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Question

A solid body starts rotating about a stationary axis with an angular acceleration β=β0cosφ, where β0 is a constant vector and φ is an angle of rotation from the initial position. If the angular velocity of the body as a function of the angle φ is ωz=±xβ0sinφ. Find x.

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Solution

Let us choose the positive direction of z-axis (stationary rotation axis) along the vector β0. In accordance with the equation
dωzdt=βz or ωzdωzdφ=βz
or, ωzdωz=βzdφ=βcosφdφ,
Integrating this equation within its limit for ωz(φ)
or, ωz0dωz=β0φ0cosφdφ
or, ω2z2=β0sinφ
Hence ωz=±2β0sinφ
The plot ωz(φ) is shown in figure below. It can be seen that as the angle φ grows, the vector ω first increases, coinciding with the direction of the vector β0(ωz>0), reaches the maximum at φ=φ2, then starts decreasing and finally turns into zero at φ=π. After that the body starts rotating in the opposite direction in a similar fashion (ωz<0). As a result, the body will oscillate about the position φ=φ2 with an amplitude equal to π2.
220897_129705_ans.png

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