A plank P is placed on a solid cylinder S, which rolls on a horizontal surface. The two are of equal mass. There is no slipping at any of the surfaces in contact. The ratio of kinetic energy of P to the kinetic energy of S is 

8 : 3

Understanding the Problem:

We have a plank (P) placed on a solid cylinder (S). Both have equal mass and roll without slipping. We need to find the ratio of their kinetic energies.

Key Points:

1. No Slipping: This implies that the point of contact between the plank and the cylinder, and between the cylinder and the ground, has zero relative velocity.
2. Equal Mass: Both objects have the same mass, which we can denote as 'm'.

Solution:

Let's denote:

• v: Linear velocity of the center of mass of the cylinder (S)
• ω: Angular velocity of the cylinder

Kinetic Energy of the Plank (P): Since the plank is moving without slipping on the cylinder, its linear velocity will be twice that of the cylinder's center of mass.

• Velocity of the plank (P) = 2v
• Kinetic energy of P = (1/2)m(2v)² = 2mv²

Kinetic Energy of the Cylinder (S): The cylinder possesses both translational and rotational kinetic energy.

• Translational kinetic energy of S = (1/2)mv²
• Rotational kinetic energy of S = (1/2)Iω²
  1. For a solid cylinder, the moment of inertia (I) = (1/2)mr²
  • Angular velocity (ω) = v/r (due to no slipping)
  • So, rotational kinetic energy of S = (1/2)(1/2)mr²(v/r)² = (1/4)mv²
• Total kinetic energy of S = (1/2)mv² + (1/4)mv² = (3/4)mv²

Ratio of Kinetic Energies:

• Ratio = (Kinetic energy of P) / (Kinetic energy of S)
• Ratio = (2mv²) / ((3/4)mv²)
• Ratio = 8/3

Therefore, the correct answer is (c) 8:3.