Solution:
First, let's analyze the situation and the forces involved:

1. Chain hanging over the edge:
When the chain hangs over the edge of the table, it creates a torque due to its weight. The torque causes a rotational force on the chain, making it slide on the table.

2. Friction force:
The friction force between the table and the chain opposes the motion of the chain. This force is responsible for preventing the chain from sliding completely off the table.

Calculating the coefficient of friction:

To calculate the coefficient of friction between the table and the chain, we need to consider the forces involved and their magnitudes.

1. Torque:
The torque created by the chain hanging over the edge can be calculated as the weight of the hanging chain multiplied by the distance from the point of rotation (edge of the table).

Given that the chain hangs 25 cm (or 0.25 m) over the edge, and assuming the chain's weight is uniformly distributed, we can calculate the torque as follows:

Torque = weight of the hanging chain x distance from the point of rotation

The weight of the hanging chain can be calculated using the formula:

Weight = mass x acceleration due to gravity

Since the chain is metallic, it has a high density and a typical density of around 7,800 kg/m³ can be assumed. The mass of the chain can be calculated as:

Mass = density x volume

Since the chain is in the shape of a cylinder, the volume is given by:

Volume = πr²h

Given that the chain has a length of 1 m and a radius of r, the volume can be calculated as:

Volume = πr² x 1

Therefore, the mass of the chain is:

Mass = 7,800 x πr²

Now that we have the mass, we can calculate the weight of the chain:

Weight = (7,800 x πr²) x 9.8

Finally, the torque can be calculated as:

Torque = [(7,800 x πr²) x 9.8] x 0.25

2. Friction force:
The friction force can be calculated using the formula:

Friction force = coefficient of friction x normal force

The normal force is the force exerted by the table on the chain perpendicular to the surface. When the chain hangs over the edge, the normal force is equal to the weight of the hanging chain.

Therefore, the friction force can be calculated as:

Friction force = coefficient of friction x [(7,800 x πr²) x 9.8]

3. Equilibrium condition:
For the chain to be in equilibrium, the torque created by the hanging chain must be balanced by the friction force.

Therefore, we have the equation:

Torque = Friction force

Using the expressions derived earlier, we can substitute the values and solve for the coefficient of friction.

[(7,800 x πr²) x 9.8] x 0.25 = coefficient of friction x [(7,800 x πr²) x 9.8]

Simplifying the equation, we find:

0.25 = coefficient of friction

Conclusion:
The value of the coefficient of friction between the table and the chain is 0