Obtain the expression for the linear acceleration of a cylinder rolling down inclined place and hence find the condition for the cylinder to roll down without slipping.

In finding the linear acceleration of a rolling cylinder on an inclined plane, we have the forces involved-the weight of the cylinder and the frictional force. The weight can be divided into two: one along the incline and the other perpendicular to it. The friction force opposes the motion to avoid sliding.

Using Newton’s second law, we analyze the net force acting along the incline on the cylinder. The net force is the difference between the component of gravitational force pulling it down the slope and the frictional force acting up the slope. This net force is also related to the linear acceleration of the cylinder.

Next, we analyze the cylinder’s rotational motion where the frictional force causes a torque around the center of mass. Torque affects the cylinder’s angular acceleration. Linear acceleration and angular acceleration for rolling without slipping are in fact related.

Integrating these considerations shows that the linear acceleration of the rolling cylinder down the incline is a fraction of gravitational acceleration. The no slipping requirement for the rolling cylinder further depends on the coefficient of static friction and the angle of the incline so that the frictional force is always quite sufficient to prevent slipping.

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