A block of mass m is attached with spring of spring constant K. There is a wall at distance 2 A from the natural length of spring. Block is displaced such that compression produced in spring is A. I

Question

A block of mass m is attached with spring of spring constant K. There is a wall at distance 2 A ​ from the natural length of spring. Block is displaced such that compression produced in spring is A. If T = 2 π k m ​ ​ then find the time period of this motion. All collision are elastic and spring is ideal. If your answer is n m ​ T where m & n are lowest possible integer, fill value of ( m + n )

TestHub · Accepted Answer

From the figure the wall is at extension $\frac{A}{2}$ ; the spring stays attached, so motion is SHM between compression $A$ and the wall

$x = - A cos ω t$ ; block reaches $x = + \frac{A}{2}$ when $cos ω t = - \frac{1}{2}$ , i.e. $t = \frac{T}{3}$

Elastic collision at the wall reverses the velocity, so the motion repeats after $2 \times \frac{T}{3}$

$T_{m o t i o n} = \frac{2}{3} T$ , so $m = 2$ , $n = 3$ and $m + n = 5$

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