# Q26  A 1 kg block situated on a rough incline is connected to a spring of spring constant $100 N m^{-1}$ as shown in Fig. 6.17. The block is released from rest with the spring in the unstretched position. The block moves 10 cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

D Devendra Khairwa

Displacement (x) of the block is given as :   =  0.1 m.

Using equilibrium conditions we can write :

$R\ =\ mg \cos 37^{\circ}$

and                                          $\mu R\ =\ mg \sin 37^{\circ}$                                 ( $\mu R$ is the frictional force).

We can write work done in terms of potential energy as :

$mg\ \left ( \sin 37^{\circ}\ -\ \mu \cos 37^{\circ} \right )x\ =\ \frac{1}{2}kx^2$

or                                       $1\times g\ \left ( \sin 37^{\circ}\ -\ \mu \cos 37^{\circ} \right )x\ =\ \frac{1}{2}100\times (0.1)^2$

or                                                     $\mu\ =\ 0.125$.

Thus the coefficient of friction is 0.125.

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