Problem — Glass of water on the swing

Problem

A child is sitting on a swing of length $\ell = 2{,}5\;\text{m}$. Under the seat hangs another string, $0{,}5\;\text{m}$ long, with a glass of water. Which of the two “pendulums” oscillates faster? By how much? Discuss the effect of the superposition of the two rhythms on the contents of the glass.

Solution

Both systems are simple pendulums, with period $T = 2\pi\sqrt{\ell/g}$.

Swing:

$T_a = 2\pi\sqrt{\frac{2{,}5}{9{,}81}} \approx 2\pi \cdot 0{,}505\;\text{s} \approx 3{,}17\;\text{s}$

$\ev{T_a \approx 3{,}17\;\text{s}}$

Glass:

$T_b = 2\pi\sqrt{\frac{0{,}5}{9{,}81}} \approx 2\pi \cdot 0{,}226\;\text{s} \approx 1{,}42\;\text{s}$

$\ev{T_b \approx 1{,}42\;\text{s}}$

The ratio of the periods depends only on the ratio of the lengths:

$\frac{T_a}{T_b} = \sqrt{\frac{2{,}5}{0{,}5}} = \sqrt{5} \approx 2{,}24$

The glass oscillates about 2,2 times faster than the swing.

Discussion: the two rhythms have frequencies that are different and not commensurable with each other; the superposition of the two movements makes the water slosh about irregularly. The swing could “excite” the glass strongly only if the two frequencies were close together (resonance), but here they are very different, so the transfer of energy from the swing to the glass remains modest.

Links

Topics: Oscillations and simple harmonic motion Concepts: Simple pendulum · Period and frequency Objects: Simple pendulum