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Work Energy Theorem Application

Two blocks of...

Question

Two blocks of masses $m_{1}$ and $m_{2}$ are connected by a spring of spring constant $k$ . The block of mass $m_{2}$ is given a sharp impulse so that it aquires a velocity $v_{o}$ towards right.

Velocity of centre of mass will be

\frac{m_{2} v_{0}}{m_{1} + m_{2}}

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Velocity of centre of mass will be

\frac{m_{1} v_{0}}{m_{1} + m_{2}}

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Maximum elongation that the spring will suffer

x_{m a x} = \sqrt{\frac{m_{2}}{k (m_{1} + m_{2})}} . v_{0}

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Maximum elongation that the spring will suffer

x_{m a x} = \sqrt{\frac{m_{1} m_{2}}{k (m_{1} + m_{2})}} . v_{0}

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Solution

The correct option is D Maximum elongation that the spring will suffer $x_{m a x} = \sqrt{\frac{m_{1} m_{2}}{k (m_{1} + m_{2})}} . v_{0}$
Given, Initial velocity of $m_{2} = v_{0}$
Initial velocity of $m_{1} = 0$
As we know, $V_{c o m} = \frac{m_{1} v_{1} + m_{2} v_{2}}{m_{1} + m_{2}}$
$V_{c o m} = \frac{m_{1} (0) + m_{2} (v_{0})}{m_{1} + m_{2}} = \frac{m_{2} v_{0}}{m_{1} + m_{2}}$

The spring will have maximum elongation when velocities of both blocks becomes equal to the velocity of COM of the two blocks.
As we know,
Change in KE = Potential energy stored in the spring.
Assume $x_{m a x} =$ maximum elongation of the spring
Hence, $K E_{i} - K E_{f}$ = Potential energy stored
$\frac{1}{2} m_{2} v_{0}^{2} - [\frac{1}{2} (m_{1} + m_{2}) v_{c o m}^{2}] = \frac{1}{2} k x_{m a x}^{2}$
$\Rightarrow \frac{1}{2} m_{2} v_{0}^{2} - [\frac{1}{2} (m_{1} + m_{2}) {(\frac{m_{2} v_{0}}{m_{1} + m_{2}})}^{2}] = \frac{1}{2} k x_{m a x}^{2}$
$\Rightarrow m_{2} v_{0}^{2} [1 - \frac{m_{2}}{m_{1} + m_{2}}] = k x_{m a x}^{2}$
$\Rightarrow x_{m a x} = \sqrt{\frac{m_{1} m_{2}}{k (m_{1} + m_{2})}} v_{0}$

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Work Energy Theorem Application

Standard XII Physics

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Two blocks of masses m1 and m2 are connected by a spring of spring constant k. The block of mass m2 is given a sharp impulse so that it aquires a velocity vo towards right.

Two blocks of masses $m_{1}$ and $m_{2}$ are connected by a spring of spring constant $k$ . The block of mass $m_{2}$ is given a sharp impulse so that it aquires a velocity $v_{o}$ towards right.