Question

A wheel is at rest. Its angular velocity increases uniformly and becomes $80$ radians second after $5$ seconds. the total angular displacement is:

Answer 1

Step 1: Concept used

The angular velocity $\left(\mathrm{\omega }\right)$ is assumed to rise evenly.

The angular acceleration $\left(\mathrm{\alpha }\right)$ must always be the same.

Given that it begins at rest and that the total angular velocity is$80$ rad/sec in $5$ seconds,
${\mathrm{\omega }}_{\mathrm{i}}=0$

${\mathrm{\omega }}_{\mathrm{f}}=80\mathrm{rad}/\sec$

$\mathrm{t}=5\sec$

Step 2: Calculation of circular motion

By the equation of circular motion,

${\mathrm{\omega }}_{\mathrm{f}}={\mathrm{\omega }}_{\mathrm{i}}+\mathrm{\alpha t}$

$\mathrm{\alpha }=\frac{80}{5}$

$=16\mathrm{rad}/{\sec }^2$

Step 3: Find the total angular displacement

The expression to find the angular displacement is given as :

$\mathrm{\theta }=12{\mathrm{\alpha t}}^2$

$=12\times 16\times 25$

$=200\mathrm{rad}\mathrm{as}({\mathrm{\omega }}_{\mathrm{o}}=0)$

Therefore, the total angular displacement is $200\mathrm{rad}$.