Question
A carpet of mass M made of inextensible material is rolled along its length in the form of a cylinder of radius R and kept on the floor. The carpet starts unrolling without sliding on the floor, when a negligible small push is given to it. Calculate the horizontal velocity of the axis of a cylindrical part of carpet, when its radius decreases to R/2.
Solution
The situation is illustrated as in figure below.
Here, mass of the carpet is M.
When the carpet unrolls and radius decreases to half, the mass of left unrolled carpet is,
Initial total energy of carpet is,
Let, v be the linear velocity, and
be the angular velocity of unrolled carpet, when it becomes a cylinder of radius R/2.
Now the total energy,
Using conservation of energy:
This is the required horizontal velocity of the axis of a cylindrical part of carpet.
Here, mass of the carpet is M.
When the carpet unrolls and radius decreases to half, the mass of left unrolled carpet is,
Initial total energy of carpet is,
Let, v be the linear velocity, and
be the angular velocity of unrolled carpet, when it becomes a cylinder of radius R/2. Now the total energy,
Using conservation of energy:
This is the required horizontal velocity of the axis of a cylindrical part of carpet.
