Question

A flywheel rotates about a fixed axis and slows down from $400$ r.p.m to $300$ r.p.m in two minutes. Calculate
(i) Angular acceleration
(ii) How many rotations does the wheel complete in two minutes?

Answer 1

We have established and are aware of the three equations of motion. Although those equations of motion are made for linear motion, with some little tweaks and adjustments, we can apply those equations for rotational motion as well. The velocity in linear motion is replaceable by the angular velocity and similarly, the angular acceleration takes the place of linear acceleration. Let’s see the detailed solution for a better understanding.
Formula Used: ${{\omega }_{2}}-{{\omega }_{1}}=\alpha t$, $\theta ={{\omega }_{1}}t+\dfrac{1}{2}\alpha {{t}^{2}}$

Complete step by step solution:
We have been provided with the initial and the final angular velocities of the flywheel and the time for which it rotates, and we have to calculate the angular acceleration of the flywheel and also the number of rotations it makes in the given time.
The initial angular velocity of the flywheel $({{\omega }_{1}})=400$ r.p.m where r.p.m means rotations per minute
The final angular velocity of the flywheel $({{\omega }_{2}})=300$ r.p.m
Applying the first equation of motion for circular motion, similar to the first equation for linear motion, we have ${{\omega }_{2}}-{{\omega }_{1}}=\alpha t$ where $\alpha$ denotes the angular acceleration of the flywheel
Substituting the values of angular acceleration and time $t=2\min$ , we get

& 300-400=\alpha \times 2 \\\ & \Rightarrow \alpha =-\dfrac{100}{2}=-50 \\\ \end{aligned}$$ The angular acceleration of the flywheel is thus $$-50$$ rotations per square minute. The negative sign tells us that the flywheel is retarding instead of accelerating. We can apply the second equation for a circular motion to find the rotations made by the flywheel in the given time. The second equation of circular motion, similar to the second equation for linear motion, is as follows $$\theta ={{\omega }_{1}}t+\dfrac{1}{2}\alpha {{t}^{2}}$$ where $$\theta $$ is the rotations made by the flywheel and the meanings of the other symbols have been discussed above Substituting the values, we get $$\begin{aligned} & \theta =400\times 2+\dfrac{1}{2}\times (-50)\times {{(2)}^{2}} \\\ & \Rightarrow \theta =800-100=700 \\\ \end{aligned}$$ **Hence the flywheel makes $$700$$ rotations in two minutes.** **Note:** If in the given question, you were asked to find the angle covered by the flywheel in the given time, you would have to multiply the obtained number of rotations by $$2\pi $$ as the flywheel covers $$360{}^\circ $$ angle in each rotation. We could have also used the third equation for a rotational motion to find the number of rotations made as both the initial and the final angular velocity is known to us.