### Episode 301: Recognising SHM

This episode allows you to familiarize your students with the main features of SHM, before going on to a more mathematical description.

Summary

• Demonstration: Observing SHM in the lab. (20 minutes)
• Discussion: The importance of SHM. (10 minutes)
• Demonstration: SHM is isochronous. (10 minutes)
• Student activity: Identifying SHM. (20 minutes)
• Demonstration: Sinusoidal graphs on a CRO. (10 minutes)

Demonstration: Observing SHM in the lab
Start by talking about oscillations. Oscillatory motion is a type of motion that repeats itself in a cyclic fashion.

Set up some visual aids: a pendulum and a mass on a spring. Set them oscillating.

It is easier to see what is going on with a slower oscillation. Tether a dynamics trolley by springs between two retort stands. Add weights to the trolley, so that its period of oscillation is a couple of seconds. Set it oscillating; you will be able to hear the trolley as it speeds up and slows down. If you use a motion sensor connected to a computer it is easy to see the displacement time graph and velocity time graph. You could do the same for a long pendulum.

Set up an oscilloscope with its timebase switched off, so that the spot is central on the screen. Connect to a very slow signal generator, set at a frequency of less than 1 Hz. Watch the up-and-down motion of the spot on the screen. Ask about the shared characteristics of these different oscillations. Points to bring out:

The oscillatory motion repeats itself and thus has similarities with circular motion which also repeatedly returns to its starting point. (This fact will be fruitfully exploited later.)

• Displacement is the greatest at the extremes of the oscillation.
• Velocity is the greatest at the midpoint.
• The oscillating mass is in equilibrium at the midpoint of its oscillation; show this by stopping the mass at this point – it remains stationary.
• When the mass is oscillating, its inertia carries it through the midpoint.

So we have the following pattern of motion:

The mass speeds up as it heads towards the midpoint, has its greatest speed as it passes through the midpoint, and slows down as it continues towards the other extreme of the oscillation. Here, it reverses and starts to accelerate again towards the midpoint.

Of course, not all oscillations are as simple as this, but this is a particularly simple kind, known as simple harmonic motion (SHM). It is relatively easy to analyze mathematically, and many other types of oscillatory motion can be broken down into a combination of SHMs.

Discussion: SHM is isochronous
At this point, it is worth saying a few words about the importance of SHM.

All clocks (starting with the pendulum in a grandfather clock) are essentially simple harmonic oscillators; all transmitters and receivers of waves are essentially simple harmonic oscillators; many aspects of engineering design from the massive to the microscopic require a detailed knowledge of SHM (e.g. bridges, earthquake protection of buildings, atomic force microscopes for imaging single atoms). Detailed theories of the behaviour of atoms and molecules (in solids and gases) are applications of SHM. Other aspects of SHM are at the heart of unsolved questions in physics today. Thus although the model oscillators that students meet appear rather basic, they mirror pretty well applications and problems far beyond the school laboratory.

The key aim is to get across how to recognize SHM experimentally as well as the features that are built into the theoretical model of SHM.

Demonstration: Identifying SHM
Using the apparatus of the earlier demonstration, set one oscillator in motion. Now change the amplitude. What do you notice about the time period of the oscillation? (It doesn’t change.) This is obvious if a motion sensor is used.

Try with the others.

This is a characteristic of SHM; the period is independent of the amplitude, and we say that the motion is isochronic.

Student activity: Identifying SHM
Your students can try out some other oscillating systems; are they isochronic? Do they appear to be SHM?

Set up a circus of experiments (see below). Students in pairs try out a selection and report back to a plenary session.

 TAP 301-1: Oscillation circusMass on a spring (vertical)Mass (large cube of polystyrene) on the end of a slinky spring suspended from the ceilingMass between two springs (vertical, both springs in tension when the mass is at rest)Mass between two springs (horizontal, both springs in tension when the mass is at rest – use an air track slider for the mass to have a low friction system)Air track slider moving between two ‘buffer’ springs, or rebounding due to magnetic repulsionVibrating cantileverSimple pendulumSimple ‘half’ pendulum (one that bounces off a hard surface when its string is vertical)

TAP 301-2: Mass oscillating between elastic barriers

TAP 301-3: Oscillating ball

• Simple pendulum whose string intercepts a peg when vertical, so the length of the pendulum gets shorter for one half of its cycle
• A torsion pendulum

TAP 301-4: Swinging bar or torsion pendulum

• A ball bouncing off a hard surface
• Ball moving in a semi-circular shaped track (curtain rail)
• Ball moving in a vertical V shaped track (rounded enough at the point of the V to let the ball pass easily)
• Ball moving in a vertical parabolic shaped track (draw out the parabola on a large piece of paper to aid the bending of the track into the parabolic shape)
• A right circular cone on an inclined plane
• A rectangular or square section bar balanced on top of a cylinder – the length of the bar at right angles to the axis of the cylinder
• Water in a U tube
• Hydrometer in water

Demonstration: Sinusoidal graphs on a CRO
Set one or more of the masses oscillating. Ask: what would a displacement-time graph for this look like? You can show the graph using the oscilloscope. Switch on the time base to see the sinusoidal graph as it is traced out. Emphasize that the graph produced is ‘a sine wave’ or ‘a sinusoidal variation’.  Alternatively show the motion with a computer and a motion sensor. A pendulum with about a 2 m string and a suitably large bob, (large coffee can with some sand) works well. The Trolley and spring system works well for slow oscillations. A card may be needed on the trolley so it is picked up correctly by the sensor. Be safe: when fixing a string to a high point, two persons are needed – one to hold the ladder or steps and one to do the job.

(If your signal generator will also generate triangular and square waves, show the oscillations these produce with the timebase off. Ask for predictions of the displacement-time graphs.)

A more challenging question, which you may wish to leave until later: What would a velocity-time graph look like? (Also sinusoidal, but maximum value when displacement = 0.) And acceleration? (Sinusoidal, but 180° out of phase with displacement.)