Mechanical Engineering Chapter 16 Problem The Coefcient Restitution The Same For Both Impacts Show That The

Problem 16.82 If the coefﬁcient of restitution is the

same for both impacts, show that the cue ball’s path

after two banks is parallel to its original path.

Solution: The strategy is to treat the two banks as two successive

oblique central impacts. Denote the path from the cue ball to the ﬁrst

294

Problem 16.83 The velocity of the 170-g hockey puck

is vP=10i−4j(m/s). If you neglect the change in the

velocity vS=vSjof the stick resulting from the impact,

and if the coefﬁcient of restitution is e=0.6, what should

vSbe to send the puck toward the goal? 20°

Direction

of goal

x

y

vP

vS

Solution: The strategy is to treat the collision as an oblique cen-

Problem 16.84 In Problem 16.83, if the stick responds

to the impact the way an object with the same mass as

the puck would, and if the coefﬁcient of restitution is

e=0.6, what should vSbe to send the puck toward

the goal?

Problem 16.85 At the instant shown (t1=0), the posi-

tion of the 2-kg object’s center of mass is r=6i+4j+

2k(m) and its velocity is v=−16i+8j−12k(m/s).

No external forces act on the object. What is the object’s

angular momentum about the origin Oat t2=1s?

O

r

x

y

z

Problem 16.86 Suppose that the total external force

(a) Use Newton’s second law to determine the object’s

Solution:

v=t2

296

Problem 16.87 A satellite is in the elliptic earth orbit

shown. Its velocity at perigee Ais 8640 m/s. The radius

of the earth is 6370 km.

(a) Use conservation of angular momentum to deter-

mine the magnitude of the satellite’s velocity at

apogee C.

(b) Use conservation of energy to determine the mag-

nitude of the velocity at C.

(See Example 16.8.)

B

A

C

13,900 km

Solution:

Problem 16.88 For the satellite in Problem 16.87,

determine the magnitudes of the radial velocity vrand

transverse velocity vθat B. (See Example 16.8.)

Solution: Use conservation of energy to ﬁnd the velocity

Problem 16.89 The bar rotates in the horizontal plane

about a smooth pin at the origin. The 2-kg sleeve A

slides on the smooth bar, and the mass of the bar is

negligible in comparison to the mass of the sleeve.

The spring constant k=40 N/m, and the spring is

unstretched when r=0. At t=0, the radial position

of the sleeve is r=0.2 m and the angular velocity of

the bar is w0=6 rad/s. What is the angular velocity of

the bar when r=0.25 m?

r

A

v0

k

Problem 16.90 At t=0, the radial position of the

sleeve Ain Problem 16.89 is r=0.2 m, the radial

velocity of the sleeve is vr=0 and the angular velocity

of the bar is w0=6 rad/s. What are the angular velocity

of the bar and the radial velocity of the sleeve when

r=0.25 m?

298

Problem 16.91 A 2-kg disk slides on a smooth

horizontal table and is connected to an elastic cord whose

tension is T=6rN, where ris the radial position of

the disk in meters. If the disk is at r=1 m and is given

an initial velocity of 4 m/s in the transverse direction,

what are the magnitudes of the radial and transverse

components of its velocity when r=2 m? (See Active

Example 16.7.)

r

Solution: The strategy is to (a) use the principle of conservation

Problem 16.92 In Problem 16.91, determine the max-

imum value of rreached by the disk.

Problem 16.93 A 1-kg disk slides on a smooth

horizontal table and is attached to a string that passes

through a hole in the table.

(a) If the mass moves in a circular path of constant

radius r=1 m with a velocity of 2 m/s, what is

the tension T?

(b) Starting from the initial condition described in

part (a), the tension Tis increased in such a way

that the string is pulled through the hole at a

constant rate until r=0.5 m. Determine the value

of Tas a function of rwhile this is taking place.

r

T

Solution:

T=−mv2/rer

|T|=(1)(2)2/1=4N

(b) By conservation of angular momentum,

mr0v0=mrvT,∴vT=r0v0

r

and from Newton’s second law

|T|=mv2

T/r =mr0v0

r2

/r

T=(1)(1)(2)

r2#r=4/r3N

r

V

er

m

Problem 16.94 In Problem 16.93, how much work is

done on the mass in pulling the string through the hole

as described in part (b)?

