A rotating wheel changes angular speed from 1800 rpm to 3000 rpm in 20 s. What is the angular acceleration assuming to be uniform? [KCET 2014]

We know that,   $\omega =2\pi n$

Given,  $n_1=1800\mathrm{rpm},n_2=3000\mathrm{rpm},\Delta t=20 \mathrm{s}$

            ${\omega }_1=2\pi n_1=2\pi \times \frac{1800}{60}=2\pi \times 30=60\pi \mathrm{rad}/\mathrm{s}$

Similarly,    ${\omega }_2=2\pi n_2=2\pi \times \frac{3000}{60}=2\pi \times 50=100\pi \mathrm{rad}/\mathrm{s}$

If the angular velocity of a rotating wheel about an axis is changed by change in angular velocity in a time interval $\Delta$t,  then the angular acceleration of rotating wheel about that axis is given by

$\alpha =\frac{\text{ Change in angular velbcity }}{\text{ Time interval }}$

$=\frac{{\omega }_2-{\omega }_1}{\Delta t}=\frac{100\pi -60\pi }{20}$

$=\frac{40\pi }{20}=2\pi {\mathrm{rads}}^{-2}$