Mechanical Engineering Chapter 15 Problem The Mass The Ball And The Strings Length The Ball Released

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subject Pages 12
subject Words 4665
subject Authors Anthony M. Bedford, Wallace Fowler

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Problem 15.84 The mass of the ball is m=2 kg and
the string’s length is L=1 m. The ball is released from
rest in position 1. When the string is vertical, it hits the
xed peg shown.
(a) Use conservation of energy to determine the min-
imum angle θnecessary for the ball to swing to
position 2.
(b) If the ball is released at the minimum angle θdeter-
mined in part (a), what is the tension in the string
just before and just after it hits the peg?
m=2kg
L=1m
L
1
2
L
1
2
u
Solution: Energy is conserved. v1=v2=0 Use θ=90as the
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Problem 15.85 A small pellet of mass m=0.2kg
starts from rest at position 1 and slides down the
smooth surface of the cylinder to position 2. The radius
R=0.8 m. Use conservation of energy to determine
the magnitude of the pellet’s velocity at position 2 if
θ=45.m
2
1
u
R
20
Problem 15.86 In Problem 15.85, what is the value of
the angle θat which the pellet loses contact with the
surface of the cylinder?
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Problem 15.87 The bar is smooth. The 10-kg slider at
Ais given a downward velocity of 6.5 m/s.
(a) Use conservation of energy to determine whether
the slider will reach point C. If it does, what is the
magnitude of its velocity at point C?
(b) What is the magnitude of the normal force the bar
exerts on the slider as it passes point B?
B
1 m
1 m
A
2 m
10 kg
CD
Solution:
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Problem 15.88 The bar is smooth. The 10-kg slider at
Ais given a downward velocity of 7.5 m/s.
(a) Use conservation of energy to determine whether
the slider will reach point D. If it does, what is the
magnitude of its velocity at point D?
(b) What is the magnitude of the normal force the bar
exerts on the slider as it passes point B?
B
1 m
1 m
A
2 m
10 kg
CD
Solution:
(a) We will rst nd the velocity at the highest point (half way
Problem 15.89 In Active Example 15.7, suppose that
you want to increase the value of the spring constant k
so that the velocity of the hammer just before it strikes
the workpiece is 4 m/s. Use conservation of energy to
determine the required value of k.
400
1
kk
Hammer
Solution:
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Problem 15.90 A rock climber of weight Whas a rope
attached a distance hbelow him for protection. Suppose
that he falls, and assume that the rope behaves like a lin-
ear spring with unstretched length hand spring constant
k=C/h, where Cis a constant. Use conservation of
energy to determine the maximum force exerted on the
climber by the rope. (Notice that the maximum force
is independent of h, which is a reassuring result for
climbers: The maximum force resulting from a long fall
is the same as that resulting from a short one.)
h
Solution: Choose the climber’s center of mass before the fall as
the datum. The energy of the climber before the fall is zero. As the
climber falls, his energy remains the same:
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Problem 15.91 The collar Aslides on the smooth hori-
zontal bar. The spring constant k=40 lb/ft. The weights
are WA=30 lb and WB=60 lb. As the instant shown,
the spring is unstretched and Bis moving downward
at 4 ft/s. Use conservation of energy to determine the
velocity of Bwhen it has moved downward 2 ft from
its current position. (See Example 15.8.)
A
B
k
Solution: Notice that the collars have the same velocity
Problem 15.92 The spring constant k=700 N/m. The
masses mA=14 kg and mB=18 kg. The horizontal bar
is smooth. At the instant shown, the spring is unstretched
and the mass Bis moving downward at 1 m/s. How fast
is Bmoving when it has moved downward 0.2 m from
its present position?
A
k
0.15 m
0.3 m
Solution: The unstretched length of the spring is
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Problem 15.93 The semicircular bar is smooth. The
unstretched length of the spring is 10 in. The 5-lb collar
at Ais given a downward velocity of 6 ft/s, and when it
reaches Bthe magnitude of its velocity is 15 ft/s. Deter-
mine the spring constant k.
