Problem 27 - University Physics with Modern Physics

Question

A cylinder with mass \(M\) and radius \(R\) is rolling without slipping through a distance \(s\) along an inclined plane that makes an angle \(\theta\) with respect to the horizontal. Calculate the work done by (a) gravity, (b) the normal force, and (c) the frictional force.

Step-by-Step Solution

Answer

Based on the provided solution, calculate the work done by the gravitational force, normal force, and frictional force on a cylinder rolling without slipping down an inclined plane with the following characteristics: 1. Angle of inclination (θ) = 30 degrees 2. Mass (M) of the cylinder = 5 kg 3. Radius (R) of the cylinder = 0.5 meters 4. The distance (s) the cylinder rolls = 10 meters Work done by gravity (Wg) = M * g * sin(θ) * s = 5 * 9.81 * sin(30) * 10 = 245.25 J Work done by normal force (Wn) = 0 Work done by frictional force (Wf) = 0 The work done by the gravitational force is 245.25 J, while the work done by normal force and frictional force are both 0 J.

Step 1: Calculate the Force Components

First, let's identify the force components based on the given information: - Gravitational force, Fg = M * g - Normal force, Fn, acts perpendicular to the inclined plane and is equal in magnitude to the component of gravitational force acting perpendicular to the inclined plane. - Frictional force, Ff, opposes the motion and acts tangentially to the inclined plane.

Step 2: Calculate the Work Done by Gravity (Wg)

To calculate the work done by gravity, we need to find the component of gravitational force acting parallel to the inclined plane and multiply it with the displacement (s). The angle between the displacement and the force component will be 0 degrees. - Gravitational force component parallel to inclined plane: Fg_parallel = Fg * sin(θ) - Work done by gravity: Wg = Fg_parallel * s * cos(0) - Wg = M * g * sin(θ) * s

Step 3: Calculate the Work Done by Normal Force (Wn)

Since the normal force acts perpendicular to the inclined plane, the angle between the displacement vector and the normal force vector is 90 degrees. - Work done by normal force: Wn = Fn * s * cos(90) - Wn = 0 The work done by the normal force on the cylinder is 0.

Step 4: Calculate the Work Done by Frictional Force (Wf)

The frictional force acts tangentially to the point of contact between the cylinder and the inclined plane. Since the cylinder is rolling without slipping, the frictional force will not affect the linear displacement. Consequently, the work done by frictional force is 0. Wf = 0