Problem:
A homogeneous and inextensible chain of length 2 m and mass 100 g lies on a smooth table. A small portion of chain of length 0.5 m hangs from the table. Initially, the part of the chain lying on the table held and then released. Calculate the velocity with which the chain leaves the table. (g=10 ms-2).

Solution:
To solve the problem, we will use the principle of conservation of energy. Initially, the chain is at rest on the table, and after being released, it falls freely due to gravity. The potential energy of the chain is converted into kinetic energy as it falls. We can equate these two energies to find the velocity with which the chain leaves the table.

Step 1: Calculate the potential energy of the hanging part of the chain.
The potential energy of an object is given by the product of its mass, the acceleration due to gravity, and its height above a reference point. In this case, the reference point can be taken as the bottom of the hanging chain.

Potential energy of the hanging chain = mgh
where m = mass of the hanging chain = 0.1 kg
g = acceleration due to gravity = 10 m/s^2
h = height of the hanging chain = 0.5 m

Potential energy of the hanging chain = (0.1 kg)(10 m/s^2)(0.5 m) = 0.5 J

Step 2: Calculate the kinetic energy of the falling chain.
The kinetic energy of an object is given by the product of one-half its mass and the square of its velocity.

Kinetic energy of the falling chain = (1/2)mv^2
where m = mass of the entire chain = 0.1 kg + (length of chain on table x linear density)
v = velocity of the chain as it leaves the table

To find the mass of the entire chain, we need to know its linear density. Since the chain is homogeneous, its linear density is given by its total mass divided by its length.

Linear density of the chain = mass/length = 0.1 kg/2 m = 0.05 kg/m

The mass of the chain on the table is given by the product of its length and linear density.

Mass of the chain on the table = (2 m - 0.5 m)(0.05 kg/m) = 0.075 kg

The total mass of the chain is the sum of the mass of the hanging chain and the mass of the chain on the table.

Total mass of the chain = 0.1 kg + 0.075 kg = 0.175 kg

Kinetic energy of the falling chain = (1/2)(0.175 kg)v^2

Step 3: Equate the potential energy and kinetic energy of the chain.
According to the principle of conservation of energy, the potential energy of the hanging chain is converted into kinetic energy as the chain falls. Therefore, we can equate the potential energy of the hanging chain to the kinetic energy of the falling chain.

Potential energy of the hanging chain = Kinetic energy of the falling chain
0.5 J = (1/2)(0.175 kg)v^2

Solving for v, we get:
v =