Mechanical Engineering Chapter 13 Homework A girl having a mass of 25 kg sits at the edge of

305

13–61.

At the instant the boy’s center of mass Gis

momentarily at rest. Determine his speed and the tension in

each of the two supporting cords of the swing when

The boy has a weight of 60 lb. Neglect his size and

the mass of the seat and cords.

u=90°.

u=60°,

SOLUTION

10 ft

G

u

Ans:

306

13–62.

A girl having a mass of 25 kg sits at the edge of the merry-

go-round so her center of mass Gis at a distance of 1.5 m

from the axis of rotation. If the angular motion of the

platform is slowly increased so that the girl’s tangential

component of acceleration can be neglected, determine the

maximum speed which she can have before she begins to

slip off the merry-go-round.The coefficient of static friction

between the girl and the merry-go-round is ms=0.3.

SOLUTION

z

1.5 m

G

Ans:

307

13–63.

The pendulum bob has a weight of 5 lb and is released from

rest in the position shown, .Determine the tension in

string just after the bob is released,,and also at the

instant the bob reaches point ,.Take .r=3ftu=45°D

u=0°BC

u=0°

B

SOLUTION

Equation of Motion: Since the bob is just being release,.Applying Eq. 13–8 to

FBD(a), we have

y=0

C

r

B

Ans:

308

*13–64.

SOLUTION

Equation of Motion: Since the bob is just being release,.Applying Eq. 13–8 to

FBD(a), we have

y=0

The pendulum bob Bhas a mass mand is released from rest

when .Determine the tension in string BC immediately

afterwards, and also at the instant the bob reaches the arbitrary

position .u

u=0°

C

r

B

Ans:

309

13–65.

SOLUTION

Determine the constant speed of the passengers on the

amusement-park ride if it is observed that the supporting

cables are directed at from the vertical. Each chair

including its passenger has a mass of 80 kg. Also, what are

the components of force in the n, t, and bdirections which

the chair exerts on a 50-kg passenger during the motion?

u=30°

6 m

4 m

n

b

t

u

Ans:

SOLUTION

13–66.

A motorcyclist in a circus rides his motorcycle within the

confines of the hollow sphere. If the coefficient of static

friction between the wheels of the motorcycle and the

sphere is m

s=0.4,

determine the minimum speed at which

he must travel if he is to ride along the wall when

u=90°.

The mass of the motorcycle and rider is 250 kg, and the

radius of curvature to the center of gravity is r

=20 ft.

Neglect the size of the motorcycle for the calculation. u

311

13–67.

The vehicle is designed to combine the feel of a motorcycle

with the comfort and safety of an automobile. If the vehicle

is traveling at a constant speed of 80 km h along a circular

curved road of radius 100 m, determine the tilt angle of

the vehicle so that only a normal force from the seat acts on

the driver. Neglect the size of the driver.

u

>u

SOLUTION

Free-Body Diagram: The free-body diagram of the passenger is shown in Fig. (a).

Ans:

312

*13–68.

The 0.8-Mg car travels over the hill having the shape of a

parabola. If the driver maintains a constant speed of 9 ms,

determine both the resultant normal force and the

resultant frictional force that all the wheels of the car exert

on the road at the instant it reaches point A.Neglect the

size of the car.

>y

Ax

y20 (1 )

x2

6400

SOLUTION

Ais given by

and the radius of curvature at point Ais

Ans:

313

13–69.

The 0.8-Mg car travels over the hill having the shape of a

parabola. When the car is at point A,it is traveling at 9 ms

and increasing its speed at .Determine both the

resultant normal force and the resultant frictional force that

all the wheels of the car exert on the road at this instant.

Neglect the size of the car.

3m>s2>

SOLUTION

y

Ax

y20 (1 )

x2

6400

Ans:

314

13–70.

SOLUTION

The package has a weight of 5 lb and slides down the chute.

When it reaches the curved portion AB, it is traveling at

If the chute is smooth, determine the speed

of the package when it reaches the intermediate point

and when it reaches the horizontal plane

Also, find the normal force on the package at C.1u=45°2.

