A solid body rotates about a stationary axis so that the rotation angle `theta` varies with time as `theta=6t-2t^(3)` radian. Find
(a) the angular acceleration at the moment when the body stops and
(b) the average value of angular velocity and angular acceleration averaged over the time interval between `t=0` and the complete stop.
(a) the angular acceleration at the moment when the body stops and
(b) the average value of angular velocity and angular acceleration averaged over the time interval between `t=0` and the complete stop.
`theta=6t-2t^(3)`
`omega=(d theta)/(dt)=6-6t^(2)`
`alpha=(domega)/(dt)=-12t`
(a) When the body stops,`omega=0=6-6t^(2)impliest=1 s`
`alpha=-12t=-12 rad//s^(2)`
`t=0, theta_(1)=0`
`t=1. theta_(2)=6-2(1)^(3)=4`
(b) `undersetomega(-)=(theta_(2)-theta_(1))/(t_(2)-t_(1))`
`=(4-0)/(1-0)=4 rad//s`
`t=0, omega_(1)=6`
`t=1 s, omega_(2)=0`
`undersetalpha(-)=(omega_(2)-omega_(1))/(t_(2)-t_(1))`
`(6-0)/(1-0)=6 rad//s^(2)`