weight of the automobile is 2250 lb and neglecting the effect of friction, determine
(a) the distance the automobile will move into the cushion before it comes to rest, (b) the maximum
deceleration of the automobile.
SOLUTION
PROBLEM 13.34
Two types of energy-absorbing fenders designed to be used on a
pier are statically loaded. The force-deflection curve for each type
of fender is given in the graph. Determine the maximum
deflection of each fender when a 90-ton ship moving at 1 mi/h
strikes the fender and is brought to rest.
SOLUTION
3
PROBLEM 13.34 (Continued)
PROBLEM 13.35
Nonlinear springs are classified as hard or soft, depending upon the curvature of their force-deflection curve
(see figure). If a delicate instrument having a mass of 5 kg is placed on a spring of length l so that its base is
just touching the undeformed spring and then inadvertently released from that position, determine the
maximum deflection
m
x
of the spring and the maximum force
m
F
exerted by the spring, assuming (a) a linear
spring of constant k = 3 kN/m, (b) a hard, nonlinear spring, for which
2
(3 kN/m)( 160 )F xx= +
.
SOLUTION
(5 kg)
49.05 N
W mg g
W
= =
=
Since
1 2 1 12 2 12
0, yields 0TT TU T U
−−
== += =
12 00
49.05 0
mm
xx
mm
U Wx Fdx x Fdx
−
=−= −=
∫∫
(1)
(a) For
(300 N/m)F kx x= =
Eq. (1):
0
49.05 3000 0
m
x
m
x x dx−=
∫
2
49.05 1500 0
mm
xx−=
3
32.7 10 m 32.7 mm
m
x
−
=×=
3
3000 3000(32.7 10 )
mm
Fx
−
= = ×
98.1 N
m
=F
(b) For
2
(3000 N/m) (1 160 )F xx= +
Eq. (1)
3
0
49.05 3000( 160 ) 0
m
x
m
x x x dx− +=
∫
24
1
49.05 3000 40 0
2
m mm
x xx
− +=
(2)
Solve by trial:
3
30.44 10 m
m
x
−
= ×
30.4 mm
m
x=
3 32
(3000)(30.44 10 )[1 160(30.44 10 ) ]
m
F
−−
= ×+ ×
104.9 N
m
=F
PROBLEM 13.36
A meteor starts from rest at a very great distance from the earth. Knowing that the radius of the earth is 6370
km and neglecting all forces except the gravitational attraction of the earth, determine the speed of the meteor
(a) when it enters the ionosphere at an altitude of 1000 km, (b) when it enters the stratosphere at an altitude of
50 km, (c) when it strikes the earth’s surface.
PROBLEM 13.37
Express the acceleration of gravity
h
g
at an altitude h above the surface of the earth in terms of the
acceleration of gravity
0
g
at the surface of the earth, the altitude h, and the radius R of the earth. Determine
the percent error if the weight that an object has on the surface of the earth is used as its weight at an altitude
of (a) 0.625 mi, (b) 625 mi.
( )
2
625
3960
1
+
PROBLEM 13.38
A golf ball struck on earth rises to a maximum height of 60 m
and hits the ground 230 m away. How high will the same golf
ball travel on the moon if the magnitude and direction of its
velocity are the same as they were on earth immediately after
the ball was hit? Assume that the ball is hit and lands at the
same elevation in both cases and that the effect of the
atmosphere on the earth is neglected, so that the trajectory in
both cases is a parabola. The acceleration of gravity on the
moon is 0.165 times that on earth.
m
e
m
PROBLEM 13.39
The sphere at A is given a downward velocity
0
v
of magnitude
5 m/s and swings in a vertical plane at the end of a rope of
length
2ml=
attached to a support at O. Determine the angle
θ
at which the rope will break, knowing that it can withstand a
maximum tension equal to twice the weight of the sphere.
SOLUTION
22
10
1
11
(5)
22
12.5 m
T mv m
T
= =
=
(6)(9.81)
PROBLEM 13.40
The sphere at A is given a downward velocity
0
v
and swings
in a vertical circle of radius l and center O. Determine the
smallest velocity
0
v
for which the sphere will reach Point B as
it swings about Point O (a) if AO is a rope, (b) if AO is a
slender rod of negligible mass.
0
0
PROBLEM 13.41
A bag is gently pushed off the top of a wall at A and swings in a vertical
plane at the end of a rope of length l Determine the angle
θ
for which the
rope will break, knowing that it can withstand a maximum tension equal to
twice the weight of the bag.
SOLUTION
PROBLEM 13.42
A roller coaster starts from rest at A, rolls down the
track to B, describes a circular loop of 40-ft diameter,
and moves up and down past Point E. Knowing that h =
60 ft and assuming no energy loss due to friction,
determine (a) the force exerted by his seat on a 160-lb
rider at B and D, (b) the minimum value of the radius of
curvature at E if the roller coaster is not to leave the
track at that point.
PROBLEM 13.42 (Continued)
EE
PROBLEM 13.43
In Problem 13.42, determine the range of values of h
for which the roller coaster will not leave the track at D
or E, knowing that the radius of curvature at E is
75 ft.
ρ
=
Assume no energy loss due to friction.
PROBLEM 13.42 A roller coaster starts from rest at A,
rolls down the track to B, describes a circular loop of
40-ft diameter, and moves up and down past Point E.
Knowing that h = 60 ft and assuming no energy loss
due to friction, determine (a) the force exerted by his
seat on a 160-lb rider at B and D, (b) the minimum
value of the radius of curvature at E if the roller coaster
is not to leave the track at that point.
SOLUTION
25
D
hr
N mg r
−
=
PROBLEM 13.43 (Continued)
PROBLEM 13.44
A small block slides at a speed v on a horizontal surface. Knowing that
h = 0.9 m, determine the required speed of the block if it is to leave the
cylindrical surface BCD when
θ
= 30°.
SOLUTION
At Point C where the block leaves the surface BCD the contact force is reduced to zero. Apply Newton’s
B
PROBLEM 13.45
A small block slides at a speed
8v=
ft/s on a horizontal surface at a
height
3h=
ft above the ground. Determine (a) the angle
θ
at which it
will leave the cylindrical surface BCD, (b) the distance x at which it
will hit the ground. Neglect friction and air resistance.
SOLUTION
3
C
PROBLEM 13.45 (Continued)
