Pendulum on a Moving Cart

A pendulum hangs from a cart that rolls without friction. Let the bob swing and the cart rolls the other way: the rod only pushes and pulls along itself, and those are internal forces, so they cannot move the system's center of mass. Released from rest the total horizontal momentum is zero and stays zero, which pins the center of mass to one vertical line (the dashed marker) while the cart and bob trade places around it. The scene shows the swing and the recoil; the diagnostic plots the cart and bob horizontal positions over time, straddling that flat center-of-mass line. A light cart recoils far, a heavy one barely moves, set by the mass ratio $m/M$.

Figure 1. Pendulum on a frictionless moving cart. Top: the cart recoils as the bob swings, the center of mass fixed on the dashed line. Bottom: cart and bob horizontal positions versus time, straddling the constant center of mass. Method: RK4 on the coupled Lagrange equations.

theta_0 (rad)0.80

cart M (kg)2.0

bob m (kg)0.5

WHAT TO TRY

Watch the dashed center-of-mass line: the cart and bob both move, but their mass-weighted average never leaves it.
Lighten the cart and it recoils much further per swing; make it heavy and the cart barely twitches while the bob does almost all the moving.
On the diagnostic the cart and bob curves are mirror images about the flat center-of-mass line, with amplitudes in the ratio of the opposite masses.
Release from a larger angle for a wider swing; the period stretches as the amplitude grows (the pendulum is not quite simple harmonic).