A uniform disc of mass M and its rim. The coefficient of...

A uniform disc of mass $M$ and its rim. The coefficient of friction radius $R$, is resting on a table on between disc and table is $\mu$. Now the disc is pulled with a force $F$ as shown in the figure. What is the maximum value of $F$ for which the disc rolls without slipping? (a) $\mu \mathrm{Mg}$ (b) $2 \mu M g$ c) $3 \mu \mathrm{Mg}$ (d) $4 \mu \mathrm{Mg}$

A uniform disc of mass $M$ and
its rim. The coefficient of friction radius $R$, is resting on a table on
between disc and table is $\mu$. Now the disc is pulled with a force $F$ as shown in the figure. What is the maximum value of $F$ for which the disc rolls without
slipping?
(a) $\mu \mathrm{Mg}$
(b) $2 \mu M g$
c) $3 \mu \mathrm{Mg}$
(d) $4 \mu \mathrm{Mg}$

Key Concepts

Rolling Without Slipping

This concept signifies that the point of contact between a rolling object and the surface does not slide. It requires a strict relation between translational and rotational motion such that the linear velocity of the center of mass equals the product of the angular velocity and the radius. This condition ensures that the kinematics of rolling are coupled, which is essential in analyzing motions where friction prevents slipping.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. In rolling scenarios, static friction provides the necessary force and torque to enable the object to roll without slipping. It reaches a maximum value, proportional to the normal force and the coefficient of static friction, which limits the applied force before the rolling motion transitions to sliding.

Moment of Inertia

The moment of inertia quantifies an object's resistance to angular acceleration about an axis. For an object like a uniform disc (or a disc with a rim), it plays a crucial role in relating the net torque to the angular acceleration. The distribution of mass relative to the rotation axis determines the moment of inertia, and thus it is central to accurately calculating the dynamics of rotational motion under applied forces.

Newton's Second Law (Translation)

In the context of rolling motion, Newton's second law for translation is used to sum the forces acting on the center of mass, thereby relating the net force to the mass and the linear acceleration of the object. It helps in determining how an applied force and the opposing frictional force collectively influence the acceleration of the object's center of mass.

Newton's Second Law (Rotation) and Torque Analysis

This principle involves summing the torques acting on a rolling object to relate them with the product of the moment of inertia and the angular acceleration. In problems involving rolling without slipping, both the applied force (when acting at a lever arm) and friction contribute torques, making it essential to analyze these moments together with the rotational equation of motion to ensure consistency with the rolling constraint.

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