Explanation:

Introduction:
When a circular loop of rope rotates with a uniform angular velocity about an axis through its center on a horizontal smooth platform, a pulse is produced due to slight radial displacement. This pulse travels along the length of the rope with a certain velocity.

Velocity of Pulse:
The velocity of the pulse, with respect to the rope, is given by the formula:

v = ωr

where v is the velocity of the pulse, ω is the angular velocity of the loop, and r is the radial displacement of the rope.

Derivation:
To understand this formula, let's consider a point on the loop of rope that is displaced radially by a small distance r. As the loop rotates, this point moves in a circular path. The velocity of this point can be given by the formula:

v = ωr

where v is the tangential velocity of the point, ω is the angular velocity of the loop, and r is the radius of the circular path.

Now, when the point is displaced radially, it creates a pulse that travels along the length of the rope. This pulse has a velocity equal to the tangential velocity of the point. Therefore, the velocity of the pulse is also given by the formula:

v = ωr

Explanation:
The formula v = ωr shows that the velocity of the pulse is directly proportional to the angular velocity of the loop and the radial displacement of the rope. This means that if the angular velocity increases or the radial displacement increases, the velocity of the pulse will also increase. Similarly, if the angular velocity decreases or the radial displacement decreases, the velocity of the pulse will decrease.

Conclusion:
In conclusion, the velocity of the pulse produced due to slight radial displacement in a rotating circular loop of rope is given by the formula v = ωr, where v is the velocity of the pulse, ω is the angular velocity of the loop, and r is the radial displacement of the rope. This formula shows that the velocity of the pulse is directly proportional to the angular velocity and the radial displacement.