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

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

ﬁxed 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=90◦as the

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?

220

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

A

2 m

10 kg

CD

Solution:

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

A

2 m

10 kg

CD

Solution:

(a) We will ﬁrst ﬁnd 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:

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

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)2−0.3−0.1=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=r−r0.

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

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:

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

230

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

16,000 km

13,900 km

Solution: We have

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-

ﬁrm 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.

232

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/RE≪1, 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:

Problem 15.111 An object is subjected to the force

F=yi−xj(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.)

234

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

rer,

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,

Problem 15.116 By substituting Eqs. (15.27) into

Eq. (15.30), conﬁrm that ∇×F=0ifFis conservative.

Solution: Eq. 15.30 is



ijk

∂





Problem 15.117 Determine which of the following are

conservative.

(a) F=(3x2−2xy)i−x2j;

(b) F=(x −xy2)i+x2yj;

(c) F=(2xy2+y3)i+(2x2y−3xy2)j.

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