Solution: The work done is

0r2

0

300

Problem 16.95 Two gravity research satellites (mA=

250 kg, mB=50 kg)are tethered by a cable. The

satellites and cable rotate with angular velocity ω0=

0.25 revolution per minute. Ground controllers order

satellite Ato slowly unreel 6 m of additional cable. What

is the angular velocity afterward?

A

B

0

12 m

ω

Solution: The satellite may be rotating in (a) a vertical plane, or

from which WA=−mAg′jand WB=−mBg′j(or, alternatively).

The angular momentum impulse is

A repeat of the argument above for any additional length of cable

ω=H

32mA+152mB=0.0116 rad/s =0.1111 rpm

Case (b): Assume that the system rotates in the x–zplane, with

Problem 16.96 The astronaut moves in the x–y plane

at the end of a 10-m tether attached to a large space

station at O. The total mass of the astronaut and his

equipment is 120 kg.

(a) What is the astronaut’s angular momentum about

Obefore the tether becomes taut?

(b) What is the magnitude of the component of his

velocity perpendicular to the tether immediately

after the tether becomes taut?

y

2i (m /s)

6 m

Ox

(120)200

=−1440k(kg-m2/s).

(b) From conservation of angular momentum,

(b) H=−1440k=−(10)(120)vk, from which

v=1440

(10)(120)=1.2 m/s

6 m

10 m

Problem 16.97 The astronaut moves in the x–yplane

at the end of a 10-m tether attached to a large space

station at O. The total mass of the astronaut and his

equipment is 120 kg. The coefﬁcient of restitution of the

“impact” that occurs when he comes to the end of the

tether is e=0.8. What are the xand ycomponents of

his velocity immediately after the tether becomes taut?

302

Problem 16.98 A ball suspended from a string that

goes through a hole in the ceiling at Omoves with

velocity vAin a horizontal circular path of radius rA. The

string is then drawn through the hole until the ball moves

with velocity vBin a horizontal circular path of radius

rB. Use the principle of angular impulse and momentum

to show that rAvA=rBvB.

Strategy: Let ebe a unit vector that is perpendicular

to the ceiling. Although this is not a central-force

problem — the ball’s weight does not point toward

O— you can show that e·(r×(F)=0, so that e·HO

is conserved.

O

B

rB

rA

A

Problem 16.99 The Cheverton ﬁre-ﬁghting and rescue

boat can pump 3.8 kg/s of water from each of its two

pumps at a velocity of 44 m/s. If both pumps point in

the same direction, what total force do they exert on the

boat.

Problem 16.100 The mass ﬂow rate of water through

the nozzle is 1.6 slugs/s. Determine the magnitude of the

horizontal force exerted on the truck by the ﬂow of the

water.

20 ft

35 ft

12 ft

20°

Solution: We must determine the velocity with which the water

304

Problem 16.101 The front-end loader moves at a

constant speed of 2 mi/h scooping up iron ore. The

constant horizontal force exerted on the loader by the

road is 400 lb. What weight of iron ore is scooped up

in 3 s?

Problem 16.102 The snowblower moves at 1 m/s and

scoops up 750 kg/s of snow. Determine the force exerted

by the entering ﬂow of snow.

Problem 16.103 The snowblower scoops up 750 kg/s

of snow. It blows the snow out the side at 45◦above

the horizontal from a port 2 m above the ground and the

snow lands 20 m away. What horizontal force is exerted

on the blower by the departing ﬂow of snow?

vf=13.36 m/s.

306

Problem 16.104 A nozzle ejects a stream of water hor-

izontally at 40 m/s with a mass ﬂow rate of 30 kg/s, and

the stream is deﬂected in the horizontal plane by a plate.

Determine the force exerted on the plate by the stream

in cases (a), (b), and (c). (See Example 16.11.)

x

45°

y

(a)

x

y

(b)

x

y

(c)

Solution: Apply the strategy used in Example 16.7. The exit veloc-

Problem 16.105* A stream of water with velocity

80i(m/s)and a mass ﬂow of 6 kg/s strikes a turbine

blade moving with constant velocity 20i(m/s).

