A spring is loaded with two blocks $m_{1}$ and $m_{2}$ where $m_{1}$ is rigidly fixed with the spring and $m_{2}$ is just kept on the block $m_{1}$ as shown in the figure. The maximum energy of oscillation that is possible for the system having the block $m_{2}$ in contact with $m_{1}$ is

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Solution

The correct option is C

Let the required amplitude be A. For this amplitude of oscillation the normal contact force between the blocks can be given as
$m_{2} g - N = m_{2} a w h e r e a = ω^{2} A \Rightarrow m_{2} g - N = m_{2} ω^{2} A P u t t i n g N = 0 f o r j u s t l o s i n g c o n t a c t f o r m a x i m u m a m p l i t u d e, w e o b t a i n A = \frac{g}{ω^{2}} w h e r e ω = a n g u l a r f r e q u e n c y o f o s c i l l a t i o n o f (m_{1} + m_{2}) a n d t h e s p r i n g . P u t t i n g ω^{2} = \frac{k}{m_{1} + m_{2}}, w e o b t a i n A = \frac{(m_{1} + m_{2}) g}{K} ∴ U_{m a x} = \frac{1}{2} k A^{2} = \frac{(m_{1} + m_{2})^{2} g^{2}}{2 k}$

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A spring is loaded with two blocks $m_{1}$ and $m_{2}$ where $m_{1}$ is rigidly fixed with the spring and $m_{2}$ is just kept on the block $m_{1}$ as shown in the figure. The maximum energy of oscillation that is possible for the system having the block $m_{2}$ in contact with $m_{1}$ is