Question

A car wheel rotates with an angular acceleration $\alpha =4t^3\left(\text{in rad}/s^2\right)$ where $t$ is the time taken (in seconds). If the wheel has initial angular velocity ${\omega }_0=20rad/s$, then the angular velocity $\omega \left(t\right)$ after $2s$ is

• A. $20rad/s$
• B. $36rad/s$
• C. $4rad/s$
• D. $16rad/s$

Answer 1

The correct option is B $36rad/s$

Initial angular velocity $w_0=20rad/s$
Given $\alpha =4t^3$
$\because \alpha$ is a function of time
$\frac{d\omega }{dt}=4t^3$
$d\omega =4t^{3}dt$
Integrating on both sides, ${\int }_{{\omega }_0}^{\omega }d\omega ={\int }_0^{t}4t^{3}dt$
$\omega -{\omega }_0={\left[t^4\right]}_0^2$
$\Rightarrow \omega ={\omega }_0+{\left[t^4\right]}_0^2$
$=20+16=36rad/s$