(a) What force is exerted on the blade by the water?

(b) What is the magnitude of the velocity of the water

as it leaves the blade?

y

x

70°

20 m/s

80 m/s

Solution: Denote the ﬁxed reference frame as the nozzle frame,

and the moving blade frame as the blade rest frame. Assume that the

in the blade rest frame is not modiﬁed by the velocity of the blade.

Denote the velocity of the blade by vB. The inlet velocity of the water

vB

θ′

308

Problem 16.106 At the instant shown, the nozzle Aof

the lawn sprinkler is located at (0.1,0,0)m. Water exits

each nozzle at 8 m/s relative to the nozzle with a mass

ﬂow rate of 0.22 kg/s. At the instant shown, the ﬂow

relative to the nozzle at Ais in the direction of the unit

vector

e=1

√3i−1

√3j+1

√3k.

Determine the total moment about the zaxis exerted on

the sprinkler by the ﬂows from all four nozzles. x

y

A

Solution:

f

Problem 16.107 A 45-kg/s ﬂow of gravel exits the

chute at 2 m/s and falls onto a conveyor moving at

0.3 m/s. Determine the components of the force exerted

θ

x

0.3 m/s

Solution: The horizontal component of the velocity of the gravel

ﬂow is vx=2 cos 45◦=√2 m/s. From Newton’s second law, (using

Problem 16.108 Solve Problem 16.107 if θ=30◦.

Solution: Use the solution to Problem 16.107 as appropriate. From

The angle of impact:

Problem 16.109 Suppose that you are designing a toy

car that will be propelled by water that squirts from

an internal tank at 10 ft/s relative to the car. The total

weight of the car and its water “fuel” is to be 2 lb. If

you want the car to achieve a maximum speed of 12 ft/s,

what part of the total weight must be water?

20°

Problem 16.110 The rocket consists of a 1000-kg

payload and a 9000-kg booster. Eighty percent of the

booster’s mass is fuel, and its exhaust velocity is

1200 m/s. If the rocket starts from rest and external

forces are neglected, what velocity will it attain? (See

310

Problem 16.111* The rocket consists of a 1000-kg

payload and a booster. The booster has two stages whose

total mass is 9000 kg. Eighty percent of the mass of

each stage is fuel, and the exhaust velocity of each stage

is 1200 m/s. When the fuel of stage 1 is expended,

it is discarded and the motor of stage 2 is ignited.

Assume that the rocket starts from rest and neglect

external forces. Determine the velocity attained by the

rocket if the masses of the stages are m1=6000 kg and

m2=3000 kg. Compare your result to the answer to

Problem 16.110.

Payload

12

Solution:

Problem 16.112 A rocket of initial mass m0takes off

straight up. Its exhaust velocity vfand the mass ﬂow rate

of its engine mf=dm

f/dt are constant. Show that, dur-

ing the initial part of the ﬂight, when aerodynamic drag

is negligible, the rocket’s upward velocity as a function

of time is

v=vfln m0

m0−˙mft−gt.

Problem 16.113 The mass of the rocket sled in

Active Example 16.9 is 440 kg. Assuming that the only

signiﬁcant force acting on the sled in the direction of its

motion is the force exerted by the ﬂow of water entering

it, what distance is required for the sled to decelerate

from 300 m/s to 100 m/s?

312

Problem 16.114* Suppose that you grasp the end of a

chain that weighs 3 lb/ft and lift it straight up off the

ﬂoor at a constant speed of 2 ft/s.

(a) Determine the upward force Fyou must exert as

a function of the height s.

(b) How much work do you do in lifting the top of the

chain to s=4 ft?

Strategy: Treat the part of the chain you have lifted

as an object that is gaining mass.

s

Solution: The force is the sum of the “mass ﬂow” reaction and

Problem 16.115* Solve Problem 16.114, assuming

that you lift the end of the chain straight up off the

ﬂoor with a constant acceleration of 2 ft/s2.

Solution: Assume that the velocity is zero at s=0. The mass

from which