Question

If the rotating mass of a rim type flywheel is distributed on a second rim type flywheel whose mean radius is half the mean radius of the former, then the energy stored in the second flywheel at the same speed will be ------- times that of the first flywheel

• A. \( \dfrac{1}{4} \)
• B. \( \dfrac{1}{2} \)
• C. \( 2 \)
• D. \( 4 \)

Hint:

For rim type flywheels at the same speed, energy stored is proportional to the square of radius.

Answer 1

The Correct Option is A

Step 1: Formula for energy stored in a flywheel.
The kinetic energy stored in a flywheel is given by: \[ E = \frac{1}{2} I \omega^2 \] For a rim type flywheel, moment of inertia \( I = mr^2 \).
Step 2: Comparing the two flywheels.
Let the radius of the first flywheel be \( r \). Then, radius of the second flywheel is \( \frac{r}{2} \).
Step 3: Ratio of energies.
\[ \frac{E_2}{E_1} = \frac{m\left(\frac{r}{2}\right)^2}{mr^2} = \frac{1}{4} \]
Step 4: Conclusion.
The energy stored in the second flywheel is one-fourth of that stored in the first flywheel.

Answer 2

For a rim-type flywheel, essentially all of the mass is concentrated at the mean radius, so its kinetic energy at a given speed scales with the square of that radius, since \( E = \tfrac{1}{2}I\omega^2 = \tfrac{1}{2}mr^2\omega^2 \). Halving the radius (while keeping the same mass and the same speed) therefore does not halve the energy; it reduces it according to the square of the radius ratio. Testing each option:

1. \( \tfrac14 \): Since energy scales as \( r^2 \), halving the radius multiplies the energy by \( (\tfrac12)^2 = \tfrac14 \), which is exactly the expected scaling for a quantity that depends on the square of the radius.
2. \( \tfrac12 \): This would be correct only if energy scaled linearly with radius, but kinetic energy of a rotating rim depends on \( r^2 \), not \( r \), so this understates how strongly radius affects stored energy.
3. \( 2 \): This would require the smaller-radius flywheel to store more energy than the larger one at the same speed, which contradicts the fact that reducing the radius (with the same mass) reduces the moment of inertia and hence the stored energy.
4. \( 4 \): This is the reciprocal of the correct scaling; it would be the ratio if the radius had instead been doubled rather than halved.

Since the mass and speed are unchanged and only the radius is halved, the quadratic dependence of energy on radius gives a ratio of \( \tfrac14 \).

Therefore, the correct answer is \( \tfrac14 \).