A solid body starts rotating about a stationary axis with an angular acceleration `beta=beta_0cos varphi`, where `beta_0` is a constant vector and `varphi` is an angle of rotation from the initial position. Find the angular velocity of the body as a function of the angle `varphi`. Draw the plot of this dependence.

Explanation

Let us choose the positive direction of z-axis (stationary rotation axis) along the vector `beta_0`. In accordance with the equation
`(domega_z)/(dt)=beta_z` or `omega_z(domega_z)/(dvarphi)=beta_z`
or, `omega_zdomega_z=beta_zdvarphi=betacos varphi d varphi`,
Integrating this Eq. within its limit for `omega_z(varphi)`
or, `underset(0)overset(omega_z)intdomega_z=beta_0 underset0overset(varphi) int cos varphi d varphi`
or, `(omega_z^2)/(2)=beta_0 sin varphi`
Hence `omega_z=+-sqrt(2 beta_0 sin varphi)`
The plot `omega_z(varphi)` is shown in the figure. It can be seen that as the angle `varphi` grows, the vector `vecomega` first increases, coinciding with the direction of the vector `vec(beta_0)(omega_zgt0)`, reaches the maximum at `varphi=varphi//2`, then starts decreasing and finally turns into zero at `varphi=pi`. After that the body starts rotating in the opposite direction in a similar fashion `(omega_zlt0)`. As a result, the body will oscillate about the position `varphi=varphi//2` with an amplitude equal to `pi//2`.