Question

A chain of length L is placed on a horizontal surface as shown in figure. At any instant x is the length of chain on rough surface and the remaining portion lies on smooth surface. initially x = 0. A horizontal force P is apllied to the chain (as shown in figure). In the duration x changes from x = 0 to x = L, for chain to move with constant speed.

• A. The magnitude of P should increase with time.
• B. The magnitude of P should decrease with time.
• C. The magnitude of P should increase first time and then decrease with time.
• D. The magnitude of P should decrease first and then increase with time.

Answer 1

Answer: (A)

The magnitude of P should increase with time.

In order for the chain to move with constant acceleration, the net force acting on it should be constant. The forces acting on the portion of the chain on the rough table are:
i) force P (towards the right)
ii) friction, f (towards the left)
ii) weight, mg (downwards)
iv) normal force, N (upwards)
N = mg (since no acceleration in y- direction) Here, m = mass of the portion of chain on the rough surface.
0 = P - f (since no acceleration in x - direction)
Since the chain is moving, the friction acting will be equal to limiting friction which is $\mu N$. Further, since it moves with constant speed, a = 0.
$f=\mu mg$
$P=\mu mg$
As the chain moves, the mass of the portion lying on the rough surface increases. therefore P increases.