Mechanical Engineering Chapter 15 Problem The Driver Car Moving Mih Applies Increasing Force The Brake Pedalthe

Problem 15.118 The driver of a 3000-lb car moving at

40 mi/h applies an increasing force on the brake pedal.

The magnitude of the resulting frictional force exerted

on the car by the road is f=250 +6slb, where sis the

car’s horizontal position (in feet) relative to its position

when the brakes were applied. Assuming that the car’s

tires do not slip, determine the distance required for the

car to stop

(a) by using Newton’s second law and

(b) by using the principle of work and energy.

Solution:

3=41.67,c =−Wv2

6g=−53493.6.

(b) Principle of work and energy: The energy of the car when the

b=125

3=41.67,c =−Wv2

1

6g=−53493.6.

Problem 15.119 Suppose that the car in Prob-

lem 15.118 is on wet pavement and the coefﬁcients of

friction between the tires and the road are µs=0.4 and

Problem 15.120 An astronaut in a small rocket vehicle

(combined mass =450 kg) is hovering 100 m above the

surface of the moon when he discovers that he is nearly

out of fuel and can exert the thrust necessary to cause

the vehicle to hover for only 5 more seconds. He quickly

considers two strategies for getting to the surface:

(a) Fall 20 m, turn on the thrust for 5 s, and then fall

the rest of the way;

(b) fall 40 m, turn on the thrust for 5 s, and then fall

the rest of the way.

Which strategy gives him the best chance of surviving?

How much work is done by the engine’s thrust in each

case? (gmoon =1.62 m/s2)

where F=mg, acting upward, h3is the altitude at the end of

the thrusting phase. The energy condition at the end of the thrust-

m=11.8 m/s .

He should choose strategy (b) since the impact velocity is reduced

238

Problem 15.121 The coefﬁcients of friction between

the 20-kg crate and the inclined surface are µs=0.24

and µk=0.22. If the crate starts from rest and the hor-

izontal force F=200 N, what is the magnitude of the

velocity of the crate when it has moved 2 m?

F

(F cos 30◦−mg sin 30◦−µkN)(2m) =1

2mv2

2−0,

we obtain v2=1.77 m/s.

x

Problem 15.122 The coefﬁcients of friction between

izontal force F=40 N. What is the magnitude of the

velocity of the create when it has moved 2 m?

Solution: See the solution of Problem 15.121. The normal force is

The friction force necessary for equilibrium is

Problem 15.123 The Union Paciﬁc Big Boy locomo-

tive weighs 1.19 million lb, and the traction force (tan-

gential force) of its drive wheels is 135,000 lb. If you

neglect other tangential forces, what distance is required

for the train to accelerate from zero to 60 mi/h?

Problem 15.124 In Problem 15.123, suppose that the

acceleration of the locomotive as it accelerates from

zero to 60 mi/h is (F0/m)(1−v/88), where F0=

135,000 lb, mis the mass of the locomotive, and vis

its velocity in feet per second.

(a) How much work is done in accelerating the train

to 60 mi/h?

(b) Determine the locomotive’s velocity as a function

of time.

Solution: [Note: Fis not a force, but an acceleration, with the

Check: To demonstrate that this is a correct expression, it is used

240

Problem 15.125 A car traveling 65 mi/h hits the crash

barrier described in Problem 15.14. Determine the max-

imum deceleration to which the passengers are subjected

if the car weighs (a) 2500 lb and (b) 5000 lb.

s

Solution: From Problem 15.14 we know that the force in the crash

2mv2+s

0

1

(120s+40s3)ds=0

s=13.6ft,a =F

32.2 ft/s2and solving, we ﬁnd that

s=16.2ft,a =F

m=120s+40s3

m=1110 ft/s2

(a)1320 ft/s2,(b)1110 ft/s2.

0

0.1

0.2

0.05 0.1

0.15

Problem 15.126 In a preliminary design for a mail-

sorting machine, parcels moving at 2 ft/s slide down a

smooth ramp and are brought to rest by a linear spring.

