Relative Velocity and Acceleration with Translating and Rotating Frames

Example: (continued)

Strategy: Apply the general relative acceleration equation for a translating/rotating frame of reference.

a_B = a_A + a_B|₁ + α x r_AB - ω² r_AB + 2 ω x v_B|₁

Relative Acceleration Equation

For the given data: a_A = 0, a_B|₁ = a_o i₁, ω = ω_o k, r_AB = r_o i₁, α = α_o k , v_B|₁ = v_o i₁

So a_B |_F = a_o i₁ + α_o k x r_o i₁ - ω_o² r_o i₁ + 2 ω_o k x v_o i₁ or

a_B |_F = i₁ [ a_o - ω_o² r_o ] + j₁ [ r_o α_o + 2 ω_o v_o ]

The final step is to express this result in terms of x and y components

by transforming from i₁ j₁ to i j . Click here, if needed, to review transformations.

In this example: i₁ = i cos θ + j sin θ and j₁ = - i sin θ + j cos θ

a_B |_F = [ a_o - ω_o² r_o ] [i cos θ + j sin θ ] + [ r_o α_o + 2 ω_o v_o ] [- i sin θ + j cos θ ] collect terms

a_B |_F = { [ a_o - ω_o² r_o ] cos θ - [ r_o α_o + 2 ω_o v_o ] sin θ } i +

{ [ a_o - ω_o² r_o ] sin θ + [ r_o α_o + 2 ω_o v_o ] cos θ } j and putting in the values

a_B |_F = [ -5 √ 3 – 16 ] i + [ - 5 + 16 √ 3 ] j ft/sec²

Click here for another example.

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Copyright © 2019 Richard C. Coddington
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