A block of mass 2 kg attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m spring constnt is 200 N/m. The block is pushed such that

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Solution

Q.
A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is 200 N/m. The block is pushed such that the length of the spring becomes 1 m and then released. At distance xm (x < 2) from the wall the speed of the block will be : [2025]

1 $10 [10 - {(2 \times x)}^{3 / 2}] m / s$

2 $10 {[1 - {(2 - x)}^{2}]}^{1 / 2} m / s$

3 $[1 - {(2 - x)}^{2}] m / s$

4 $10 [1 - {(2 - x)}^{2}] m / s$

Ans.
(2)

Given, Natural length of spring = 2 m

Initial compression is spring $(x_{i}) = 1 m$

Final compression in spring $(x_{f}) = (2 - x) m$

Using energy conservation: $K_{i} + U_{i} = K_{f} + U_{f}$

$0 + \frac{1}{2} K x_{i}^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} K x_{f}^{2}$

$\frac{1}{2} m v^{2} = \frac{1}{2} K (x_{i}^{2} - x_{f}^{2})$

$\frac{1}{2} \times 2 \times v^{2} = \frac{1}{2} \times 200 \times (1^{2} - {(2 - x)}^{2})$

$v^{2} = 100 [1 - {(2 - x)}^{2}]$

$\Rightarrow v = 10 {[1 - {(2 - x)}^{2}]}^{1 / 2}$

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