Question

A uniform rod of length L and mass M has been placed on a rough horizontal surface. The horizontal force F applied on the rod is such that the rod is just in the state of rest. If the coefficient of friction varies according to the relation mu=kx. Where k is a positive constant. Find the tension at midpoint of rod.​

Answer 1

The tension at midpoint of rod is Tx =  F/4

Explanation:

Let us consider an element dx at a distance x from one of the ends of the rod.

dF = μ(x) x Mg/L x dx = Mg/L kxdx

F = Mgk/L(x^2/2)L-0   = MgkL/2

K = 2F/MgL

Tx = 0.5 L = Mg/L (2F/MgL)(x^2/2)L/2-0

Tx =  F/4

Thus the tension at midpoint of rod is Tx =  F/4

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