A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What

step 1 In this problem, rope is wound over a hollow cylinder and one end of the rope is being pulled by

A rope of negligible mass is wound round a hollow cylinder...

A rope of negligible mass is wound round a hollow cylinder of mass $3 \mathrm{~kg}$ and radius $40 \mathrm{~cm}$. What is the angular acceleration of the cylinder if the rope is pulled with a force of $30 \mathrm{~N}$ ? What is the linear acceleration of the rope ? Assume that there is no slipping.

Key Concepts

Relationship Between Angular and Linear Acceleration

The relationship between angular and linear acceleration is given by the equation a = ?r, where a is the tangential (linear) acceleration, ? is the angular acceleration, and r is the radius at which the acceleration is measured. This connection is crucial when analyzing systems where rotational and translational motions are linked, such as when a rope unwinds without slipping.

Newton's Second Law for Rotation

Newton's second law for rotation states that the net torque acting on an object is equal to the product of its moment of inertia and its angular acceleration (? = I?). This fundamental principle is used to relate the applied torque to the resulting angular acceleration, and it is essential for analyzing rotational dynamics.

Moment of Inertia

The moment of inertia is a property of a body that quantifies its resistance to angular acceleration about a particular axis. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a hollow cylinder, the moment of inertia is typically given by the product of its mass and the square of its radius.

Torque

Torque is the measure of the force that can cause an object to rotate about an axis. It is calculated as the product of the force applied and the distance from the point of application to the axis of rotation. In rotational dynamics, torque is the driving factor behind angular acceleration, analogous to how force causes linear acceleration in translational motion.

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