A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is

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Solution

Q.
A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is [2024]

1 1 m

2 $\sqrt{6}$ m

3 $\sqrt{5}$ m

4 2 m

Ans.
(4)

$\sin 30 ° = \frac{h}{10} \Rightarrow h = 5$

At maximum compression

using Work-Energy Theorem,

$W_{gravity} + W_{friction} + W_{spring} = Δ K E$

$m g h - μ m g (2 + x) - \frac{1}{2} k x^{2} = 0 - 0$

$5 \times 10 \times 5 - \frac{1}{2} \times 5 \times 10 (2 + x) - \frac{1}{2} 100 \times x^{2} = 0$

$250 - 50 x^{2} - [50 + 25 x] = 0$

$200 - 50 x^{2} - 25 x = 0$

$2 x^{2} + x - 8 = 0$

$x = \frac{- 1 \pm \sqrt{1^{2} + 4 \times 2 \times 8}}{2 \times 2} \Rightarrow x = \frac{- 1 \pm \sqrt{65}}{4} \approx 1.77 m$

Nearest answer is 2m.

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