
Concept explainers
The plate shown moves in the xy plane. Knowing that (vA)x=250 mm/s, (vB)y=−450 mm/s and (vc)x=−500 mm/s, determine (a) the angular velocity of the plate, (b) the velocity of point A.

(a)
The angular velocity of the plate.
Answer to Problem 15.46P
The angular velocity of the plate is −5 rad/s.
Explanation of Solution
Given Information:
The figure below shows a plate moving in x-y plane.
Figure-(1)
The velocity along x- direction of the point A is 250 mm/s, the velocity of point B along y -direction is −450 mm/s and the velocity of point C along x -direction is −500 mm/s.
Write the expression for position vector of AB.
rB/A=(Bx−Ax)i+(By−Ay)j ...... (I)
Here, the coordinates of point A is (Ax, Ay) and the coordinates of point B is (Bx, By).
Write the expression for the velocity at point A.
vA=(vA)xi+(vA)yj ...... (II)
Here, the velocity of point A along x -direction is (vA)x, the velocity of point A along y -direction is (vA)y.
Write the expression for the velocity at point B.
vB=(vB)xi+(vB)yj ...... (III)
Here, the velocity of point B along x -direction is (vB)x, the velocity of point B along y -direction is (vB)y.
Write the expression for the velocity at point C.
vC=(vC)xi+(vC)yj ...... (IV)
Here, the velocity of point C along x -direction is (vC)x, the velocity of point C along y -direction is (vC)y.
Write the expression to calculate the velocity of point B.
vB=vA+ωk(rB/A) ..... (V)
Here, the angular velocity of the plate is ω.
Write the expression for velocity at point C.
vC=(vA)x+ωk(rC/A) ...... (VI)
Write the expression for position vector of CA.
rC/A=(Cx−Ax)i+(Cy−Ay)j ...... (VII)
Here, the coordinates of point C is (Cx, Cy).
Calculation:
Substitute 0 mm for Ax, 150 mm for Ay, 150 mm for Bx and 150 mm for By in Equation (I).
rB/A=(150 mm−0 mm)i+(150 mm−150 mm)j=(150 mm)i
Substitute 0 mm for Ax, 150 mm for Ay, 200 mm for Cx and 0 mm for Cy in Equation (VII).
rC/A=(200 mm−0 mm)i+(0 mm−150 mm)j=(200 mm)i−(150 mm)j
Substitute (vC)xi+(vC)yj for (vC), (vA)xi+(vA)yj for vA and (200 mm)i−(150 mm)j for rC/A in Equation (VI).
(vC)xi+(vC)yj=(vA)xi+(vA)yj+ωk((200 mm)i−(150 mm)j) ...... (VIII)
Substitute −500 mm/s for (vC)x and 250 mm/s for (vA)x in Equation (VIII).
(−500 mm/s)i+(vC)yj=[(250 mm/s)i+(vA)yj+ωk((200 mm)i−(150 mm)j)](−500 mm/s)i+(vC)yj=[(250 mm/s)i+(vA)yj+((200ω mm)j−(150ω mm)i)] ...... (IX)
Equate the coefficients of i in Equation (IX).
(−500 mm/s)=(250 mm/s)+(150ω mm)ω=−750 mm/s150 mmω=−5 rad/s
Conclusion:
The angular velocity of the plate is −5 rad/s and the negative sign indicates the clockwise direction of the angular velocity.

(b)
The velocity of point A.
Answer to Problem 15.46P
The velocity of point A is 390.51 mm/s.
Explanation of Solution
Write the expression for velocity of point A.
vA=√(vA)2x+(vA)2y ...... (X)
Calculation:
Substitute (vB)xi+(vB)yj for vB
(vA)xi+(vA)yj for vA and (150 mm)i for rB/A in Equation (V).
(vB)xi+(vB)yj= [(vA)xi+(vA)yj+ωk(150 mm)i](vB)xi+(vB)yj= [(vA)xi+(vA)yj+ω(150 mm)j] ...... (XI)
Substitute −450 mm/s for (vB)y
250 mm/s for (vA)x and −5 rad/s for ω in Equation (XI).
(vB)xi+(−450 mm/s)j=[(250 mm/s)i+(vA)yj+(150(−5 rad/s) mm)j](vB)xi+(−450 mm/s)j=[(250 mm/s)i+(vA)yj+(−750 mm/s)j] ...... (XII)
Equate the coefficients of j in Equation (XII).
(−450 mm/s)=(vA)y−750 mm/s(vA)y=300 mm/s
Substitute 300 mm/s for (vA)y and 250 mm/s for (vA)x in Equation (X).
va=√(250 mm/s)2+(300 mm/s)2=√152500 mm2/s2=390.51 mm/s
Conclusion:
The velocity of point A is 390.51 mm/s.
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Chapter 15 Solutions
Vector Mechanics for Engineers: Dynamics
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