A light string is wound round a cylinder and carries a mass tied to it at the free end. When the mass is released, calculate (a) the linear acceleration of the descending mass and (b) the angular acceleration of the cylinder and (c) the tension in the string. Show that the acceleration of mass is less than g.

A mass is tied to a thin, light string that is wound around a cylinder. The mass is released. There is a gravitational force pulling it downward, and the tension in the string is pulling it upward. This difference between the gravitational force and the tension can be calculated to find the net force on the mass. By Newton’s second law, this net force must equal the product of the mass and its linear acceleration.

There’s also tension in the string creating a torque in the cylinder which can be represented in terms of tension and radius of the cylinder. The angular acceleration is found to be associated with the linear acceleration of mass by the radius of the cylinder.

We can obtain the linear acceleration of the falling mass and the tension in the string by considering these relationships. The acceleration can be written as a fraction of the acceleration due to gravity, showing that it is smaller than the acceleration due to gravity. This happens because part of the gravitational force is being utilized to generate rotational motion in the cylinder, meaning that there will be a smaller linear acceleration of the mass than if it were falling freely. This whole system demonstrates principles of rotational dynamics and linear motion.

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