1 ft
A
2 in
5 in
B
k
Solution: The stretch distances for the spring at Aand Bare
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Problem 15.94 The mass m=1 kg, the spring con-
stant k=200 N/m, and the unstretched length of the
spring is 0.1 m. When the system is released from rest
in the position shown, the spring contracts, pulling the
mass to the right. Use conservation of energy to deter-
mine the magnitude of the velocity of the mass when
the string and spring are parallel.
k
0.3 m
0.25 m
Solution: The stretch of the spring in position 1 is
S2=(0.3+0.15)2+(0.25)20.30.1=0.115 m.
0.3 m
2
1
Datum
β
Problem 15.95 In problem 15.94, what is the tension
in the string when the string and spring are parallel?
Solution: The free body diagram of the mass is: Newton’s second
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Problem 15.96 The force exerted on an object by
anonlinear spring is F=−[k(r r0)+q(r r0)3]er,
where kand qare constants and r0is the unstretched
length of the spring. Determine the potential energy of
the spring in terms of its stretch S=rr0.
k
θ
r
Solution: Note that dS =dr. The work done in stretching the
Problem 15.97 The 20-kg cylinder is released at
the position shown and falls onto the linear spring
(k =3000 N/m). Use conservation of energy to
determine how far down the cylinder moves after
contacting the spring. 2 m
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Problem 15.98 The 20-kg cylinder is released at the
position shown and falls onto the nonlinear spring.
In terms of the stretch Sof the spring, its potential
energy is V=1
2kS2+1
2qS4, where k=3000 N/m and
q=4000 N/m3. What is the velocity of the cylinder
when the spring has been compressed 0.5 m?
Problem 15.99 The string exerts a force of constant
magnitude Ton the object. Use polar coordinates to
show that the potential energy associated with this force
is V=Tr.
T
θ
r
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Problem 15.100 The system is at rest in the position
shown, with the 12-lb collar Aresting on the spring
(k =20 lb/ft), when a constant 30-lb force is applied to
the cable. What is the velocity of the collar when it has
risen 1 ft? (See Problem 15.99.)
30 lb
3 ft
A
Problem 15.101 A 1-kg disk slides on a smooth hori-
zontal table and is attached to a string that passes through
a hole in the table. A constant force T=10 N is exerted
on the string. At the instant shown, r=1 m and the
velocity of the disk in terms of polar coordinates is
v=6eθ(m/s). Use conservation of energy to determine
the magnitude of the velocity of the disk when r=2m.
(See Problem 15.99.)
r
T
Solution:
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Problem 15.102 A 1-kg disk slides on a smooth hori-
zontal table and is attached to a string that passes through
a hole in the table. A constant force T=10 N is exerted
on the string. At the instant shown, r=1 m and the
velocity of the disk in terms of polar coordinates is
v=8eθ(m/s). Because this is central-force motion, the
product of the radial position rand the transverse com-
ponent of velocity vθis constant. Use this fact and con-
servation of energy to determine the velocity of the disk
in terms of polar coordinates when r=2m.
r
T
Problem 15.103 A satellite initially is inserted into
orbit at a distance r0=8800 km from the center of
the earth. When it is at a distance r=18,000 km from
the center of the earth, the magnitude of its velocity is
v=7000 m/s. Use conservation of energy to determine
its initial velocity v0. The radius of the earth is 6370 km.
(See Example 15.9.)
v
v0
r0
r
RE
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Problem 15.104 Astronomers detect an asteroid
100,000 km from the earth moving at 2 km/s relative to
the center of the earth. Suppose the asteroid strikes the
earth. Use conservation of energy to determine the mag-
nitude of its velocity as it enters the atmosphere. (You
can neglect the thickness of the atmosphere in compari-
son to the earth’s 6370-km radius.)
Solution: Use the solution to Problem 15.103. The potential
Problem 15.105 A satellite is in the elliptic earth orbit
shown. Its velocity in terms of polar coordinates when
it is at the perigee Ais v=8640eθ(m/s). Determine the
velocity of the satellite in terms of polar coordinates
when it is at point B.
