The block shown in the figure is connected by a spring which is in a relaxed state. Now the block is displaced a distance ‘ A ‘ towards left and released. It oscillates many times. A weak friction force F is acting on the block. It is given that F/k=A/10. When the block comes to a state of rest for the first two times the deformation of the spring is X1 and X2. The A-X1X1-X2 is

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The block shown in the figure is connected by a spring which is in a relaxed state. Now the block is displaced a distance ‘ A ‘ towards left and released. It oscillates many times. A weak friction force F is acting on the block. It is given that $F / k = A / 10 .$ When the block comes to a state of rest for the first two times the deformation of the spring is $X_{1}$ and $X_{2} .$ The $\frac{(A - X_{1})}{(X_{1} - X_{2})}$ is

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Detailed Solution

Apply energy conservation from 2 to 3 , $\frac{1}{2} k A^{2} = \frac{1}{2} K x_{1}^{2} + μ m g (A + x_{1})$

$\frac{1}{2} k (A^{2} - x_{1}^{2}) = μ m g (A + x_{1}), \frac{1}{2} k (A^{2} - x_{1}) = μ m g______(1)$

Apply energy conservation from 3 to 4

$\frac{1}{2} k x_{1}^{2} = k x_{2}^{2}) + μ m g (x_{1} + x_{2}), \frac{1}{2} k (x_{1}^{2} - x_{2}^{2}) = μ m g (x_{1} + x_{2}), \frac{1}{2} k (x_{1} - x_{2}) + μ m g________(2)$