Physics · Oscillations and waves · Grade 10

Swing

When is a swing pushed so that it goes higher?

The situation

At the playground you are pushing your little brother on a swing. Each time you push, he goes higher — but only if you push at the right moment. Push at the wrong moment and the swing slows down instead. A swing has its own “rhythm”. What does it depend on: the length of the chain, the child's weight, or how high he swings?

Topic

Mechanical oscillations, pendulum. A swing on a playground

Key ideas

Simple pendulum
Conservation of energy
Resonance

Formulas

\ddot{θ} + \frac{g}{L} θ = 0, T_{0} = 2 π \sqrt{\frac{L}{g}}

Equation and period of harmonic oscillations for small displacements — there is no mass in it!

T \approx T_{0} (1 + \frac{θ_{0}^{2}}{16})

At large amplitude the period becomes slightly longer ( $θ_{0}$ in radians).

E = \frac{m v^{2}}{2} + m g L (1 - \cos θ) = const

Without resistance, kinetic and potential energy change into each other.

F_{t} = m g \cos θ + \frac{m v^{2}}{L}

Tension in the chain: at the lowest point you feel heavier.

A_{i t} = \int F v d t > 0 ⟺ F ↑ ↑ v

A push adds energy only if the force acts in the direction of the velocity.

Theory

How do oscillations arise?

Mechanical oscillation is a motion that repeats around an equilibrium position. Two conditions are needed for it: when a body leaves equilibrium, a restoring force must appear, and the body must have inertia (it overshoots the equilibrium position); friction must also be small. A swing is almost a simple pendulum: a heavy load hangs on a long string, and the restoring force is the tangential component of the gravitational force, $m g \sin θ$ . For small angles $\sin θ \approx θ$ and the motion is harmonic. Types of oscillation: a swing given one push performs free (natural) oscillations; with resistance, damped oscillations; if it is pushed periodically, forced oscillations; a clock pendulum receives a push from its own source at the right moment — self-sustained oscillations. A load on a spring is in the “Spring pendulum” experiment, and the addition of oscillations in the “Coupled oscillations” experiment; when an oscillation spreads through a medium, a wave forms — see “Waves on a rope and a spring”.

Harmonic oscillation: period, frequency, phase

For small displacements, $θ = θ_{0} \cos (ω_{0} t + φ_{0})$ is the solution of the equation $\ddot{θ} = - ω_{0}^{2} θ$ . $θ_{0}$ is the amplitude, $T$ is the period (the time of one complete oscillation), $ν = 1 / T$ is the frequency (Hz), $ω_{0} = 2 π ν = \sqrt{g / L}$ is the angular frequency, and $ω_{0} t + φ_{0}$ is the phase (it shows which stage of the oscillation we are at; when released from the extreme position, $φ_{0} = 0$ ). The displacement along the arc is $s = L θ$ , the velocity is $v = - s_{0} ω_{0} \sin (ω_{0} t)$ , and the acceleration is $a = - s_{0} ω_{0}^{2} \cos (ω_{0} t) = - ω_{0}^{2} s$ : the velocity leads the displacement by a quarter of a period ( $90 °$ ), and the acceleration is opposite to the displacement ( $180 °$ ). Compare all three curves on the graph: at the extreme point $v = 0$ and $| a |$ is greatest; at the bottom $v$ is greatest and $a_{t} = 0$ .

A one-dimensional system and the physical pendulum

In any system described by a single coordinate $q$ , near the minimum of potential energy $U \approx U_{0} + \frac{1}{2} k q^{2}$ and the kinetic energy is $\frac{1}{2} m {\dot{q}}^{2}$ , so $\ddot{q} + ω^{2} q = 0$ with $ω^{2} = k / m$ — the free oscillations of a one-dimensional system are always harmonic (at small amplitude). For a pendulum $q = θ$ and $U = m g L (1 - \cos θ)$ : $k = m g L$ , the “mass” is $m L^{2}$ , and $ω^{2} = g / L$ . If the body is not a point, it is a physical pendulum: $T = 2 π \sqrt{I / (m g d)}$ , where $I$ is the moment of inertia about the axis of suspension and $d$ is the distance from the axis to the centre of mass. The child on a real swing is, in fact, also a physical pendulum; the laboratory physical (reversible) pendulum is in the laboratory mode of the “Spring pendulum” experiment.

Why does mass not matter?

The force pulling a heavier child down is also greater, but it is also harder to accelerate him — both are proportional to mass and cancel out. That is why an adult and a child on the same chain swing with the same period. The period depends only on $L$ and $g$ — so a pendulum can be used to **measure $g$ ** (lab work): measure the period at several lengths, pressing “Record measurement” each time; the graph of $T^{2}$ against $L$ is a straight line with gradient $4 π^{2} / g$ . Keep the amplitude below 10°: at large amplitude the period becomes longer and $g$ comes out too small.

Resonance

Each push adds a little energy. If the pushes come in rhythm with the swing's natural period, the energy builds up — this is resonance. If the rhythm is broken, some pushes work against the motion and take energy away.

Where does the energy go?

Air resistance and friction in the links of the chain turn part of the energy into heat with every oscillation. That is why a swing that is not pushed gradually comes to a stop.

Did you know?

According to legend, the young Galileo timed the swinging of a chandelier in Pisa Cathedral against his own pulse and noticed that the period hardly depends on the amplitude.
In 1851 Léon Foucault showed the rotation of the Earth in the Panthéon in Paris using a 28 kg pendulum hung on a 67 m wire.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Spring pendulum — Grade 10
Waves on a rope and a spring — Grade 10
Sound: speed, pitch, ultrasound — Grade 10
The dutar string and sound — Grade 11
Waves and the Doppler effect — Grade 11
Bridge resonance — Grade 11
Coupled oscillations — Grade 11
Nonlinear oscillations — Year 2
Fourier: the spectrum of sound — Year 3