A carpet of mass `M` is rolled along its length so as to from a cylinder of radius `R` and is kept on a rough floor. When a negligibly small push is given to the cylindrical carpet, it stars unrolling itself without sliding on the floor. Calculate horizontal velocity of cylindrical part of the carpet when its radius reduces to `R//2`.

A floor-mat of mass M made up of extensible material, is rolled along its length so as to form a cylinder of radius R and kept on a rough horizontal surface. If the mat is now unrolled, without sliding, to a radius (R )/(2) , the decrease in potential energy is

A solid cylinder of mass M and radius R rolls without slipping on a flat horizontal surface. Its moment of inertia about the line of contact is ?

A bullet of mass m moving with a velocity of u just grazes the top of a solid cylinder of mass M and radius R resting on a rough horizontal surface as shown and is embedded in the cylinder after impact. Assuming that the cylinder rolls without slipping, find the angular velocity of the cylinder and the final velocity of the bullet.

A sphere of mass m and radius R is kept on a rough floor. A sharp impulse is applied in the horizontal direction at the height of centre of sphere so that sphere acquires a linear velocity v_(0) without any angular velocity. Calculate the velocity of sphere when pure rolling starts.

A cylinder of mass m and radius R is kept on a rough surface after giving its centre a horizontal speed v_(0) . Find the speed of the centre of the cylinder when it stops slipping.

A particle of mass M is moving in a horizontal circle of radius R with uniform speed V . When it moves from one point to a diametrically opposite point , its

A particle of mass M is moving in a horizontal circle of radius R with uniform speed V. When it moves from one point to a diametrically opposite point, its

A thin carpet of mass 2m is rolled over a hollow cylinder of mass m . The cylinder wall is thin and radius of the cylinder is R. The carpet rolled over it has outer radius 2R (see figure). This roll is placed on a rough horizontal surface and given gentle push so that the carpet begins to roll and unwind. Friction is large enough to prevent any slipping of the carpet on the floor. Also assume that the carpet does not slip on the surface of the cylinder. The entire carpet is laid out on the floor and the hollow cylinder rolls out with speed V. Find V.

A cylinder is rotating with angular velocity omega_(0) and is gently put on a rough horizontal floor. Assume mass of the cylinder is m and radius R. Calculate the velocity of cylinder when it starts pure rolling on the surface.