Question

A string is wrapped around the rim of a wheel of moment of inertia $0.20kg-m^2$ and radius $20\text{cm}$. The wheel is free to rotate about its axis. Initially, the wheel is at rest. The string is now pulled by a force of $20\text{N}$. Find the angular velocity (rad/s) of the wheel after 5 seconds.

Answer 1

Given, $t=5sec,I=0.2Kg-m^2,r=20cm,F=20N$
using kinamatics equation:$\omega ={\omega }_0+\alpha t$ (where, ${\omega }_0=0$)
$\omega =\alpha t$ .....(1)

as we know; $\tau =\overrightarrow{r}\times \overrightarrow{F}\tau =I\times \alpha$
so, $\overrightarrow{r}\times \overrightarrow{F}=I\times \alpha$
here, $sin\alpha =\sin {90}^o=1$
so, $\overrightarrow{r}\times \overrightarrow{F}=I\alpha \Rightarrow \alpha =\frac{rF}{I}=\frac{20\times {10}^{-2}\times 20}{0.2}=20rad/s^2$

from eq. (1).
$\omega =\alpha t$
$\omega =20\times 5\omega =100rad/s$