Question #42356

A rigid cylindrical tube of mass $M$ and length $L$ is initially at rest in deep space, free from any external forces. Inside the tube, a non-uniform liquid of mass $M$ initially occupies the left half of the tube ($0 \le x \le L/2$, where $x$ is the distance from the left end of the tube). The linear mass density of the liquid is given by $\lambda(x) = \lambda_0 \sin\left(\frac{\pi x}{L}\right)$. An internal pump transfers the entire liquid to the right half of the tube ($L/2 \le x \le L$), where it settles with the density profile $\lambda(x) = \lambda_0 \sin\left(\frac{\pi x}{L}\right)$. What is the magnitude of the absolute displacement of the tube?