What should the spring constant be if you don’t want the

10-lb parcel to be subjected to a maximum deceleration

greater than 10g’s?

2 ft/s

3 ft

k

242

Problem 15.127 When the 1-kg collar is in position 1,

the tension in the spring is 50 N, and the unstretched

length of the spring is 260 mm. If the collar is pulled

to position 2 and released from rest, what is its velocity

when it returns to position 1? 300 mm

600 mm

12

v1=k

m(S2

2−S2

1)=14.46 m/s

600 mm

Problem 15.128 When the 1-kg collar is in position 1,

the tension in the spring is 100 N, and when the collar

is in position 2, the tension in the spring is 400 N.

(a) What is the spring constant k?

(b) If the collar is given a velocity of 15 m/s at position

1, what is the magnitude of its velocity just before

it reaches position 2?

k,0.3−S0=100

k.

from which

Problem 15.129 The 30-lb weight is released from

rest with the two springs (kA=30 lb/ft, kB=15 lb/ft)

unstretched.

(a) How far does the weight fall before rebounding?

(b) What maximum velocity does it attain?

kA

kB

244

Problem 15.130 The piston and the load it supports

are accelerated upward by the gas in the cylinder. The

total weight of the piston and load is 1000 lb. The cylin-

der wall exerts a constant 50-lb frictional force on the

piston as it rises. The net force exerted on the piston

by pressure is (p2−patm)A, where pis the pressure of

the gas, patm =2117 lb/ft2is the atmospheric pressure,

and A=1ft

2is the cross-sectional area of the piston.

Assume that the product of pand the volume of the

cylinder is constant. When s=1 ft, the piston is sta-

tionary and p=5000 lb/ft2. What is the velocity of the

piston when s=2 ft?

s

Piston

Gas

The potential energy due to gravity is

Vgravity =−

s

s0

(−W)ds =W(s −s0).

The work done by the friction is

Ufriction =s

s0

(−f)ds =−f(s−s0), where f=50 lb.

from which v=2(298.7)g

W=4.39 ft/s

Problem 15.131 When a 22,000-kg rocket’s engine

burns out at an altitude of 2 km, the velocity of the

rocket is 3 km/s and it is traveling at an angle of 60◦

relative to the horizontal. Neglect the variation in the

gravitational force with altitude.

(a) If you neglect aerodynamic forces, what is the

magnitude of the velocity of the rocket when it

reaches an altitude of 6 km?

(b) If the actual velocity of the rocket when it reaches

an altitude of 6 km is 2.8 km/s, how much work is

done by aerodynamic forces as the rocket moves

from 2 km to 6 km altitude?

Problem 15.132 The 12-kg collar Ais at rest in the

position shown at t=0 and is subjected to the tangen-

tial force F=24 −12t2N for 1.5 s. Neglecting friction,

what maximum height hdoes the collar reach?

h

A

F

m[12t2−t4]1.5

Problem 15.133 Suppose that, in designing a loop for

a roller coaster’s track, you establish as a safety criterion

that at the top of the loop the normal force exerted on a

passenger by the roller coaster should equal 10 percent of

the passenger weight. (That is, the passenger’s “effective

weight” pressing him down into his seat is 10 percent

of his actual weight.) The roller coaster is moving at

62 ft/s when it enters the loop. What is the necessary

instantaneous radius of curvature ρof the track at the

top of the loop?

ρ

50 ft

246

Problem 15.134 A 180-lb student runs at 15 ft/s, grabs

a rope, and swings out over a lake. He releases the rope

when his velocity is zero.

(a) What is the angle θwhen he releases the rope?

(b) What is the tension in the rope just before he

release it?

(c) What is the maximum tension in the rope?

θ

30 ft

Problem 15.135 If the student in Problem 15.134

releases the rope when θ=25◦, what maximum height

does he reach relative to his position when he grabs the

rope?

Problem 15.136 A boy takes a running start and jumps

on his sled at position 1. He leaves the ground at

position 2 and lands in deep snow at a distance of

b=25 ft. How fast was he going at 1?

