PROBLEM 15.38

PROBLEM 15.38
An automobile travels to the right at a constant speed of
48 mi/h. If the diameter of a wheel is 22 in., determine the
velocities of Points B, C, D, and E on the rim of the wheel.
SOLUTION
vA = 48 mi/h = 70.4 ft/s
d = 22 in. r =
ω=
vC = 0 
d
= 11 in. = 0.91667 ft
2
vA
70.4
=
= 76.8 rad/s
0.91667
r
vB/A = vD/A = vE/A = rω
= (0.91667)(76.8) = 70.4 ft/s
vB = vA + vB/A = [70.4 ft/s
] + [70.4 ft/s
]
vB = 140.8 ft/s
vD = vA + vD/A = [70.4 ft/s
vE = vA + vE/A = [70.4 ft/s

] + [70.4 ft/s

30°]
vD = 136.0 ft/s
15.0° 
vE = 99.6 ft/s
45.0° 
] + [70.4 ft/s ]
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1052
PROBLEM 15.50
Arm AB rotates with an angular velocity of 20 rad/s counterclockwise. Knowing that the outer gear C is stationary, determine
(a) the angular velocity of gear B, (b) the velocity of the gear tooth
located at Point D.
SOLUTION
Arm AB:
Gear B:
(a)
BE = 0.05 m:
v B = vD + vB/E = 0 + ( BE )ωB
2.4 m/s = 0 + (0.05 m)ω B
ωB = 48 rad/s
(b)
DE = (0.05 2):
ω B = 48 rad/s

vD = vE + vD/E = 0 + ( DE )ωB
vD = 0 + (0.05 2)(48)
vD = 3.39 m/s
vD = 3.39 m/s
45° 
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1067
PROBLEM 15.57
A straight rack rests on a gear of radius r and is attached to a
block B as shown. Denoting by ωD the clockwise angular
velocity of gear D and by θ the angle formed by the rack and
the horizontal, derive expressions for the velocity of block B
and the angular velocity of the rack in terms of r, θ , and ωD .
SOLUTION
Gear D:
Rotation about D. Tooth E is in contact with rack AB.
vE = rωD
lEB =
Rack AB:
Plane motion
=
θ
r
tan θ
+
Translation with E
vB = vE + vB/E
[ vB
Rotation about E.
] = [ vE
θ ] + [vB/E θ ]
Draw velocity vector diagram.
vB =
vE
rωD
=
cos θ cos θ
vB =
rωD
cos θ

vB/E = vE tan θ
= rωD tan θ
ω AB =
=
vB/E
lEB
rω D tan θ
r
tan θ
= ω D tan 2 θ
ω AB = ωD tan 2 θ

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1074
PROBLEM 15.76
A 60-mm-radius drum is rigidly attached to a 100-mm-radius drum as
shown. One of the drums rolls without sliding on the surface shown,
and a cord is wound around the other drum. Knowing that end E of
the cord is pulled to the left with a velocity of 120 mm/s, determine
(a) the angular velocity of the drums, (b) the velocity of the center of
the drums, (c) the length of cord wound or unwound per second.
SOLUTION
Since the drum rolls without sliding, its instantaneous center lies at D.
v E = v B = 120 mm/s
v A = v A /Dω ,
(a)
ω=
vB = rB/Dω
vB
120
=
= 3 rad/s
vB/D 100 − 60
ω = 3.00 rad/s 
(b)
v A = (100)(3) = 300 mm/s
v A = 300 mm/s

Since v A is greater than vB , cord is being wound.
v A − vB = 300 − 120 = 180 mm/s
Cord wound per second = 180.0 mm 
(c)
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1104
PROBLEM 15.83
Rod ABD is guided by wheels at A and B that roll in horizontal and
vertical tracks. Knowing that at the instant shown β = 60° and the
velocity of wheel B is 40 in./s downward, determine (a) the angular
velocity of the rod, (b) the velocity of Point D.
SOLUTION
Rod ABD:
We locate the instantaneous center by drawing lines perpendicular to vA and vD.
(a)
Angular velocity.
vB = ( BC )ω
40 in./s = (12.99 in.)ω
ω = 3.079 rad/s
(b)
ω = 3.08 rad/s

Velocity of D:
In ΔCDE:
γ = tan −1
7.5
25.98
= 16.1°; CD =
= 27.04 in.
25.98
cos γ
vD = (CD )ω = (27.04 in.)(3.079 rad/s) = 83.3 in./s
vD = 83.3 in./s
vD = 83.3 in./s
16.1°
73.9° 
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reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited
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1112
PROBLEM 15.88
Rod AB can slide freely along the floor and the inclined
plane. Denoting by vA the velocity of Point A, derive an
expression for (a) the angular velocity of the rod, (b) the
velocity of end B.
SOLUTION
Locate the instantaneous center at intersection of lines drawn perpendicular to vA and vB .
Law of sines.
AC
BC
=
sin[90° − ( β − θ )] sin(90° − θ )
l
=
sin β
AC
BC
=
cos( β − θ ) cos θ
l
=
sin β
cos( β − θ )
AC = l
sin β
cos θ
BC = l
sin β
(a)
Angular velocity:
vA = ( AC )ω = l
cos( β − θ )
ω
sin β
(b)
Velocity of B:
vB = ( BC )ω = l
cos θ  vθ
sin β 
⋅ ⋅

sin β  l cos( β − θ ) 
ω=
vA
sin β
⋅

l cos( β − θ )
vB = vA
cos θ

cos( β − θ )
PROPRIETARY MATERIAL. © 2013 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited
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1118