Question

A wheel starts rotating from rest at time t=0 with a angular acceleration of 50 radians/s2. The angular acceleration ($\alpha$) decreases to zero value after 5 seconds. During this interval, $\alpha$varies according to the question:
$\alpha = {\alpha _0}\left( {1 - \dfrac{t}{5}} \right)$ .The angular velocity at T=5s will be:
a. 10rad/s
b. 250rad/s
c. 125rad/s
d. 100rad/s

Answer 1

Be it angular acceleration or linear both are defined as :
Rate of change of velocity is called acceleration. In the question given above angular acceleration is mentioned which is mathematically defined as:
$\alpha = \dfrac{{d\omega }}{{dt}}$ ($\alpha$is the angular acceleration, t is the time and $\omega$ is the angular velocity). Using the above relation we will calculate the angular velocity.

Complete step by step answer:
We have;
$\alpha = {\alpha _0}\left( {1 - \dfrac{t}{5}} \right)$ ----(1)
According to equation 1 at t=0 $\alpha$= ${\alpha _0}$, so ${\alpha _0}$=50rad/s.
As per the definition change in angular velocity will give acceleration, therefore change in angular velocity is equal to equation 1 (as it is angular acceleration).
$\Rightarrow \dfrac{{d\omega }}{{dt}} = {\alpha _0}\left( {1 - \dfrac{t}{5}} \right)$
On integrating both the sides,
$\Rightarrow \int {d\omega = \int {{\alpha _0}} } \left( {1 - \dfrac{t}{5}} \right)dt \\\ \Rightarrow \omega = {\alpha _0}\left( {t - \dfrac{{{t^2}}}{{10}}} \right) \\\$ (done the simple integration with respect to t)
Now substituting the values of all the terms;
$\Rightarrow \omega = 50\left( {5 - \dfrac{{25}}{{10}}} \right) \\\ \Rightarrow \omega = 125rad/s \\\$ (50 multiplies inside the bracket)

Hence, the correct answer is option (C).

Note: Angular velocity: angular velocity is defined as how fast an object rotates or revolves relative to another point that is how fast the angular position of an object changes with time.
Angular acceleration: Time rate of change of angular velocity is called angular acceleration.

We have seen numerous daily life examples of angular acceleration like when we open any door or window in our house we observe angular acceleration as the door is bolted on hinges and makes angular motion on opening or closing, similarly doors of the car, our wall clocks, wheel of the bicycle, scooter etc.