The Swing

The swing as a pendulum

A swing with rope length $L$ and a person sitting still is a pendulum with the period

$$T\approx 2\pi\sqrt{\frac{L_{\rm eff}}{g}},$$

where $L_{\rm eff}$ is the distance from the suspension to the common centre of mass – a little shorter than the rope, because the person sits above the seat. For a swing 2.2 m long, $T\approx 3$ s. The period hardly depends on how heavy the person is, nor on the amplitude, as long as it is below about 20–30°.

How to pump without touching the ground

The person cannot change the total angular momentum about the suspension with internal forces. Only gravity can. But the person can move their own centre of mass and push on the ropes. When the person leans back with a jerk, the upper body turns one way, and the reaction turns the ropes and the seat the other way, a little forward. Repeated in time with the swing, the small pushes add up.

The equations of motion

Let $\varphi$ be the angle of the ropes with the vertical and $\alpha$ the person's angle, so that the person's tilt relative to the ropes is $\psi=\alpha-\varphi$. The seat is at $\mathbf S=L(\sin\varphi,-\cos\varphi)$, and the parts of the person's body at $\mathbf S+\mathsf R(\alpha)\mathbf b_i$. The Lagrange equations become

$$\begin{aligned} M_{11}\ddot\varphi+M_{12}\ddot\alpha&=-M g\,S_x+\dot\alpha^2C-\tau-\gamma M_{11}\dot\varphi,\\ M_{12}\ddot\varphi+M_{22}\ddot\alpha&=-g\textstyle\sum_i m_iB_{i,x}-\dot\varphi^2C+\tau, \end{aligned}$$

with $M_{11}=ML^2$, $M_{22}=\sum m_i|\mathbf b_i|^2$, $M_{12}=\sum m_i\,\mathbf S\cdot\mathbf B_i$ and $C=\sum m_i\,\mathbf S\times\mathbf B_i$, where $\mathbf B_i=\mathsf R(\alpha)\mathbf b_i$. The muscles' torque $\tau$ acts with opposite signs on the ropes and on the person, and this is where the energy comes in:

$$\tau=-\kappa\left(\psi-\psi_{\rm target}(t)\right)-c\left(\dot\psi-\dot\psi_{\rm target}\right).$$

The muscles behave like a stiff spring with damping that pulls the tilt towards the desired $\psi_{\rm target}$. The term $M_{12}$ is the coupling: if the person accelerates their body, it pushes the swing, and vice versa.

When should you lean back?

The simulation's automatic person uses the rule

$$\psi_{\rm target}=A\tanh\left(\dot\varphi/\omega_c\right):$$

lean back and straighten the legs while the swing moves forwards, and lean forward and bend the legs while it moves backwards. The switch happens at the turning points. The muscles' work per swing is $W=-\oint\tau\,d\psi$. It is positive when the tilt is a quarter of a swing out of phase with the swing's displacement – exactly as when you push a swing by hand as it passes the bottom. Reverse the sign, and the person brakes the swing instead.

Two coupled pendulums

If the person holds their muscles still, $\psi_{\rm target}=0$, swing and person are two coupled pendulums. The person is an inverted pendulum, with the centre of mass above the seat, held up by the muscle stiffness $\kappa$. Linearise the equations about equilibrium, and you get two normal modes, $\mathsf K\mathbf u=\omega^2\mathsf M\mathbf u$, as shown in the panel:

The swing mode is slow. The ropes and the person swing almost as one rigid pendulum, and the person tilts only slightly relative to the ropes.

The body mode is fast. The person rocks back and forth about the seat, and the ropes swing slightly in antiphase.

The person drives the second degree of freedom, and the coupling transfers the motion to the first. It is a driven oscillating system with a resonance at the frequency of the swing mode. Choose Rhythmic and try different pumping frequencies: the swing only really grows when the pumping frequency matches the frequency of the swing mode, about 0.34 Hz for a swing 2.2 m long. Choose Limp, lower the muscle stiffness, and press Push the person: now the two pendulums exchange energy without help from the muscles, just as on the page on coupled pendulums.

Standing up: parametric resonance

If you stand on the swing, you pump in a different way: you bend your knees at the turning points and straighten up as the swing passes the bottom. This raises the centre of mass just when the rope tension is greatest, and changes the pendulum's effective length twice per swing. That is parametric resonance. It only works if the swing is already moving a little, and it gives a more exponential growth. Seated pumping, on the other hand, is a driven oscillation: it can start from rest, and the amplitude grows roughly linearly at first. Press Stop the swing in the simulation and see that the seated person can start the swing from complete rest.

The period. Sit on the swing, get a push and sit completely still. Time 10 full swings and compare with $2\pi\sqrt{L/g}$, where $L$ is measured from the suspension to your navel, roughly your centre of mass.

Start from rest. Sit still, with your feet off the ground. Lean back and straighten your legs, then lean forward and bend your legs, again and again. Find the rhythm in which the swing grows fastest. Time it, and compare with the swing's period.

Wrong phase. Once the swing is well under way, swap round: lean back while you are moving backwards. The swing slows down.

Stand up. Stand on the seat and try pumping by bending your knees at the turning points and straightening up at the bottom. Pump twice per swing. Try starting from complete rest: it is almost impossible.

References

• W. B. Case & M. A. Swanson, ‘The pumping of a swing from the seated position’, Am. J. Phys. 58, 463 (1990).
• W. B. Case, ‘The pumping of a swing from the standing position’, Am. J. Phys. 64, 215 (1996).
• S. M. Curry, ‘How children swing’, Am. J. Phys. 44, 924 (1976).
• S. Wirkus, R. Rand & A. Ruina, ‘How to pump a swing’, College Math. J. 29, 266 (1998).
• A. A. Post, G. de Groot, A. Daffertshofer & P. J. Beek, ‘Pumping a playground swing’, Motor Control 11, 136 (2007).