A cylinder of mass M, radius R is resting on a horizontal...

A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $X Y$ -plane) with its axis fixed along the $y$ -axis and free to rotate about its axis. The platform is given a motion in $X$ -direction given by $x=A \cos \omega t$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is (a) $\frac{1}{2} M R A \omega^{2}$ (b) $M R A \omega^{2}$ (c) $2 M R A \omega^{2}$ (d) $M R A \omega^{2} \times \cos \omega t$

A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $X Y$ -plane) with its axis fixed along the $y$ -axis and free to rotate about its axis. The platform is given a motion in $X$ -direction given by $x=A \cos \omega t$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is
(a) $\frac{1}{2} M R A \omega^{2}$
(b) $M R A \omega^{2}$
(c) $2 M R A \omega^{2}$
(d) $M R A \omega^{2} \times \cos \omega t$

Key Concepts

Torque

Torque is the rotational equivalent of force. It is defined as the product of the applied force and the lever arm (the perpendicular distance from the axis of rotation to the line of action of the force). In problems with rotational motion, torque determines the angular acceleration of the body according to Newton’s second law for rotation.

Moment of Inertia

The moment of inertia quantifies a body’s resistance to angular acceleration about a given axis and depends on how the mass is distributed relative to the axis. In rotational dynamics, the moment of inertia plays a role analogous to mass in linear dynamics, linking the net applied torque to the angular acceleration of the body.

Rolling Without Slipping

Rolling without slipping is the condition where the point of contact between a rolling object and the surface is momentarily at rest relative to the surface. This condition imposes a constraint that links the translational motion of the center of mass with the rotational motion of the object, typically expressed as a = R?, where a is linear acceleration, R is the radius, and ? is the angular acceleration.

Newton’s Second Law for Rotational Motion

Newton’s second law for rotation states that the net torque acting on a rigid body is equal to the product of its moment of inertia and its angular acceleration (? = I?). This principle is fundamental in analyzing how torques affect rotational dynamics, similar to how forces affect linear motion.

Static Friction and Its Role in Rolling

Static friction is the force that prevents relative motion between surfaces in contact and is crucial in rolling without slipping. In the context of rotational dynamics, static friction can provide the necessary torque to initiate or sustain rotational motion when an external acceleration is applied, even if no deformation occurs.

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