Question

Two solid cylinders P and Q of same mass and same radius, start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which of the following statements is correct?
a. Both the cylinders P and Q reach the ground at the same time.
b. Cylinder P has larger linear acceleration than cylinder Q
c. Both cylinders reach the ground with same translational kinetic energy.
d. Cylinder Q reaches the ground with larger angular speed.

• A. c
• B. a
• C. d
• D. b

Hint:

Rolling down an incline, the body with the smaller moment of inertia has the larger acceleration and reaches the bottom with the larger speed and angular speed.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
For rolling without slipping, \(a=\frac{g\sin\theta}{1+I/mR^{2}}\) and \(v^{2}=\frac{2gh}{1+I/mR^{2}}\). A larger \(I\) means a smaller \(a\) and \(v\).

Step 2: Calculation
P has its mass near the surface, so \(I_{P}>I_{Q}\). Hence \(a_{Q}>a_{P}\), Q reaches the ground first, and \(v_{Q}>v_{P}\). The radii are equal, so \(\omega=\frac{v}{R}\) gives \(\omega_{Q}>\omega_{P}\).

Step 3: Compare the options
Statement a is wrong (different times). Statement b is wrong (P has the smaller acceleration). Statement c is wrong because the translational kinetic energies \(\frac{1}{2}mv^{2}\) differ as the speeds differ. Statement d is correct.

Final Answer:
Only statement d is correct: Q reaches the ground with the larger angular speed. \[ \boxed{\text{(C) }\text{d}} \]