Question

All the surfaces shown in figure (12-E15) are frictionless. The mass of the car is M, that of the block is m and the spring has spring constant k. Initially, the car and the block are at rest and the spring is stretched through a length $$x_0$$ when the system is released.
Find the amplitudes of the simple harmonic motion of the block and of the car as seen from the road.

Answer 1

Let the amplitude of oscillation of $$m$$ and $$M$$ be $$x_{1}$$ and $$x_{2}$$ respectively.

a) From law of conservation of momentum,

$$mx_{1}=Mx_{2}$$ ...(1) [because only internal forces are present]

Again $$(1/2)kx_{0}^{2}=(1/2)k(x_{1}+x_{2})^{2}$$

[Block and mass oscillates in opposite direction, But $$x\rightarrow$$ stretched part]

From equation $$(1)$$ and $$(2)$$

$$\therefore x_{0}=x_{1}+\dfrac{m}{M}x_{1}=\left(\dfrac{M+m}{M}\right)x_{1}$$

$$\therefore x_{1}\dfrac{Mx_{0}}{M+m}$$

So, $$x_{2}=x_{0}=x_{0}\left[1-\dfrac{M}{M+m}\right]=\dfrac{mx_{0}}{M+m}$$ respectively.