Question

A stationary wheel starts rotating about its own axis with an angular acceleration of $5.5rad/{{s}^{2}}$ .To acquire an angular velocity $420$ revolutions per minute, the number of rotations made by the wheel is:
A.$14$
B.$21$
C.$28$
D.$35$

Answer 1

First step we need to do is to convert all the parameters into a single unit system. Next, the number of rotations can be calculated by dividing the total distance travelled by the wheel with twice the pi value. Next, find out the distance travelled. Eventually, the number of rotations can also be calculated.
Formulas used:
$\begin{aligned} & {{\omega }_{1}}-{{\omega }_{2}}=2\alpha s \\\ & n=\dfrac{s}{2\pi } \\\ \end{aligned}$

Complete answer:
Let us assume the initial angular velocity of the wheel as ${{\omega }_{1}}$ , the final angular velocity as ${{\omega }_{2}}$ , the angular acceleration as $\alpha$ and the distance travelled as $s$. Now, the number of rotations can be calculated as $\dfrac{s}{2\pi }$ . Now, the first step to be made is convert the revolutions per minute to radians per second. It is done by multiplying the value of revolutions per minute with $\dfrac{2\pi }{60}$.
So, the value of the angular velocity will turn out to be
$\begin{aligned} & 420\times \dfrac{2\pi }{60} \\\ & 14\pi \\\ \end{aligned}$
Therefore, we get,
$\begin{aligned} & {{\omega }_{1}}=0,{{\omega }_{2}}=14\pi rad/s,\alpha =5.5rad/{{s}^{2}} \\\ & {{\omega }_{2}}^{2}-{{\omega }_{1}}^{2}=2\alpha s \\\ & {{(14\pi )}^{2}}-{{(0)}^{2}}=2\times 5.5\times s \\\ & s=\dfrac{98{{\pi }^{2}}}{5.5} \\\ \end{aligned}$
Now, number of rotations by the wheel can be calculated as
$\dfrac{s}{2\pi }=\dfrac{98{{\pi }^{2}}}{5.5\times 2\pi }=28$

Therefore, the number of rotations made by the wheel to acquire the given angular velocity is $28$, option c.

Additional information:
Angular acceleration is the rate of change of angular velocity with a time of an object in motion. Whenever the object moves in a circular direction, then the acceleration can be called angular acceleration. It is also known as rotational acceleration. It is a vector quantity and all the magnitude or length is directly proportional to the rate of change of the angular velocity.Angular velocity is the rate of velocity at which the object or particle is moving around a centre or specific point in a given period of time. Similar to angular acceleration, angular velocity can also be called rotational velocity.

Note:
The given values in the question are in different units. Don’t forget to convert them into a single unit system. Also, the number of the rotations must be calculated using the above formula, the distance travelled will not be equal to the number of rotations. The question said it’s a stationary wheel, so the initial angular velocity is taken as zero.