C1u=30°2

1u=0°2.8ft>s

20 ft

45°

=30°

θ

B

A

8 ft/s

C

Ans:

315

Ans:

13–71.

SOLUTION

The150-lb man lies against the cushion for which the

coefficient of static friction is Determine the

resultant normal and frictional forces the cushion exerts on

him if,due to rotation about the zaxis,hehasaconstant

speed Neglect the size of the man.Take u=60°.v=20 ft>s.

ms=0.5. z

G

8 ft

316

*13–72.

The 150-lb man lies against the cushion for which the

coefficient of static friction is If he rotates about

the zaxis with a constant speed determine the

smallest angle of the cushion at which he will begin to

slip off.

u

v=30 ft>s,

ms=0.5.

SOLUTION

+©Fn=man;

z

G

8 ft

u

Ans:

317

13–73.

Determine the maximum speed at which the car with mass

mcan pass over the top point Aof the vertical curved road

and still maintain contact with the road. If the car maintains

this speed, what is the normal reaction the road exerts on

the car when it passes the lowest point Bon the road? rr

rr

A

B

SOLUTION

Free-Body Diagram: The free-body diagram of the car at the top and bottom of the

Equations of Motion: When the car is on top of the vertical curved road, it is

required that its tires are about to lose contact with the road surface.Thus,.

N=0

Ans:

318

Ans:

SOLUTION

Geometry. The radius of curvature of the road at A must be determined first. Here

13–74.

Determine the maximum constant speed at which the 2-Mg

car can travel over the crest of the hill at A without leaving

the surface of the road. Neglect the size of the car in the

calculation.

y

A

x

y  20 (1  )

100 m

x2

10 000

319

13–75.

The box has a mass mand slides down the smooth chute

having the shape of a parabola. If it has an initial velocity of

at the origin, determine its velocity as a function of x.

Also, what is the normal force on the box, and the

tangential acceleration as a function of x?

v0

SOLUTION

x=-

1

2x2

y

x

y=– 0.5 x

2

Ans:

320

*13–76.

Prove that if the block is released from rest at point Bof a

smooth path of arbitrary shape, the speed it attains when it

reaches point Ais equal to the speed it attains when it falls

freely through a distance h; i.e., v=22gh.

SOLUTION

A

B

h

Ans:

SOLUTION

12–77.

The cylindrical plug has a weight of 2 lb and it is free to

move within the confines of the smooth pipe. The spring

has a stiffness k

=

14 lb

>

ft and when no motion occurs

the distance d

=

0.5 ft. Determine the force of the spring

on the plug when the plug is at rest with respect to the pipe.

The plug is traveling with a constant speed of 15 ft

>

s, which

is caused by the rotation of the pipe about the vertical axis.

3 ft

d

G

13–78.

When crossing an intersection, a motorcyclist encounters

the slight bump or crown caused by the intersecting road.

If the crest of the bump has a radius of curvature r

=50 ft,

determine the maximum constant speed at which he can

travel without leaving the surface of the road. Neglect the

size of the motorcycle and rider in the calculation. The rider

and his motorcycle have a total weight of 450 lb.

Ans:

SOLUTION

323

13–79.

The airplane, traveling at a constant speed of is

executing a horizontal turn. If the plane is banked at

when the pilot experiences only a normal force on

the seat of the plane, determine the radius of curvature of

the turn. Also, what is the normal force of the seat on the

pilot if he has a mass of 70 kg.

r

u=15°,

50 m>s,

SOLUTION

u

r

Ans:

324

*13–80.

The 2-kg pendulum bob moves in the vertical plane with

a velocity of 8 m

>

s when

u=0°.

Determine the initial

tension in the cord and also at the instant the bob reaches

u=30°.

Neglect the size of the bob.

2 m



SOLUTION

Equations of Motion. Referring to the FBD of the bob at position

u=0°

, Fig. a,

T=v2–19.62 sin u

(1)

Kinematics. The velocity of the bob at the position

u=30°

can be determined by

integrating

vdv=at

ds.

However,

ds =rdu=2du.

u=30°

Ans: