PROBLEM 15.44
The plate shown moves in the xy plane. Knowing that
(v A ) x = 120 mm/s, (vB ) y = 300 mm/s, and (vC ) y = −60 mm/s,
determine (a) the angular velocity of the plate, (b) the velocity
of Point A.
SOLUTION
rC/B = (180 mm)i + (360 mm) j
ω =ωk
vB = (vB ) x i + (300 mm/s) j
vC = (vC ) x i − (60 mm/s) j
vC = vB + vC/B
(a)
(vC ) x i − (60 mm/s) j = (vB ) x i + (300 mm/s) j + ω × rC/B
(vC ) x i − 60 j = (vB ) x i + 300 j + ω k × (180i + 360 j)
(vC ) x i − 60 j = (vB ) x i + 300 j + 180ω j − 360ω i
Coefficients of j:
−60 = 300 + 180ω
ω = −2 rad/s
(b)
Velocity of A:
ω = 2 rad/s

rA/B = −(180 mm)i + (180 mm) j
vA = vB + vA/B = vB + ω × rA/B
120i + (v A ) y j = (vB ) x i + 300 j + (−2k ) × (−180i + 180 j)
120i + (v A ) y j = (vB ) x i + 300 j + 360 j + 360i
Coefficients of j:
(v A ) y = 300 + 360 = 660 mm/s
vA = (120 mm/s)i + (660 mm/s) j 
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PROBLEM 15.45
In Problem 15.44, determine (a) the velocity of Point B,
(b) the point of the plate with zero velocity.
PROBLEM 15.44 The plate shown moves in the xy
plane. Knowing that (v A ) x = 120 mm/s, (vB ) y = 300 mm/s,
and (vC ) y = −60 mm/s, determine (a) the angular velocity
of the plate, (b) the velocity of Point A.
SOLUTION
rB/A = (180 mm)i − (180 mm) j
From the answer of Problem 15.44, we have
ω = −(2 rad/s)k
vA = (120 mm/s)i + (660 mm/s) j
(a)
Velocity of B:
vB = vA + vB/A = vA + ω × rB/A
= 120i + 660 j − 2k × (180i − 180 j)
= 120i + 660 j − 360 j − 360i
vB = −(240 mm/s)i + (300 mm/s) j 
(b)
Point with v = 0:
Let P = xi + yj be an arbitrary point.
Thus
rP/A = (180 + x)i + yj
vP = vA + v P/A = vA + ω × rP/A
vP = 120i + 660 j + (−2k ) × [(180 + x)i + yj]
vP = 120i + 660 j − (360 + 2 x) j + 2 yi
vP = (120 + 2 y )i + (300 − 2 x) j
For v P = 0:
120 + 2 y = 0 and 300 − 2 x = 0
v = 0 at:
y = −60 mm, x = 150 mm 
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
distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual,
you are using it without permission.
1060
PROBLEM 15.46
The plate shown moves in the xy plane. Knowing that (v A ) x = 250 mm/s,
(vB ) x = −450 mm/s, and (vC ) x = −500 mm/s, determine (a) the angular
velocity of the plate, (b) the velocity of Point A.
SOLUTION
ω =ωk
Angular velocity:
rB/A = (150 mm)i
Relative position vectors:
rC/A = (200 mm)i + (150 mm) j
v A = (250 mm/s)i + (v A ) y j
Velocity vectors:
v B = (vB ) x i − (450 mm/s) j
vC = −(500 mm/s)i + (vC ) y j
Unknowns are ω, (v A ) y , (vB ) x , and (vC ) y .
v B = v A + v B/A = v A + ω k × rB/A
(vB ) x i − 450 j = 250i + (v A ) y j + ω k × 150i
= 250i + (v A ) y j + 150ω j
i:
(vB ) x = 250
(1)
j:
−450 = (v A ) y + 150ω
(2)
vC = v A + v C/A = v A + ω k × rC/A
−500i + (vC ) y j = 250i + (v A ) y j + ω k × (200i − 150 j)
= 250i + (v A ) y j + 200ω j + 150ω i
(a)
i:
−500 = 250 + 150ω
(3)
j:
(vC ) y = (v A ) y + 150ω
(4)
Angular velocity of the plate.
From Eq. (3),
ω=−
750
= −5
150
ω = −(5.00 rad/s)k = 5.00 rad/s
(b)

Velocity of Point A.
From Eq. (2), (v A ) y = −450 − 150ω = −450 − (150)(−5) = 300 mm/s
v A = (250 mm/s)i + (300 mm/s) j 
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
distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual,
you are using it without permission.
1061