# Two masses connected by a spring

• grusini

## Homework Statement

Two masses ##m_1=2kg##, ##m_2=2m_1## are placed on the opposite edges of a spring of constant ##k_1=3N/m##, compressed of a length ##x_1=1.73cm##. The system is located on a smooth plane. At the right end of the plane there is a second spring of constant ##k_2=12N/m##. Once the first spring is at rest the masses are free to move (they are not fixed to the spring). So the situation is as follows:

_________________M1 spring1 M2___________spring2 ||

a) Find the velocity ##v_1,v_2## of masses ##m_1,m_2## respectively, when the first spring is at rest.

b) Find the maximum compression ##\Delta x## of the second spring.

## Homework Equations

Conservation of energy, conservation of momentum, potential energy of a spring

## The Attempt at a Solution

a) The velocities must satisfy the system:
$$\begin{cases}\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2=\frac{1}{2}k_1x_1^2 \\ m_1v_1+m_2v_2=0\end{cases}$$
##\Rightarrow v_2=\sqrt{\frac{m_1k_1x_1^2}{m_2^2+m_1m_2}}=##
##\sqrt{\frac{2\cdot 3\cdot (0.0173)^2}{4^2+4\cdot 2}}=0.008 m/s##.
Then ##v_1=-\frac{m_2}{m_1}v_2=-2v_2=-0.016m/s##.

b) The value ##\Delta x## must satisfy:
##\frac{1}{2}m_2v_2^2=\frac{1}{2}k_2(\Delta x)^2 \Rightarrow \Delta x=\sqrt{\frac{m_2v_2^2}{k_2}}=\sqrt{\frac{4\cdot (0.008)^2}{12}}=0.0046m##

Values are wrong on my textbook..