# Episode 305: Energy in simple harmonic motion

Qualitatively, students will appreciate that there is a continuous interchange between potential and kinetic energy during simple harmonic motion (SHM). Here, they can also learn about the mathematical basis for calculating energy.

**Summary**

- Demonstration: An experimental displacement-time graph (10 minutes)
- Discussion: Maximum values of quantities in SHM (15 minutes)
- Student questions: Practice with the equations (30 minutes)
- Discussion: Energy changes in SHM (20 minutes)
- Student questions: Energy of a pendulum (20 minutes)

**Demonstration: An experimental displacement-time graph**

Use a ‘water pendulum’ to draw a large displacement-time graph for a pendulum. You could ask a group of students to prepare this in advance and demonstrate it to the class.

Episode 305-1: The water pendulum (Word, 36 KB)

**Discussion: Maximum values of quantities in SHM**

Refer back to the sin and cos equations for SHM. Show that the maximum values of displacement, velocity and acceleration are given by (the term in front of sin or cos):

- Maximum displacement = A
- Maximum velocity = Aw
- Maximum acceleration = Aw
^{2} - Compare these relationships with the equations for circular motion:
- Displacement = r
- Velocity = rw
- Acceleration = rw
^{2}

If you have adopted the ‘auxiliary circle’ approach earlier, the parallels should be clear.

**Student questions: Practice with the equations**

It will help to provide some more practice in using the equations and analyzing motion.

Episode 305-2: Oscillators (Word, 70 KB)

**Discussion: Energy changes in SHM**

Think about the energy changes in a mechanical oscillator. Recap that, as it passes through its equilibrium position, its speed and hence its kinetic energy are a maximum. At the maximum displacement, the speed and hence the kinetic energy are both zero. The potential energy will be a maximum when the speed is zero and vice versa. Assuming that there is no friction or air drag the total energy input *E* of the oscillator must remain constant

For the mass and spring system, the work done stretching a spring by an amount *x* is the area under the force extension graph = 1/2 *kx*^{2}. The PE-extension graph is a parabola.

The kinetic energy will be zero at +*A* and a maximum when *x* = 0, so its graph is an inverted version of the strain energy graph. At any position kinetic + elastic strain energy is a constant *E*, where *E* = KE_{max} = PE_{max}.

PE_{max} µ *A*^{2}, so the total energy *E* of SHM is proportional to (amplitude) ^{2}.

Episode 305-3: Elastic energy (Word, 79 KB)

Draw a graph from *x* = {–*A* to +*A}*, to show kinetic energy, strain energy and total energy. You can also draw graphs of KE and PE against time.

Episode 305-4: Energy flow in an oscillator (Word, 71 KB)

**Student questions: Energy of a pendulum**

Some useful questions on energy of a pendulum.

Episode 305-5: Energy and pendulums (Word, 54 KB)

**Download this episode**

Episode 305: Energy in SHM (Word, 273 KB)