# Episode 305: Energy in simple harmonic motion

Lesson for 16-19

- Activity time 95 minutes
- Level Advanced

Qualitatively, students will appreciate that there is a continuous change in the ways that energy is stored during simple harmonic motion (SHM). Here, they can also learn about the mathematical basis for calculating energy.

Lesson 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: Changes in the ways energy is stored in SHM (20 minutes)
- Student questions: Changes in the ways energy is stored 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 sine and cosine 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 =
*ω*×*A* - Maximum acceleration =
*ω*^{2}×*A*

Compare these relationships with the equations for circular motion:

- Displacement =
*r* - Velocity =
*ω*×*r* - Acceleration =
*ω*^{2}×*r*

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 changes in the ways energy is stored in a mechanical oscillator. Recap that, as it passes through its equilibrium position, its speed and hence its energy stored kinetically are a maximum. At the maximum displacement, the speed and hence the energy stored kinetically are both zero. The potential energy

will be a maximum when the speed is zero and vice versa. In a spring, the potential energy

is the energy stored elastically; in a pendulum it is energy stored gravitationally. Assuming that there is no friction or air drag the total energy *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, 12 *k**x*^{ 2}. The PE-extension graph is a parabola.

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

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

Episode 305-3: Energy stored in a stretched spring (Word, 79 KB)

Draw a graph from *x* from − *A* to + *A*
, to show energy stored kinetically, potential energy (energy stored elastically or gravitationally), 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)