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

A rope of negligible mass is wound around 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 $30 \mathrm{~N}$ ? Assume that there is no slipping: (a) $10 \mathrm{rad} / \mathrm{s}^{2}$ (b) $15 \mathrm{rad} / \mathrm{s}^{2}$ (c) $20 \mathrm{rad} / \mathrm{s}^{2}$ (d) $25 \mathrm{rad} / \mathrm{s}^{2}$

A rope of negligible mass is wound around 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 $30 \mathrm{~N}$ ? Assume that there is no slipping:
(a) $10 \mathrm{rad} / \mathrm{s}^{2}$
(b) $15 \mathrm{rad} / \mathrm{s}^{2}$
(c) $20 \mathrm{rad} / \mathrm{s}^{2}$
(d) $25 \mathrm{rad} / \mathrm{s}^{2}$

Key Concepts

No-Slip Condition

The no-slip condition occurs when there is sufficient friction between surfaces (or between a rope and a cylinder in this context) such that there is no relative motion at the point of contact. This condition ensures a fixed relationship between the linear motion of a point on the object and its angular motion, which is key in relating the applied force to the object's angular acceleration.

Newton's Second Law for Rotation

Newton’s second law for rotational motion states that the net torque acting on a body is equal to the product of its moment of inertia and its angular acceleration (net torque = moment of inertia × angular acceleration). This law provides the foundation for analyzing the rotational dynamics of objects when subjected to torques.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. In problems involving cylinders, the moment of inertia plays a crucial role in determining how the object accelerates when a torque is applied.

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 and the lever arm distance from the axis of rotation. This concept is essential because it quantifies the rotational effect produced by a force applied perpendicular to the radius of a rotating object.

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