1

2

15 ft 35°

v1=4.72 ft/s The other values were v2=25.8 ft/s, and the time in the

air before impact was timpact =1.182 s. Check: An analytical solution

is found as follows: Combine (1) and (2)

2sin θcos θ

Problem 15.137 In Problem 15.136, if the boy starts

at 1 going 15 ft/s, what distance bdoes he travel through

the air?

248

Problem 15.138 The 1-kg collar Ais attached to the

linear spring (k =500 N/m)by a string. The collar starts

from rest in the position shown, and the initial tension in

the spring is 100 N. What distance does the collar slide

up the smooth bar?

A

k

k=0.2m.

Problem 15.139 The masses mA=40 kg and mB=

60 kg. The collar Aslides on the smooth horizontal bar.

The system is released from rest. Use conservation of

energy to determine the velocity of the collar Awhen it

has moved 0.5 m to the right.

A

B

Problem 15.140 The spring constant is k=850 N/m,

mA=40 kg, and mB=60 kg. The collar Aslides on

the smooth horizontal bar. The system is released from

rest in the position shown with the spring unstretched.

Use conservation of energy to determine the velocity of

the collar Awhen it has moved 0.5 m to the right.

A

k

0.4 m

B

moved 0.5 m. The component of A′svelocity parallel to the cable

equals B′svelocity: vAcos 45◦=vB.B

′sdownward displacement

Problem 15.141 The yaxis is vertical and the curved

bar is smooth. If the magnitude of the velocity of the

4-lb slider is 6 ft/s at position 1, what is the magnitude

of its velocity when it reaches position 2?

y

x

1

2

2 ft

4 ft

Solution: Choose the datum at position 2. At position 2, the

250

Problem 15.142 In Problem 15.141, determine the

magnitude of the velocity of the slider when it reaches

position 2 if it is subjected to the additional force F=

3xi−2j(lb) during its motion.

Solution:

Problem 15.143 Suppose that an object of mass mis

beneath the surface of the earth. In terms of a polar

coordinate system with its origin at the earth’s center, the

gravitational force on the object is −(mgr/RE)er, where

REis the radius of the earth. Show that the potential

energy associated with the gravitational force is V=

mgr 2/2RE.

Solution: By deﬁnition, the potential associated with a force Fis

F·dr.

Problem 15.144 It has been pointed out that if tunnels

could be drilled straight through the earth between points

on the surface, trains could travel between these points

using gravitational force for acceleration and decelera-

tion. (The effects of friction and aerodynamic drag could

be minimized by evacuating the tunnels and using mag-

netically levitated trains.) Suppose that such a train travels

from the North Pole to a point on the equator. Determine

the magnitude of the velocity of the train

(a) when it arrives at the equator and

(b) when it is halfway from the North Pole to the equa-

tor. The radius of the earth is RE=3960 mi.

N

(See Problem 15.143.)

Solution: The potential associated with gravity is

Problem 15.145 In Problem 15.123, what is the max-

imum power transferred to the locomotive during its

acceleration?

252

Problem 15.146 Just before it lifts off, the 10,500-kg

airplane is traveling at 60 m/s. The total horizontal force

exerted by the plane’s engines is 189 kN, and the plane

is accelerating at 15 m/s2.

(a) How much power is being transferred to the plane

by its engines?

(b) What is the total power being transferred to the

plane?

Problem 15.147 The “Paris Gun” used by Germany in

World War I had a range of 120 km, a 37.5-m barrel, a

muzzle velocity of 1550 m/s and ﬁred a 120-kg shell.

(a) If you assume the shell’s acceleration to be

constant, what maximum power was transferred to

the shell as it traveled along the barrel?

(b) What average power was transferred to the shell?

v=F

mt+C.

just before the muzzle exit: P=F(1550)=5.96 ×109joule/s =

5.96 GW. (b) From Eq. (15.18) the average power transfer is

1

2mv2

2−1

2mv2

1