Question

A solid cylinder and a hollow cylinder, both of the same mass and same external diameter are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first?

• A. Both together only when angle of inclination of plane is $45^\circ$
• B. Both together
• C. Hollow cylinder
• D. Solid cylinder

Answer 1

The Correct Option is D

To determine which cylinder will reach the bottom of an inclined plane first, we need to consider the rotational dynamics of both the solid cylinder and the hollow cylinder, both of which roll down without slipping.

The time it takes for each cylinder to reach the bottom of the incline depends on its moment of inertia and how it converts its potential energy into translational and rotational kinetic energy. The potential energy at the top is entirely converted to kinetic energy at the bottom.

1. The moment of inertia I for each cylinder type is as follows:

   • Solid cylinder: I = \frac{1}{2} m r^2
   • Hollow cylinder: I = m r^2

2. Using the principle of conservation of energy:

   Potential energy at the top = Total kinetic energy at the bottom

   mgh = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2

   Where \omega = \frac{v}{r} is the angular velocity and v is the linear velocity.

3. For the solid cylinder, substituting the value of I:

   mgh = \frac{1}{2} mv^2 + \frac{1}{2} \left(\frac{1}{2} m r^2\right) \left(\frac{v}{r}\right)^2

   Simplifying, we find:

   mgh = \frac{3}{4} mv^2

   Thus, v = \sqrt{\frac{4}{3}gh}

4. For the hollow cylinder:

   mgh = \frac{1}{2} mv^2 + \frac{1}{2} (m r^2) \left(\frac{v}{r}\right)^2

   Simplifying, we find:

   mgh = mv^2

   Thus, v = \sqrt{gh}

5. Comparing the two results, the linear velocity of the solid cylinder (\sqrt{\frac{4}{3}gh}) is greater than that of the hollow cylinder (\sqrt{gh}). Therefore, since the solid cylinder has a greater acceleration and hence velocity, it will reach the bottom of the incline first.

Conclusion: The solid cylinder will reach the bottom of the inclined plane before the hollow cylinder.