B
A
C
8000 km
8000 km
16,000 km
13,900 km
Solution: We have
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Problem 15.106 Use conservation of energy to deter-
mine the magnitude of the velocity of the satellite in
Problem 15.105 at the apogee C. Using your result, con-
rm numerically that the velocities at perigee and apogee
satisfy the relation rAvA=rCvC.
Problem 15.107 The Voyager and Galileo spacecraft
have observed volcanic plumes, believed to consist of
condensed sulfur or sulfur dioxide gas, above the sur-
face of the Jovian satellite Io. The plume observed above
a volcano named Prometheus was estimated to extend
50 km above the surface. The acceleration due to gravity
at the surface is 1.80 m/s2. Using conservation of energy
and neglecting the variation of gravity with height, deter-
mine the velocity at which a solid particle would have
to be ejected to reach 50 km above Io’s surface.
Problem 15.108 Solve Problem 15.107 using conser-
vation of energy and accounting for the variation of
gravity with height. The radius of Io is 1815 km.
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Problem 15.109* What is the relationship between
Eq. (15.21), which is the gravitational potential energy
neglecting the variation of the gravitational force
with height, and Eq. (15.23), which accounts for the
variation? Express the distance from the center of the
earth as r=RE+y, where REis the earth’s radius and
yis the height above the surface, so that Eq. (15.23) can
be written as
V=−mgRE
1+y
RE
.
By expanding this equation as a Taylor series in terms
of y/REand assuming that y/RE1, show that you
obtain a potential energy equivalent to Eq. (15.21).
Problem 15.110 The potential energy associated with
a force Facting on an object is V=x2+y3N-m, where
xand yare in meters.
(a) Determine F.
(b) Suppose that the object moves from position 1 to
position 2 along path A, and then moves from posi-
tion 1 to position 2 along path B. Determine the
work done by Falong each path.
y
x
1B
A
2
(1, 1) m
Solution:
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Problem 15.111 An object is subjected to the force
F=yixj(N), where xand yare in meters.
(a) Show that Fis not conservative.
(b) Suppose the object moves from point 1 to point 2
along the paths Aand Bshown in Problem 15.110.
Determine the work done by Falong each path.
Problem 15.112 In terms of polar coordinates, the
potential energy associated with the force Fexerted on
an object by a nonlinear spring is
V=1
2k(r r0)2+1
4q(r r0)4,
where kand qare constants and r0is the unstretched
length of the spring. Determine Fin terms of polar
coordinates. (See Active Example 15.10.)
Problem 15.113 In terms of polar coordinates, the
force exerted on an object by a nonlinear spring is
F=−(k(r r0)+q(r r0)3)er,
where kand qare constants and r0is the unstretched
length of the spring. Use Eq. (15.36) to show that Fis
conservative. (See Active Example 15.10.)
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Problem 15.114 The potential energy associated with
a force Facting on an object is V=−rsin θ+
r2cos2θft-lb, where ris in feet.
(a) Determine F.
(b) If the object moves from point 1 to point 2 along
the circular path, how much work is done by F?
y
x
1
2
1 ft
Solution: The force is
Check: Since the force is derivable from a potential, the system is con-
Problem 15.115 In terms of polar coordinates, the
force exerted on an object of mass mby the gravity
of a hypothetical two-dimensional planet is
F=−mgTRT
rer,
where gTis the acceleration due to gravity at the surface,
RTis the radius of the planet, and ris the distance from
the center of the planet.
(a) Determine the potential energy associated with this
gravitational force.
(b) If the object is given a velocity v0at a distance r0,
what is its velocity vas a function of r?
RT
r0
0
Solution:
(Note: Alternatively, the choice of r=1length-unit as the datum,
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Problem 15.116 By substituting Eqs. (15.27) into
Eq. (15.30), conrm that ∇×F=0ifFis conservative.
Solution: Eq. 15.30 is
ijk
Problem 15.117 Determine which of the following are
conservative.
(a) F=(3x22xy)ix2j;
(b) F=(x xy2)i+x2yj;
(c) F=(2xy2+y3)i+(2x2y3xy2)j.
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