Periodic And Oscillatory Motions

Step 1: When an object rolls without slipping, its total kinetic energy is the sum of its translational kinetic energy and its rotational kinetic energy. The total kinetic energy is: Step 2: For a solid cylinder rolling without slipping, the rotational kinetic energy is related to the translational kinetic energy by: where is the moment of inertia for a solid cylinder and is the angular velocity. Thus, Therefore, the total kinetic energy becomes: Step 3: Given that , the total kinetic energy is: Thus, the total kinetic energy of the cylinder is 210 J.

Using the kinematic equation for rotational motion: Given: Initial angular speed, rev/min rad/s Final angular speed, rad/s Angular acceleration, rad/s Solving for : Since 1 revolution corresponds to radians, Thus, the total number of revolutions is 157.

Step 1: The two blocks are connected by a string, so they will move with the same acceleration. Let the acceleration of the blocks be . The forces on the blocks are: For block : For block : Step 2: Adding these two equations: Step 3: The speed of the centre of mass after time is: Thus, the speed of the centre of mass is 6 ms .

The moment of inertia for a solid cylinder about its central axis is given by: where is the mass and is the radius. For a hollow cylinder (assuming a thin-walled structure), the moment of inertia is: Clearly, This shows that the moment of inertia of a hollow cylinder is greater than that of a solid cylinder of the same mass and radius.