# An introduction to simple harmonic motion

##### Class practical

This circus of qualitative experiments provides an introductory look at simple harmonic motion.

#### Apparatus and materials

Station A

retort stand, boss and clamp

pair of 5 cm wood or metal blocks as jaws

mass to use as pendulum bob, 1 kg

G-clamp

Station B

expendable spring

S-hook

mass hanger, 10 x 100 g

retort stand, boss and clamp

g-clamp

Station C

dynamics trolley

expendable springs, 2

g-clamps, 2

retort stands, 2

#### Health & Safety and Technical notes

Avoid large amplitude oscillations.

For large hanging masses put something on the floor to protect it should the mass fall.

Station A

Attach the 1 kg mass to one end of the thread. Clamp the other end of the thread between a pair of 5 cm metal strips which act as jaws, attached to a retort stand that is fixed rigidly to the bench.

Station B
The top of the spring is suspended from retort stands. To the lower end is attached a weight hanger with a total mass of about 400 g.

Station C
Clamp two retort stands to the bench about 60 cm apart. Using expendable springs, connect the trolley between the retort stands, as shown below.

#### Procedure

Observe the motion at each station, looking for common features.

Station A: Gently set the pendulum swinging.

Station B: Experiment with vertical oscillations.

Station C: Displace the trolley from its equilibrium position so that it oscillates between the stands.

#### Teaching notes

1 These experiments can give students a qualitative appreciation of a range of oscillators. Encourage them to use their own initiative to develop a description (graphical or otherwise) of the motion of an oscillator in its cycle. Careful work will provide the basis for discussions about the displacement, velocity and acceleration of the oscillator. You could also introduce the terms displacement, amplitude, period, frequency.

2 Features common to all harmonic oscillations are:

• each complete oscillation of a system takes the same time
• a force returns the system to its equilibrium position when displaced
• an inertia factor makes the system overshoot its equilibrium position when in motion

Some systems have a period of oscillation which depends on the mass. In many systems, the amplitude of oscillation decreases with time.

The link from acceleration of an oscillator to the force acting on the oscillator should nonetheless be stressed. Later modelling depends upon consideration of the changes in the force on an oscillator during its cycle.

If the acceleration of a body is directly proportional to its distance from a fixed point, and is always directed towards that point, the motion is simple harmonic.

3 Ask: How could you test whether the decay of amplitude is exponential? [Plot a graph of amplitude against time and test the graph for exponential decay by measuring the amplitude at equal intervals of time.] Ask: How could you make the amplitude decay more quickly? [by adding paper sails, by immersing the system in water, by increasing the mass].

4 At station B, the loaded spring continually changes its mode of oscillation between vertical and horizontal. The spring acts as the coupling agent between the vertical oscillation and a sideways pendulum motion. Energy is carried from one motion to the other and back again. The closer the two frequencies are to each other, the easier it is for the load, when moving with one motion, to excite the other motion.

You might ask: How could you prevent the interchange of motion? [Change the frequency by changing the mass so making one motion much faster or slower than the other. The period of vertical bouncing of the spring is equal to the period of a simple pendulum whose length is equal to the extension of the spring by the stationary load. If the experimenter is trying to measure the period of the vertical motion this is a very irritating phenomenon. The cure is to insert a considerable length of string between the lower end of the spring and the load. This does not change the forces involved in the vertical oscillation but the period of the pendulum motion is now much longer and transfer to that motion will happen much more slowly.]

5 The experiment at station C is a good example to study because gravity is not involved. The two essential factors that control S.H.M in all its forms are both visible and variable: the spring factor and the inertia factor.