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

step 1 Christian is a uniform disk of mass and radius are rolls without slipping down a plane at an ang

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 magnitude of torque acting on the disc is $\{\mathrm{A}\} \mathrm{MgR}$ $\{\mathrm{B}\} \mathrm{MgR} \sin \theta$ $\{\mathrm{C}\}[(2 \mathrm{MgR} \sin \theta) / 3]$ $\{\mathrm{D}\}[(\mathrm{MgR} \sin \theta) / 3]$

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 magnitude of torque acting on the disc is
$\{\mathrm{A}\} \mathrm{MgR}$
$\{\mathrm{B}\} \mathrm{MgR} \sin \theta$
$\{\mathrm{C}\}[(2 \mathrm{MgR} \sin \theta) / 3]$
$\{\mathrm{D}\}[(\mathrm{MgR} \sin \theta) / 3]$

Key Concepts

Torque

Torque is a measure of the tendency of a force to produce rotation about an axis. It is calculated as the product of the force and the lever arm (the perpendicular distance from the axis of rotation to the line of action of the force). In problems involving rolling objects, torque determines how the applied forces cause angular acceleration.

Rolling Without Slipping

Rolling without slipping is the condition where the rolling object moves in such a way that its point of contact with the surface is momentarily at rest relative to the surface. This provides a kinematic constraint linking translational motion and rotational motion (a = R?), which is fundamental in analyzing the dynamics of rolling objects on inclined planes or other surfaces.

Moment of Inertia

The moment of inertia quantifies an object's resistance to angular acceleration about a given axis, depending on how its mass is distributed relative to that axis. For a uniform disc rotating about its center, the moment of inertia is a well?known quantity (I = ½MR²). This property is essential when applying rotational dynamics equations to determine angular acceleration and torque.

Rotational Dynamics

Rotational dynamics is the branch of mechanics that deals with the relationship between torque, moment of inertia, and angular acceleration (? = I?). In problems involving rolling objects, the net torque acting on the object, when combined with the moment of inertia, governs the rotational motion of the object, analogous to the role of net force in translational motion.

Please give Ace some feedback