A uniform disc of mass M and radius R rolls without slipping down a plane inclined at an angle θ

step 1 Question is a uniform test of mass M and radius R rolls without slipping down at a plane incline

A uniform disc of mass M and radius R rolls without slipping...

A uniform disc of mass $\mathrm{M}$ and radius $\mathrm{R}$ rolls without slipping down a plane inclined at an angle $\theta$ with the horizontal. The acceleration of the centre of mass of the disc is $\{\mathrm{A}\} \mathrm{g} \sin \theta$ $\{B\}[(2 g \sin \theta) / 3]$ $\{\mathrm{C}\}[(\mathrm{g} \sin \theta) / 3]$ $\{\mathrm{D}\}[(2 \mathrm{~g} \cos \theta) / 3]$

A uniform disc of mass $\mathrm{M}$ and radius $\mathrm{R}$ rolls without slipping down a plane inclined at an angle $\theta$ with the horizontal.
The acceleration of the centre of mass of the disc is
$\{\mathrm{A}\} \mathrm{g} \sin \theta$
$\{B\}[(2 g \sin \theta) / 3]$
$\{\mathrm{C}\}[(\mathrm{g} \sin \theta) / 3]$
$\{\mathrm{D}\}[(2 \mathrm{~g} \cos \theta) / 3]$

Key Concepts

Rolling Without Slipping

This concept describes a condition where a rolling object, such as a disc, does not slide across the surface it rolls on. The point of contact is momentarily at rest relative to the surface. In dynamics, this condition links the linear acceleration of the object’s center of mass to its angular acceleration through the relationship a = R?, making it a crucial constraint when analyzing systems that involve both translation and rotation.

Moment of Inertia

The moment of inertia quantifies an object's resistance to angular acceleration based on its mass distribution relative to the axis of rotation. For a uniform disc, the moment of inertia is given by I = (1/2)MR². This value is essential for calculating rotational dynamics and plays a key role in determining how the disc accelerates as it rolls down an incline.

Newton's Second Law for Translation and Rotation

Newton's second law is applied to both the translational and rotational motions of a rolling object. For translation, the net force is equal to mass times linear acceleration, while for rotation, the net torque is equal to the moment of inertia times the angular acceleration. The simultaneous application of these two forms of motion along with the no-slip condition allows us to derive the acceleration of the center of mass for rolling objects.

Dynamics on an Inclined Plane

When an object rolls down an incline, gravitational force is decomposed into components parallel and perpendicular to the plane. The component parallel to the incline (g sin?) drives the acceleration, while friction provides the necessary torque for rotation without slipping. Understanding these force components is critical for analyzing the acceleration and overall behavior of the rolling object.

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