Problem — Why the pendulum stops

Pendulum that stops. Explain why a real pendulum, sooner or later, stops, in terms of energy balance.

Solution

The ideal pendulum would never stop. With conservation of mechanical energy, energy is continuously exchanged between the two forms with no losses:

$E_\text{mecc} = \tfrac{1}{2}mv^2 + mgh = \text{constant}$

Entirely kinetic at the bottom, entirely potential at the extremes: it would swing with the same amplitude forever.

The real pendulum has dissipative forces. On every swing two friction effects remove mechanical energy:

• air resistance on the string and the mass;
• friction at the pivot of the suspension.

Both do negative work and turn mechanical energy into heat. So $E_\text{mecc}$ is no longer constant but decreases on every pass:

$\ev{E_\text{mecc} = \tfrac{1}{2}mv^2 + mgh \ \searrow\ \text{converting into heat}}$

Conclusion. The amplitude of the oscillations progressively shrinks (damped motion) until all the initial mechanical energy has become dispersed heat: the pendulum stops at the lowest point, its stable equilibrium point, where the potential energy is minimum. Total energy is always conserved — it has simply migrated from the ordered mechanical form to the disordered thermal one.

Collegamenti

Argomenti: Work and energy Concetti: Thermal energy from friction · Conservation of mechanical energy Competenze: Bubble energy balance Oggetti: Simple pendulum