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Relative Velocity and Acceleration with Translating and Rotating Frames (continued)

Example: (continued)

v_P|_F = v_Q|_F + v_P|_B+ ω_B x r_QP here QP = r = √ ( 3² + 4² ) = 5

or v_R i = v_Q i + v_P|_B i₁+ ω_B k x r i₁ = v_Q i + v_P|_B i₁+ r ω_B j₁

Next apply the coordinate transformation i₁ = i cos θ + j sin θ and j₁ = - i sin θ + j cos θ

v_R i = v_Q i + v_P|_B ( i cos θ + j sin θ ) + r ω_B ( - i sin θ + j cos θ ) and collect terms

v_R i = i [ v_Q + v_P|_B cos θ - r ω_B sin θ ] + j [v_P|_B sin θ + r ω_B cos θ ]

Next equate i and j components. Note: sin θ = 3/5 and cos θ = 4/5 and r = 5

v_R = v_Q + v_P|_B cos θ - r ω_B sin θ or v_R = v_Q + (4/5) v_P|_B - (3/5) r ω_B (1)

0 = v_P|_B sin θ + r ω_B cos θ or 0 = (3/5) v_P|_B + (4/5) r ω_B (2)

Equations (1) and (2) contain unknowns v_P|_B and ω_B . Solve (2) for ω_B in terms of v_P|_B and

put into (2). The result is

v_P|_B = ˗ 20 so v_P|_B = ˗ 20 i₁ m/sec and i₁ = (4/5) i + (3/5) j

v_P|_B = ˗ 16 i ˗ 12 j m/sec (final result)

and then put into (2) to calculate ω_B

ω_B = 3 So ω_B = 3 k rad/sec (final result)