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Introduction to Radial & Tangential Acceleration

Two masses ...

Question

Two masses $M_{1}$ and $M_{2}$ are connected with each other with the help of a massless spring of spring-constant $k$ . Mass $M_{1}$ is in contact with a wall and the system is at rest on the frictionless floor. $M_{2}$ is displaced by distance $x$ to compress the spring and released. The velocity of COM (centre of mass) of the system when $M_{1}$ is detached from wall is :

\frac{x \sqrt{k M_{1}}}{M_{1} + M_{2}}

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\frac{x \sqrt{k M_{2}}}{M_{1} + M_{2}}

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\frac{k x \sqrt{M_{2}}}{M_{1} + M_{2}}

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\frac{x}{\sqrt{k M_{1}} (M_{1} - M_{2})}

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Solution

The correct option is A $\frac{x \sqrt{k M_{2}}}{M_{1} + M_{2}}$
Block 1 will leave wall just after the spring is at its natural length.
P.E. will convert into K.E. of 2nd block.
$\frac{1}{2} k x^{2} = \frac{1}{2} M_{2} v^{2} v = x \sqrt{\frac{k}{M_{2}}} M_{2} v = x \sqrt{k M_{2}}$
velocity of centre of mass $= \frac{M_{1} v_{1} + M_{2} v_{2}}{M_{1} + M_{2}} = \frac{x \sqrt{k M_{2}}}{M_{1} + M_{2}}$

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