Question

Statement \(-1\) - Two cylinder one hollow and other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow will reach the bottom of inclined plane first. Statement \(-2-\mathrm{By}\) the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline. \(\\{\mathrm{A}\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\) \\{B \\} Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1\). \\{C\\} Statement - 1 is true, statement- 2 is false \\{D \(\\}\) Statement- 2 is false, statement \(-2\) is true

Short answer

The correct answer is \\(D\\): Statement- 1 is false, statement -2 is false.

Step-by-step solution

Calculate the Moment of Inertia

First, we'll calculate the moment of inertia for both cylinders. The moment of inertia for a solid cylinder is given by the I = 1/2 MR^2, where M is the mass of the cylinder and R is the radius. For a hollow cylinder, the moment of inertia is given by I = MR^2.

Determine Accelerations

Next, we'll use the moment of inertia (I) to find the acceleration for each cylinder. Assuming the angle of the inclined plane is θ, the equation for acceleration is a = (g * sin(θ))/(1 + (I/(MR^2))). Plug in the moment of inertia for the solid cylinder a_solid = (g * sin(θ))/(1 + (1/2)) and for the hollow cylinder a_hollow = (g * sin(θ))/(1 + 1).

Compare Accelerations

Now, compare the accelerations to determine which cylinder will reach the bottom first. Since a_solid > a_hollow, the solid cylinder will reach the bottom of the inclined plane first, not the hollow cylinder as stated in Statement 1. Thus, Statement 1 is false.

Analyze Kinetic Energies

Now let's analyze the kinetic energy of both cylinders when they reach the bottom of the incline. Using the principle of conservation of energy, the potential energy at the top of the inclined plane will be equal to the sum of translational and rotational kinetic energies at the bottom. Therefore, we have PE = KE_trans + KE_rot, which implies Mgh = 1/2MV^2 + 1/2 IV^2/R^2 for each cylinder, where h is the height of the plane, g is the gravitational force, and V is the final velocity. Solving this equation for both cylinders, we can find their final velocities and thus their kinetic energies.

Determine if Kinetic Energies are Equal

To decide whether Statement 2 is true, we need to find out if the total kinetic energies of the cylinders are equal when they reach the bottom of the inclined plane. Simplify the equation Mgh = 1/2MV^2 + 1/2 IV^2/R^2 for both cylinders and compare the results. Since Statement 1 is false and we've shown that the total kinetic energies are not equal (Statement 2 is also false), we can conclude that the correct answer is: \\{D\\} Statement- 1 is false, statement -2 is false