Two blocks with a spring and the CM graph

Problem

An isolated system made of two blocks connected by a spring oscillates on a frictionless rail. The graph shows the position $x_1(t)$ and $x_2(t)$ of the two blocks as a function of time: two sinusoids in antiphase, centred on different average values. (a) Sketch $x_\text{CM}(t)$ qualitatively and justify it. (b) Explain how the mass ratio can be recognised from the amplitudes.

Solution

(a) $x_\text{CM}(t)$ is a straight line. The system is isolated (no external force): the spring force is internal. Hence $\vec{v}_\text{CM}$ is constant and $x_\text{CM}(t) = x_\text{CM}(0) + v_\text{CM}\,t$ is a straight line. The oscillations of the two blocks, summed with weight $m_i$, cancel out: $x_\text{CM}$ does not oscillate at all. If the CM is initially at rest, the line is horizontal.

(b) Ratio of the amplitudes. Since $m_1 x_1 + m_2 x_2 = M x_\text{CM}$ (with no oscillations on the CM), the oscillations of the two blocks about the CM must compensate each other instant by instant: $m_1 A_1 = m_2 A_2$. Hence $\ev{\frac{A_1}{A_2} = \frac{m_2}{m_1}}$ The lighter block oscillates with a larger amplitude. The ratio of the amplitudes is the inverse of the mass ratio.

Links

Topics: Centre of mass Concepts: Centre of mass · Conservation of momentum Skills: Reading graphs