Question

The block of mass m₁ shown in figure is fastened to the spring and the block of mass m₂ is placed against it. (a) Find the compression of the spring in the equilibrium position. (b) The blocks are pushed a further distance (2/k) (m₁ + m₂)g sin θ against the spring and released. Find the position where the two blocks separate. (c) What is the common speed of blocks at the time of separation?

Answer 1

(a) As it can be seen from the figure,
Restoring force = kx
Component of total weight of the two bodies acting vertically downwards = (m₁ + m₂) g sin θ
At equilibrium,  
 kx = (m₁ + m₂) g sin θ

\[\Rightarrow x = \frac{\left( m_1 + m_2 \right)  g  \sin  \theta}{k}\]

(b) It is given that:
Distance at which the spring is pushed,

\[x_1  = \frac{2}{k}\left( m_1 + m_2 \right)g  \sin  \theta\]

As the system is released, it executes S.H.M.
where \[\omega = \sqrt{\frac{k}{m_1 + m_2}}\]

When the blocks lose contact, P becomes zero.                   (P is the force exerted by mass m₁ on mass m₂)

\[\therefore    m_2 g  \sin  \theta =  m_2  x_2  \omega^2  =  m_2  x_2  \times \frac{k}{m_1 + m_2}\] \[ \Rightarrow  x_2  = \frac{\left( m_1 + m_2 \right)  g  \sin  \theta}{k}\]

Therefore, the blocks lose contact with each other when the spring attains its natural length.

(c) Let v be the common speed attained by both the blocks.

\[\text { Total  compression }  =    x_1  +  x_2 \] 

\[\frac{1}{2}  \left( m_1 + m_2 \right) v^2  - 0   =   \frac{1}{2}k \left( x_1 + x_2 \right)^2  - \left( m_1 + m_2 \right)  g  \sin  \theta  \left( x + x_1 \right)                      \] 

\[ \Rightarrow \frac{1}{2}\left( m_1 + m_2 \right) v^2  = \frac{1}{2}k\left( \frac{3}{k} \right)  \left( m_1 + m_2 \right)  g  \sin  \theta - \left( m_1 + m_2 \right)  g  \sin  \theta  \left( x_1 + x_2 \right)\] 

\[ \Rightarrow \frac{1}{2}\left( m_1 + m_2 \right) v^2  = \frac{1}{2}  \left( m_1 + m_2 \right)  g  \sin  \theta \times \left( \frac{3}{k} \right)  \left( m_1 + m_2 \right)  g  sin  \theta\] 

\[ \Rightarrow v = \sqrt{\left\{ \frac{3}{k}  \left( m_1 + m_2 \right) \right\}}g  \sin  \theta\]