A solid cylinder is rolling down on an inclined plane of angle θ. The coefficient of static fric

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A solid cylinder is rolling down on an inclined plane of...

A solid cylinder is rolling down on an inclined plane of angle $\theta .$ The coefficient of static friction between the plane and cylinder is $\mu_{x}$. The condition for the cylinder not to slip is (a) $\tan \theta \geq 3 \mu_{x}$ (b) $\tan \theta>3 \mu_{2}$ (c) $\tan \theta \leq 3 \mu$, (d) $\tan \theta<3 \mu$,

A solid cylinder is rolling down on an inclined plane of angle $\theta .$ The coefficient of static friction between the plane and cylinder is $\mu_{x}$. The condition for the cylinder not to slip is
(a) $\tan \theta \geq 3 \mu_{x}$
(b) $\tan \theta>3 \mu_{2}$
(c) $\tan \theta \leq 3 \mu$,
(d) $\tan \theta<3 \mu$,

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For different shapes, the moment of inertia varies; for instance, a solid cylinder has a specific value that is used in conjunction with the torque provided by static friction to analyze its rotational acceleration.

Rotational Dynamics

Rotational dynamics involves applying Newton's second law to rotational motion, where the net torque on an object is related to its angular acceleration through its moment of inertia. This concept links the frictional force acting at the point of contact to the rotational acceleration of the rolling object.

Static Friction

Static friction is the force that resists the initiating motion of contact surfaces. In the context of rolling objects, it is responsible for providing the torque that causes rotation without slipping. Its maximum value, governed by the coefficient of static friction, establishes the limit beyond which slipping occurs.

Rolling Without Slipping

This concept refers to the condition in which a rolling object has a direct relationship between its linear and angular velocities, meaning that the point in contact with the surface is momentarily at rest. This condition is crucial for connecting translational motion to rotational motion in problems involving rolling objects.

Newton's Second Law for Translational Motion

This law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In the case of an object on an inclined plane, the gravitational force is decomposed into components parallel and perpendicular to the plane, which are used to determine the net acceleration down the slope.

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