Answer:
<em>Angular displacement=68.25 rad</em>
Explanation:
<u>Circular Motion</u>
If the angular speed varies from ωo to ωf in a time t, then the angular acceleration is given by:
![\displaystyle \alpha=\frac{\omega_f-\omega_o}{t}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Calpha%3D%5Cfrac%7B%5Comega_f-%5Comega_o%7D%7Bt%7D)
The angular displacement is given by:
![\displaystyle \theta=\omega_o.t+\frac{\alpha.t^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D%5Comega_o.t%2B%5Cfrac%7B%5Calpha.t%5E2%7D%7B2%7D)
The wheel decelerates from ωo=13.5 rad/s to ωf=6 rad/s in t=7 s, thus:
![\displaystyle \alpha=\frac{6-13.5}{7}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Calpha%3D%5Cfrac%7B6-13.5%7D%7B7%7D)
![\displaystyle \alpha=\frac{-7.5}{7}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Calpha%3D%5Cfrac%7B-7.5%7D%7B7%7D)
![\displaystyle \alpha=-1.071 \ rad/s^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Calpha%3D-1.071%20%5C%20rad%2Fs%5E2)
Thus, the angular displacement is:
![\displaystyle \theta=13.5*7+\frac{-1.071*7^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D13.5%2A7%2B%5Cfrac%7B-1.071%2A7%5E2%7D%7B2%7D)
![\displaystyle \theta=94.5-26.25](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D94.5-26.25)
![\boxed{\displaystyle \theta=68.25\ rad}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdisplaystyle%20%5Ctheta%3D68.25%5C%20rad%7D)
Angular displacement=68.25 rad