A solid cylinder of mass M and radius R rolls without...

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined plane making an angle $\theta$ with the horizontal. Then its acceleration is (a) $\frac{1}{3} g \sin \theta$ (b) $\frac{2}{3} g \sin \theta$ (c) $\frac{2}{5} g \sin \theta$ (d) $\frac{2}{7} g \sin \theta$

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined plane making an angle $\theta$ with the horizontal. Then its acceleration is
(a) $\frac{1}{3} g \sin \theta$
(b) $\frac{2}{3} g \sin \theta$
(c) $\frac{2}{5} g \sin \theta$
(d) $\frac{2}{7} g \sin \theta$

Key Concepts

Newton's Laws for Translation and Rotation

Newton’s second law for translation (F = ma) and its rotational analogue (? = I?) form the basis for analyzing the motion of rolling objects. These laws allow us to set up equations that relate the forces and torques acting on the object to its linear and angular accelerations, respectively.

Dynamics on an Inclined Plane

When an object is placed on an inclined plane, the gravitational force can be resolved into components parallel and perpendicular to the surface. The component parallel to the incline drives the motion, while the perpendicular component is balanced by the normal force. This setting is commonly used to analyze the acceleration of objects sliding or rolling on slopes.

Rolling Without Slipping

This concept describes the condition where an object rolls in such a way that at the point in contact with the surface, its instantaneous velocity is zero relative to the surface. It establishes a relation between translational motion and rotational motion, meaning the linear velocity of the center of mass is directly proportional to the angular velocity through the radius of the object.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a given axis. It depends on both the mass of the object and the distribution of that mass relative to the axis of rotation. It plays a crucial role in determining how dynamics of rotation evolve under applied torques.

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