Question

A solid body rotates with angular velocity `omega=ati+bt^2j`, where `a=0.50rad//s^2`, `b=0.060rad//s^3`, and `i` and `j` are the unit vectors of the x and y axes. Find:
(a) the moduli of the angular velocity and the angular acceleration at the moment `t=10.0s`,
(b) the angle between the vectors of the angular velocity and the angular acceleration at that moment.

Answer 1

We have `vecomega=at veci+bt^2vecj` (1)
So, `omega=sqrt((at)^2+(bt^2)^2)`, thus, `omega|_(t=10s)=7.81 rad//s`
Differentiating Eq. (1) with respect to time
`vecbeta=(dvecomega)/(dt)=aveci+2btvecj` (2)
So, `beta=sqrt(a^2+(2bt)^2)`
and `beta|_(t=10s)=1*3rad//s^2`
(b) `cos alpha=(vecomega*vecbeta)/(omegabeta)=((atveci+bt^2vecj)*(aveci+2btvecj))/(sqrt((at)^2+(bt^2)^2)sqrt(a^2+(2bt)^2))`
Putting the values of (a) and (b) and taking `t=10s`, we get
`alpha=17^@`