Question

STATEMENT-1 Two cylinders, one hollow (metal) and the 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 cylinder will reach the bottom of the inclined plane first. and STATEMENT-2 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. (A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT- 2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

Short answer

The correct answer is (D) STATEMENT-1 is False, STATEMENT-2 is True.

Step-by-step solution

Analyze Statement-1

To determine the truth of Statement-1, recall that the acceleration of a cylinder rolling down an incline without slipping can be given by the formula: \(a=\frac{g\sin(\theta)}{1+\frac{I}{mR^2}}\), where \(a\) is the acceleration, \(g\) is the acceleration due to gravity, \(\theta\) is the angle of inclination, \(I\) is the moment of inertia of the cylinder, \(m\) is the mass of the cylinder, and \(R\) is the radius. Since both cylinders have the same mass and dimensions, they differ only in their moment of inertia. The moment of inertia for a solid cylinder is \(\frac{1}{2}mR^2\), and for a hollow cylinder is \(mR^2\). Hence, the hollow cylinder has a larger moment of inertia which results in a smaller acceleration. Thus, the hollow cylinder would take longer to reach the bottom, making Statement-1 False.

Analyze Statement-2

For Statement-2, the principle of conservation of energy states that the total mechanical energy (potential plus kinetic energy) in an isolated system remains constant if only conservative forces are acting. Both cylinders are subjected to gravitational force only, which is conservative. The initial potential energy of each cylinder is the same as they start from the same height. When they reach the bottom of the incline, all their potential energy would have been converted into translational and rotational kinetic energy. For both cylinders, the total kinetic energies will thus be identical given that they have the same mass and started from the same height, regardless of their shape or distribution of mass. Therefore, Statement-2 is True.

Combine Analyses to Determine Correct Answer

With Statement-1 being False and Statement-2 being True, the correct answer is option (D), which states that STATEMENT-1 is False and STATEMENT-2 is True. Statement-2 is not a correct explanation for Statement-1 because even though the total kinetic energies are the same at the bottom, it does not explain the order in which they reach the bottom.

Key concepts

Conservation of Energy

The principle of conservation of energy is a foundational concept in physics, stating that energy cannot be created or destroyed, only transformed from one form to another. This principle applies perfectly to the scenario of cylinders rolling down an incline.

At the top, the cylinders possess gravitational potential energy, which is dependent on their mass, height, and the acceleration due to gravity. As they roll down the incline, this potential energy is converted into two kinds of kinetic energy: translational (due to linear motion) and rotational (due to spinning). The total amount of energy remains constant throughout the descent, assuming there's no energy lost to external factors like air resistance or friction other than what causes rolling without slipping.

Considering both cylinders start from the same height and have the same mass, the initial potential energy is identical for both. Upon reaching the bottom, all this energy will have been converted to kinetic energy, confirming that the total kinetic energy for each cylinder must be the same, hence verifying Statement-2.

Moment of Inertia

The moment of inertia is often described as the rotational equivalent of mass for linear motion. It is a measure of an object's resistance to changes in its rotational motion. The moment of inertia depends on both the mass of the object and the distribution of that mass relative to the axis of rotation.

In our case, we have a solid wooden cylinder and a hollow metal cylinder. For a solid cylinder, the moment of inertia is given by \(\frac{1}{2}mR^2\), whereas for a hollow cylinder it's \(mR^2\). The hollow cylinder's mass is further from its axis of rotation, which increases its moment of inertia. Consequently, a greater moment of inertia means that, under the same force, the hollow cylinder would accelerate more slowly and thus take a longer time to roll down the incline compared to the solid cylinder. This reality contradicts Statement-1, which mistakenly claims the hollow cylinder reaches the bottom first.

Rolling Without Slipping

Rolling without slipping is a term used when an object rolls on a surface and there is no relative motion between the point of contact on the object and the surface. This condition is fundamental for understanding the motion of rolling objects and establishes a crucial relationship between the translational and rotational motion of the object.

In the context of the cylinders rolling down the incline, rolling without slipping ensures that all of the gravitational potential energy is converted into kinetic energy without any energy being lost to heat through friction, as would be the case if the cylinders were slipping. Additionally, the condition of rolling without slipping implies that the linear velocity \(v\) at the point of contact with the surface is equal to the rotational velocity \(\omega\) times the radius \(R\), or \(v = \omega R\). This relationship helps us understand that the object's motion is a combination of rotational and translational kinetic energy components.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. There are two types of kinetic energy relevant to rotating objects: translational kinetic energy and rotational kinetic energy. Translational kinetic energy is associated with the linear motion of an object and is calculated by \(\frac{1}{2}mv^2\), where \(m\) is the mass and \(v\) is the linear velocity. Rotational kinetic energy, on the other hand, is associated with the object's rotation and is calculated by \(\frac{1}{2}I\omega^2\), where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

For the rolling cylinders, their total kinetic energy at the bottom of the incline is the sum of both the translational and rotational kinetic energies collected during the descent. While the shape and mass distribution of the cylinders affect their moments of inertia and thus their rotational kinetic energy, the conservation of energy principle ensures that the sum of the kinetic energies is equal for both cylinders, as stated in Statement-2.