Mechanical Engineering Chapter 19 Homework The plane has amass of 200 Mg, its center of mass

985

19–1.

SOLUTION

The rigid body (slab) has a mass mand rotates with an

angular velocity about an axis passing through the fixed

point O.Show that the momenta of all the particles

composing the body can be represented by a single vector

having a magnitude and acting through point P,called

the center of percussion,which lies at a distance

from the mass center G.Here is the

radius of gyration of the body,computed about an axis

perpendicular to the plane of motion and passing through G.

kG

rP>G=k2

G>rG>O

mvG

V

mv

G

vG

G

V

P

rP/G

rG/O

O

986

19–2.

At a given instant, the body has a linear momentum

and an angular momentum computed

about its mass center. Show that the angular momentum of

the body computed about the instantaneous center of zero

velocity IC can be expressed as , where

represents the body’s moment of inertia computed about

the instantaneous axis of zero velocity. As shown, the IC is

located at a distance away from the mass center G.rG>IC

IIC

HIC =IICV

HG=IGVL=mvG

SOLUTION

GIGV

rG/IC

IC

mvG

987

19–3.

SOLUTION

Since , the linear momentum . Hence the angular momentum

L=myG=0yG=0

Show that if a slab is rotating about a fixed axis perpendicular

to the slab and passing through its mass center G,the angular

momentum is the same when computed about any other

point P.

P

G

V

988

SOLUTION

Equilibrium. Since slipping occurs at brake pad,

F

f

=

mk

N=0.3 N

.

F

Principle of Impulse and Momentum. The mass moment of inertia of the disk about

1

It is required that v

2=0

. Assuming that

t72 s

,

Since

t72 s

, the assumption was correct.

*19–4.

The 40-kg disk is rotating at V

=100 rad>s.

When the force

P is applied to the brake as indicated by the graph. If the

coefficient of kinetic friction at B is m

k=0.3,

determine

thetime t needed to stay the disk from rotating. Neglect the

thickness of the brake.

150 mm

O

300 mm

200 mm

P

B

P (N)

500

2t

(s)

V

989

19–5.

The impact wrench consists of a slender 1-kg rod AB which

is 580 mm long, and cylindrical end weights at Aand Bthat

each have a diameter of 20 mm and a mass of 1 kg.This

which are attached to the lug nut on the wheel of a car.If

the rod AB is given an angular velocity of 4 and it

strikes the bracket Con the handle without rebounding,

determine the angular impulse imparted to the lug nut.

rad>s

SOLUTION

A

B

300 mm

C

assembly is free to turn about the handle and socket,

Ans:

19–6.

8 m

A

G

B

T

A



40 kN

TB  20 kN

The airplane is traveling in a straight line with a speed of

300 kmh, when the engines Aand Bproduce a thrust of

and , respectively. Determine the

angular velocity of the airplane in .The plane has a

mass of 200 Mg, its center of mass is located at G,and its

radius of gyration about Gis .kG=15 m

t=5 s

T

B=20 kNT

A=40 kN

>

SOLUTION

Ans:

991

19–7.

The double pulley consists of two wheels which are attached

to one another and turn at the same rate.The pulley has a

mass of 15 kg and a radius of gyration of If

the block at Ahas a mass of 40 kg, determine the speed

of the block in 3 s after a constant force of 2 kN is applied

to the rope wrapped around the inner hub of the pulley.The

block is originally at rest.

kO=110 mm.

200 mm

75 mm

O

2kN

SOLUTION

Ans:

*19–8.

1 ft

0.8 ft

G

The assembly weighs 10 lb and has a radius of gyration

about its center of mass G.The kinetic energy

of the assembly is when it is in the position shown.

If it is rolling counterclockwise on the surface without

slipping,determine its linear momentum at this instant.

31 ft#lb

kG=0.6 ft

SOLUTION

Substitute into Eq.(1),

IG=(0.6)2a10

32.2b=0.1118 slug #ft2

Ans:

993

19–9.

The disk has a weight of 10 lb and is pinned at its center O.

If a vertical force of is applied to the cord wrapped

around its outer rim, determine the angular velocity of the

disk in four seconds starting from rest. Neglect the mass of

the cord.

P=2lb

SOLUTION

c

+)IOv1+©

MOdt =IOv2

(

0.5 ft

O

P

Ans:

994

19–10.

P  200 N

BA

O

0.15 m

The 30-kg gear Ahas a radius of gyration about its center of

mass Oof .If the 20-kg gear rack Bis

subjected to a force of , determine the time

required for the gear to obtain an angular velocity of

,starting from rest.The contact surface between the

gear rack and the horizontal plane is smooth.

20 rad>s

P=200 N

kO=125 mm

SOLUTION

Kinematics: Since the gear rotates about the fixed axis,the final velocity of the gear

rack is required to be

Principle of Impulse and Momentum: Applying the linear impulse and momentum

equation along the xaxis using the free-body diagram of the gear rack shown in Fig.a,

The mass moment of inertia of the gear about its mass center is

.Writing the angular impulse and momentum

equation about point Ousing the free-body diagram of the gear shown in Fig. b,

mkO

2=30(0.1252)=0.46875 kg#m2

IO =

Ans:

19–11.

