The spring shown in figure is kept I a stretched position with extension $x_{0}$ when the system is released. Assuming the horizontal surface to be frictionless, find the frequency of oscillation.

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Solution

The correct option is B

No external force on the system. So center of mass at rest

If block m moves by x towards right and block M moves by X towards left then mx = MX

Also the spring is totally compressed by (x + X). Applying conservation of energy method.

$\frac{1}{2} k (x + X)^{2} + \frac{1}{2} m v_{1}^{2} + \frac{1}{2} M v_{2}^{2} = c o n s t a n t$

Taking derivative with respect to time on both sides

$- k (x + X) (v_{1} + v_{2}) + m v_{1} a_{1} + M v_{2} a_{2} = 0$

$- K x (1 + \frac{m}{M}) (v_{1} + v_{2}) = (m v_{1} a_{1} + M a_{2} v_{2})$

$∴ m x = M x$

$\frac{d x}{d t} = M \frac{d x}{d t}$

$\Rightarrow m v_{1} = M v_{2}$

$\Rightarrow m a_{1} = M a_{2}$

$- k x (1 + \frac{m}{M}) = m a_{1}$

$ω = m \sqrt{\frac{k (M + m)}{M m}}$

frequency $f = \frac{ω}{2 π} = \frac{1}{2 π} \sqrt{\frac{k (M + m)}{M m}}$

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The spring shown in figure is kept I a stretched position with extension x0 when the system is released. Assuming the horizontal surface to be frictionless, find the frequency of oscillation.

The spring shown in figure is kept I a stretched position with extension $x_{0}$ when the system is released. Assuming the horizontal surface to be frictionless, find the frequency of oscillation.