SOLUTION

Principle of Impulse and Momentum: The mass moment inertia of the pulley about its

The pulley has a weight of 8 lb and may be treated as a thin

disk. A cord wrapped over its surface is subjected to forces

and Determine the angular velocity

of the pulley when if it starts from rest when

Neglect the mass of the cord.

t=0.t=4s

TB=5 lb.TA=4lb

0.6ft

Ans:

996

*19–12.

The 40-kg roll of paper rests along the wall where the

coefficient of kinetic friction is m

k=0.2.

If a vertical force

of

P=40 N

is applied to the paper, determine the angular

velocity of the roll when

t=6 s

starting from rest. Neglect

the mass of the unraveled paper and take the radius of

gyration of the spool about the axle O to be

kO=80 mm.

12

5

13

O

B

120 mm

P  40 N

A

SOLUTION

Principle of Impulse and Momentum. The mass moment of inertia of the paper roll

(

S

+

)

m[(vO)x]1+Σ

L

t

2

t

Fx dt =m[(vO)x]2

(+

c

)

m[(vO)y]1+ΣL

t

2

t

1

Fy dt =m[(vO)y]2

Solving Eqs. (1) and (2)

Subsequently

Ans:

997

19–13.

The slender rod has a mass mand is suspended at its end A

byacord. If the rod receives a horizontal blow giving it an

impulse Iat its bottom B, determine the location yof the

point Pabout which the rod appears to rotate during the

impact.

SOLUTION

Principle of Impulse and Momentum:

Kinematics: Point Pis the IC.

Using similar triangles,

A

P

l

y

Ans:

19–14.

The rod of length L and mass m lies on a smooth horizontal

surface and is subjected to a force P at its end A as shown.

Determine the location d of the point about which the rod

begins to turn, i.e, the point that has zero velocity.

A

P

L

d

SOLUTION

(

S

+

)

m(vGx)1+Σ

L

Fx dt =m(vGx)2

999

19–15.

A 4-

k

g

di

s

k

A

i

s mounte

d

on arm BC,w

hi

c

h

as a neg

li

g

ibl

e

mass. If a torque of where tis in seconds,

is applied to the arm at C, determine the angular velocity of

BC in 2 s starting from rest. Solve the problem assuming

that (a) the disk is set in a smooth bearing at Bso that it

rotates with curvilinear translation, (b) the disk is fixed to

the shaft BC, and (c) the disk is given an initial freely

spinning angular velocity of prior to

application of the torque.

VD=5–80k6 rad>s

M=15e0.5t2 N #m,

SOLUTION

a)

Thus,

b)

c)

250 mm

M (5e

0.5 t

) N m

60 mm

z

C

AB

#

*19–16.

1.5 ft

AB

M 300 lb ft

The frame of a tandem drum roller has a weight of 4000 lb

excluding the two rollers. Each roller has a weight of 1500 lb

and a radius of gyration about its axle of 1.25 ft.If a torque

of is supplied to the rear roller A,determine

the speed of the drum roller 10 s later, starting from rest.

M=300 lb #ft

SOLUTION

and

Referring to the free-body diagram of the frame shown in Fig. d,

Substituting Eqs. (1) and (2) into Eq.(3),

Ans:

19–17.

SOLUTION

Principle of Impulse and Momentum: We can eliminate the force Ffrom the

analysis if we apply the principle of impulse and momentum about point A.The

Kinematics: Since the wheel rolls without slipping at point A, the instantaneous

center of zero velocity is located at point A.Thus,

The 100-lb wheel has a radius of gyration of

If the upper wire is subjected to a tension of

determine the velocity of the center of the wheel in 3 s,

starting from rest.The coefficient of kinetic friction between

the wheel and the surface is mk=0.1.

T=50 lb,

kG

=0.75 ft.

1ft

T

G

0.5ft

Ans:

1002

19–18.

SOLUTION

(+c)

m(vGy)1+©

LF

y dt =m(vGy)2

1;

+2

m(vGx)1+©

x dt =m(vGx)2

The 4-kg slender rod rests on a smooth floor. If it is kicked

so as to receive a horizontal impulse at point A

as shown, determine its angular velocity and the speed of its

mass center.

I=8 N #s

2 m

1.75 m

60

A

Ans:

19–19.

The double pulley consists of two wheels which are

attached to one another and turn at the same rate.The

pulley has a mass of 15 kg and a radius of gyration

If the block at Ahas a mass of 40 kg,

determine the speed of the block in 3 s after a constant

force is applied to the rope wrapped around the

inner hub of the pulley.The block is originally at rest.

Neglect the mass of the rope.

F=2 kN

kO=110 mm.

SOLUTION

c

+)

(H

O)1+©

LMO dt =(H

O)2

(

O

200 mm

75 mm

F

A

Ans:

*19–20.

The 100-kg spool is resting on the inclined surface for which

the coefficient of kinetic friction is m

k=0.1.

Determine the

angular velocity of the spool when

t=4 s

after it is released

from rest. The radius of gyration about the mass center is

kG=0.25 m.

G

A

0.2 m

0.4 m

SOLUTION

Kinematics. The IC of the spool is located as shown in Fig. a. Thus

Principle of Impulse and Momentum. The mass moment of inertia of the spool

Q

+

m[(vG)x]1+Σ

t

2

Fx dt =m[(vG)x]2

Solving Eqs. (1) and (2),

Ans: