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### Simple harmonic motion

- An introduction to simple harmonic motion
- Examples of simple harmonic motion
- The velocity of a very long pendulum
- Investigating a mass-on-spring oscillator
- Datalogging S.H.M. of a mass on a spring
- S.H.M. with a cantilever
- S.H.M. and circular motion
- Broomstick pendulum, sinusoidal motion
- S.H.M. on a rope
- Musical frequencies shown on a C.R.O.
- Vibrating tuning fork and rotating mirror
- Lath with a load
- Straight line graphs
- Simple pendulum experiments
- Oscillations, waves and mathematical models

## Simple harmonic motion

for 14-16

Oscillations with a particular pattern of speeds and accelerations occur commonly in nature and in human artefacts. They also happen in musical instruments making very pure musical notes, and so they are called 'simple harmonic motion', or S.H.M.

These experiments are suitable for students at an advanced level of study. For introductory and intermediate levels, see the related collection Oscillations.

**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
- Length of thread
- Pair of 5 cm wood or metal blocks as jaws
- Mass to use as pendulum bob, 1 kg

**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.

Read our standard health & safety guidance

**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

- 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.
- 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.
- 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]. - 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.] - 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.

### Up next

### Examples of simple harmonic motion

**Class practical**

A circus with many different examples of simple harmonic motion.

Apparatus and Materials

**Station A: Simple Pendulum**

- stands, 3
- clamps, 3
- bosses, 3
- G-clamps, 3
- pairs of 5 cm wood or metal blocks as jaws, 3
- pendulum bobs, 3, on strings with lengths in ratio 1:2:4

**Station B: Torsional Pendulum**

- pair of 5 cm wood or metal blocks as jaws
- stand, clamp and boss
- G-clamp
- short wooden rod
- wire, Eureka, effective length 50 cm, 26 SWG

**Station C: Vibrating Lath**

- G-clamp
- metre rule
- boss

**Station D: Oscillating water column**

- U-tube of water 2.5 m, filled halfway up
- disposable mouthpieces, to protect hygiene or use a simple puffer bottle to start the oscillations

**Station E: Rolling ball**

- steel ball or marble
- bowl, shallow and spherical

**Station F: Wig-wag**

- wig-wag, with 3 removable masses
- G-clamps, 2

**Station G**

- curtain rail, 60 cm length, 3 (circular shape, parabolic shape & V-shape) mounted on an appropriate board
- safety screen holders, 2
- steel ball or marble

**Station H: Undamped light beam galvanometer**

- light beam galvanometer
- cell holder with one cell
- switch
- resistance substitution box
- leads

Health & Safety and Technical Notes

Read our standard health & safety guidance

At station B, the rod used for the torsional pendulum must be balanced. Two rods fastened together with elastic bands or a shorter length of wire may also be tried.

At station C, provide a second boss-head so that students can investigate the effect of increasing the load. The position of the boss head along the length of the metre rule could also be varied.

At station D, have students alter the water levels by blowing into the tube or use a simple puffer bottle. The water will then perform damped harmonic motion. Obtaining a time trace is not easy, since the period is short and damping is high. One possibility would be using a light beam and a scalar timer, with repeated timings.

At station G , set up the board leaning backwards a little, at about 10° to 15°.

At station H, set up the circuit with the galvanometer on its least sensitive scale; then increase the sensitivity until, with a resistance of over 500 kΩ, the spot reaches almost a full-scale deflection with the switch closed. Then, with the galvanometer on its direct

setting, open the switch: the spot will oscillate about its central zero position.

Procedure

At each station, displace the system from its equilibrium position and carefully observe what happens. Listen to the differences if a sound is made.

Teaching Notes

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 introduce the terms displacement, amplitude, period, frequency.

**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.

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.

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 on the oscillator is obvious but should nonetheless be stressed as later modelling depends upon consideration of the changes in the force on an oscillator during its cycle.

**Expected results for some of the stations:**

- A: The periodic time,
*T*, depends on the length,*l*. (The motion is isochronous.)*T*∝*l*^{½}. - C: This behaves like a very large ticker-timer blade.
- D: The motion is damped by fluid friction but is clearly isochronous. Ask students whether period be the same if a denser liquid is used. The force tending to return the liquid to its equilibrium position will be
*rg*D*hA*, where*r*is the fluid density,*g*is gravitational field strength, D*h*is the difference in liquid column heights, and*A*is the column cross-section. - E: Listen to the sound: what does this tell you about the motion? The amplitude decreases but the frequency remains unaltered.
- F: Load the end of the wig-wag with a variable number of masses so that it oscillates sideways. Note the affect of mass on the time for one oscillation.
- G: Listen to the sound the ball makes as it rolls or slides along the tracks. The circular track will give what sounds like an isochronous motion; the parabolic track gives a frequency that increases as the amplitude decreases; the V-shaped track is not isochronous at all.
- H: A time trace for one oscillation can be obtained by photography, using a multi-slit stroboscope. Students could also record how the amplitude dies away, and isochronous property of the oscillations.

### Up next

### The velocity of a very long pendulum

**Demonstration**

Measuring the velocity of a pendulum at different points along its swing to introduce discussions of acceleration and force.

Apparatus and Materials

- Massive pendulum
- Electronic scaler/timer
- Light gate, or photo-diode assembly and light souce

Health & Safety and Technical Notes

The bob should be close to the floor or bench when at rest with a suitable cushion to catch it should the wire slip or break.

As the wire is under tension, it would be prudent to wear safety spectacles.

Read our standard health & safety guidance

The pendulum: It’s best to use a massive pendulum bob such as a large brick or a 3 kg mass, but you could use a 5 cm steel sphere. To avoid timing difficulties caused by rotation, the bob should be a large sphere or should have a cardboard cylinder fixed round it like a collar. The bob should be suspended by steel wire and have a very rigid support, if possible the ceiling so that it is as long as possible.

If you use a photo-diode assembly, connect it to the red terminals of the scaler/ timer. The pre-focused bulb illuminates the photo-diode.

Procedure

- Position the pendulum and light gate so that when the bob is at its lowest point it passes through the gate. Connect the light gate to the timer.
- Draw the massive pendulum to one side. Release it to swing through the light gate. The timer will record the time taken, enabling its velocity to be calculated.
- Reposition the light gate and time the bob at several places in the swing. Calculate its velocity at each position.

Teaching Notes

- This experiment is only suited to more able students. Begin by asking:
*"is the acceleration of a pendulum constant?"* - Help your students to appreciate that the acceleration is greatest where the velocity can only increase, at the top of the swing (here the velocity is zero). The acceleration is zero where the velocity cannot increase any more, at the bottom of the swing (here the velocity is a maximum).
- To follow this discussion, students need to clarify in their own minds the relations between bob position, velocity and acceleration. Weaker students will easily become confused.
- 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.
- Sketch (or plot) a graph of velocity against time or displacement for a few oscillations. With more advanced students, you may want to compare a graph of velocity against time with a graph of displacement against time, to introduce the idea of a phase difference.
- The force causing the acceleration is always directed inwards towards the centre of the swing. It is the vector sum of the weight of the bob and the tension in the string.

### Up next

### Investigating a mass-on-spring oscillator

**Demonstration**

A mass suspended on a spring will oscillate after being displaced. The period of oscillation is affected by the amount of mass and the stiffness of the spring. This experiment allows the period, displacement, velocity and acceleration to be investigated by datalogging the output from a motion sensor. It is an example of simple harmonic motion.

Apparatus and Materials

- Motion sensor, interface and computer
- Slotted masses on holder, 100 g-400 g
- Clamp and stand
- String
- Springs, 3
- Card

Health & Safety and Technical Notes

Unless the stand is very heavy, use a G-clamp to anchor it to the bench.

Read our standard health & safety guidance

Suspend the spring from a clamp and attach a mass to the free end. Adjust the height of the clamp so that the mass is about 30 cm above the motion sensor, which faces upwards.

The clarity of measurements depends upon the choice of spring stiffness and mass. Good results can be obtained with three springs linked in series, and masses in the range 100 to 400 g. With this choice, it is necessary to place the sensor on the floor and allow the mass and spring to overhang the edge of the bench.

When the mass is displaced and released, its vertical motion is monitored by a motion sensor connected via an interface to a computer. In general, the magnitude of the initial displacement should not exceed the extension of the spring. It is best to lift the mass to displace it, rather than pull it down.

The mass may acquire a pendulum type of motion from side to side. Eliminate this by suspending the spring from a piece of string up to 30 cm.

For the collection and analysis of the data, data-logging software is required to run on the computer. Configure the program to measure the distance of the mass from the sensor, and to present the results as a graph of distance against time. Scale the vertical axis of the graph to match the amplitude of oscillation.

Procedure

- Lift and release a 400 g mass to start the oscillation. Start the data-logging software and observe the graph for about 10 seconds.
- Before the oscillation dies away, restart the data-logging software and collect another set of data, which can be overlaid on the first set.
- Repeat the experiment with 300 g, 200 g and 100 g masses.
- The period of the sinusoidal graph may be measured using a time-interval analysis tool in the software. Measure the period from peak to peak.
- Take measurements at several different places on the time axis, and observe that the period does not vary with elapsed time.
- Take similar measurements on the set of results with a smaller amplitude, and observe that the period appears to be independent of amplitude.
- Measure the period for each of the other graphs resulting from using different masses. Plot a new graph of period against mass. (Y axis: period; X axis: mass.)
- Use a curve-matching tool to identify the algebraic form of the relationship. This is usually of the form 'period is proportional to the square root of mass'.
- Use the program to calculate a new column of data representing the square of the period. Plot this against mass on a new graph. A straight line is the usual result, showing that the period squared is proportional to the mass.
- On the 'distance vs. time' graph, the gradient at any point represents the velocity of the oscillating mass. Choose the clearest set of data and use the program to calculate the gradient at every point on the graph.
- A plot of the resulting data shows a 'velocity vs. time' graph. Note that the new graph is also sinusoidal. However, compared with the 'distance vs. time' graph, there is a phase difference - the velocity is a maximum when the displacement is zero, and vice versa.
- A similar gradient calculation based on the 'velocity vs. time' graph yields an 'acceleration vs. time' graph. Comparing this with the original 'distance vs. time' graph shows a phase difference of 180°. This indicates that the acceleration is always opposite in direction to the displacement.

**Data collection**

**Analysis**

*Measurement of period*

*Effect of mass*

*Velocity and acceleration*

Teaching Notes

- This experiment illustrates the value of rapid collection and display of data in assisting thinking about the phenomenon under investigation. Data is collected within a few seconds and the graph is presented simultaneously. Students can observe connections between features on the graph and the actual motion of the mass. For example, the crests and troughs on the graph represent the mass at the extremes of its displacement.
- The parameters suggested here usually produce displacements of a few centimetres. The motion sensors can detect these with suitable precision. Small amplitude oscillations produce rather noisy data. Starting with the largest mass, shows the clearest results first.
- Software tools for taking readings from the graph are employed: measuring gradients and time intervals. The detail available in the data allows the idea that the periodic time remains constant for a given mass to be tested.
- A particularly useful software function is that which calculates the velocity for all points on the graph and plots these as a new graph. A notable feature of the velocity graph is the phase difference from the distance data. This can provoke useful discussion about the change in magnitude and direction of velocity during each cycle of oscillation. The
noisiness

of the measurements begins to show more markedly on the velocity graph. The process by which the program calculates the velocity (usually by taking differences between distance readings) should be questioned. - The further derivation of acceleration from the gradients of the velocity graph usually shows even more measurement noise. Nevertheless, the form of the graph convincingly shows the antiphase relation with the distance graph. This is useful for prompting discussion about the conditions for simple harmonic motion (SHM). This can be reinforced by plotting a further graph of acceleration against displacement. The negative gradient straight line supports the basic condition for SHM: acceleration is proportional to displacement, but in the opposite direction.
- You could add a card to the bottom of the masses to increase the damping. Students can see if the presence or amount of damping affect the natural frequency. Secondly, the amplitude can be extracted from each peak and a damping curve plotted. This can be tested to see if it is exponential.

**Additional activities**

*This experiment was safety-checked in May 2006*

### Up next

### Datalogging S.H.M. of a mass on a spring

**Demonstration and Class practical**

To demonstrate S.H.M. of a mass on a spring and gather accurate data using a datalogger.

Apparatus and Materials

- Datalogger position sensor
- Slotted 50 g masses and hanger
- Expendable spring
- Card

Health & Safety and Technical Notes

Masses hanging from ceiling.

Read our standard health & safety guidance

Be careful to set the mass moving only vertically, not swinging side to side. Also many position sensors do not work if the object gets too close: be careful to maintain at least the minimum working distance at all times.

Procedure

- Attach an expendable spring to the ceiling or a very high retort stand and hang a 50 g slotted mass hanger from it. Place a motion sensor underneath, pointing upwards. Displace the mass a small distance downwards. Position-time data can be recorded swiftly and easily.
- The experiment can be repeated using different numbers of masses, springs in series, and adding card to the bottom of the masses to increase the damping.

Teaching Notes

- If you have motion sensors this is much easier to set up than a pendulum attached to a rotary potentiometer.
- This can work at every level. Initially data can be recorded over a few cycles and used to find the period and hence the frequency. The effect of changing the mass, springs and damping is quickly measured and compared directly to theory.
- Position-time data can be analyzed to see if it is sinusoidal and if the period is constant as the amplitude damps down. Also, period being independent of initial amplitude can be checked.
- Velocity and acceleration-time graphs can be plotted by the computer and shown to have the same shape and period but different phase to the position-time graph.
- The damping can be investigated in various ways. Initially students can look and see if the presence or amount of damping affects the natural frequency. Secondly the amplitude can be extracted from each peak and a damping curve plotted. This can be tested to see if it is exponential.
- As the data is on a computer it can be exported to a spreadsheet or mathematics package and fitted to a sinusoidal form (with exponential damping if appropriate).

#### How Science Works Extension

As well as illustrating the fundamental relationships of SHM (displacement, velocity, acceleration, time etc.), this experiment can be used as the basis of several open-ended investigations. Questions to tackle include:

- How does the period of oscillation T depend on the mass m, spring constant k and amplitudeA? (Note that, to double the spring constant, connect two springs side-by-side; to halve it, connect two end-to-end.)
- How does the amplitude of damped oscillations ‘decay’?
- Does damping affect the period of oscillation?

The mathematical relationship involved is T = 2 π√ m/k and this gives the opportunity to discuss how to choose appropriate axes to obtain a straight line graph. T 2 is proportional to m and to 1/k, so you can introduce the idea of adding columns to results tables to calculate relevant quantities. You could also extend this to consider the quantities which can be deduced from the gradients of graphs.

A graph of T 2 against 1/k has a gradient of 4π2 m; students can deduce m and compare it with the value of the mass they have used in the experiment. (Note that they may have to add the mass of the spring.)

You could point out that this is a method of determining mass which does not require gravity; fix a mass between two horizontal springs so that it oscillates from side to side, and deduce its mass from the period of oscillation. That would be useful in an orbiting spacecraft; it can also introduce the idea of inertial mass as opposed to gravitational mass.

The ‘decay’ of damped oscillations can be tested in two ways. Firstly, look for a ‘half-life’; does the decay of amplitude always halve over the same time interval? Next, plot a log-linear graph and show that it is a straight line. Points to make: a straight line graph makes use of all the data, rather than just selected points; and it will show up more clearly any errant data points.

### Up next

### S.H.M. with a cantilever

**Class practical**

This experiment could extend (or replace) the traditional pendulum or mass-on-a-spring experiments illustrating S.H.M.

Apparatus and Materials

- Metre rules
- G-clamps
- Slotted masses (100 g each)
- Sellotape
- Stopwatches
- Small, rough wooden blocks

Health & Safety and Technical Notes

It might be best advised to wear goggles in case something snaps.

Don't stand with toes underneath the slotted masses.

Read our standard health & safety guidance

Procedure

- G-clamp the metre rule securely to the bench using the wooden blocks to protect the rule and bench.
- Sellotape one or more slotted masses near the end of the rule.
- Twang and time several oscillations.
- Adjust the vibrating length (or mass attached) and repeat.

Teaching Notes

- You could use this experiment as a follow-up to the standard "g from a pendulum". There are more variables to play with so you can easily set up differentiated tasks for your students.
- In this case we can find
*E*for wood because for this cantilever we have ...
ω *x*= width of ruler*y*= thickness of ruler (scale to undersurface)*M*= mass Sellotaped on*L*= vibrating length*E*= Young modulus of wood*ω*= 2π/T- So T
^{2}v*M*(or*L*^{ 3}) gives you*E*from the gradient. - With a wide range of abilities you can have one group simply verifying it's S.H.M. (by proving -
*T*is independent of amplitude), another determining*E*, and another using log graphs to discover that*T*is proportional to -*L*^{3/2}. It's also a good one for error analysis; which term contributes the largest error in*E*(answers on a postcard)? - The vibrations are quite fast (especially at short lengths). To obtain an accurate result for
*T*, time many oscillations and find the average time for a single oscillation. - If you have the materials you can try things other than metre rules.

^{2}=

Exy

^{3}/ 4

*ML*

^{3}where

*Thank you to Wayne Morton for pointing out that there was an error in the formula that we previously printed.*

*This experiment was submitted by Jason Welch who is Head of Physics at County High School, Leftwich, Cheshire.*

### Up next

### S.H.M. and circular motion

**Demonstration**

This experiment leads towards a quantitative approach to simple harmonic motion. It shows that S.H.M., treated mathematically, is a projection of the circular motion.

Apparatus and Materials

- Turntable, with sphere attachment (see diagrams)
- Expendable springs, 2
- Mass, 0.5 kg
- S-hook
- Simple pendulum
- Retort stands, bosses and clamps, 2
- Compact light source (12 V tungsten halogen lamp), with suitable power supply
- Translucent screen
- G-clamps, 10 cm, 2
- Fractional horsepower motor, with gearbox
- Power supply for motor, low voltage, variable
- Battery, 12 V (OPTIONAL)

Health & Safety and Technical Notes

Do not use an unfiltered halogen lamp. It must have a suitable filter to reduce the U.V. emissions.

Read our standard health & safety guidance

Setting up the apparatus initially:

Join together two expendable springs and suspend them from a retort stand. Using an S-hook, attach the mass to the lower end.

Attach the turntable to a retort stand so that its plane is vertical. Connect the sphere attachment to the turntable and drive the turntable using the motor.

Put the compact light source at least 1 metre behind the mass and the sphere, so that their two shadows are projected on the screen at least 1 metre farther away. Connect the light source to its power supply.

Connect the field and armature windings of the motor in parallel to the DC terminals of the variable power supply. If fine adjustment of the motor speed is difficult, it may be helpful to connect the field terminals to the 12 volt battery and only connect the armature winding to the power supply.

It is probably easiest to get synchronization by holding the mass on the pendulum at the limit of its oscillation and releasing it as the sphere comes to the same position. If the speed of the motor is very wrong, alter the speed and try again. If the speed of the motor is slightly wrong, try chasing the shadow of the mass by altering the speed, or change the load slightly.

See the following website for a supplier of:

Procedure

- Switch on the motor. Set the spring oscillating vertically and adjust the speed of the turntable so that the shadows of the mass and sphere synchronize.
- Alter the turntable so that it now rotates in a horizontal plane. Support a simple pendulum about 1 metre long so that its shadow is just above the shadow of the sphere.
- Set the pendulum oscillating with an amplitude of about 5 cm and switch on the motor so that the sphere rotates on the turntable. Adjust the speed of the motor so that the two shadows move together.

Teaching Notes

- The main purpose of this experiment is to show that simple harmonic motion is the projection of a circular motion. As the shadow motions show, circular motion when viewed from the side exactly matches a simple harmonic oscillator.
- The experiment can be used to introduce or illustrate ideas of phase, phase difference and angular velocity.
- If the shadows move together exactly ‘in step’, they are said to be in phase. If the oscillator is released at some other instant, there will be a constant time interval between one shadow reaching the outermost limit of its swing and other reaching that position. The fraction of a complete oscillation by which one is ahead of the other is known as the phase difference. This can be expressed as a fraction of a revolution or oscillation or, more usually, as an angle. Such an angle is usually measured in radians.
- It may be worth projecting the shadow of two bobs fixed to the rotating turntable (without an accompanying oscillator) to emphasize how difference in
angle

between the positions of the two bobs corresponds to a difference ofphase

between the two oscillating shadows. - If the oscillator has a different period than the circular motion, then the movements will not say in step; the phase difference will vary continuously.
- Angular velocity, w, is the change in angle per unit time. It is usually measured in radians per second, rather than degrees per second. Angular velocity,
*ω = v / R*where*v*is the (orbital) speed along the circumference and*R*is the radius of the circle. - The experiment can also be used to introduce the relationship displacement,
*x*=*A cosωt*, where*ω*is angular velocity and*t*is time. - The variation of displacement of a harmonic oscillator with time is sinusoidal, having the general form
*s = A*sin (2π*ft*+*ø*) where*ø*is a phase angle. - The expressions
*s*=*A*sin (2π*ft*) and*s = A*cos (2π*ft*) are often convenient. - There are fixed phase relationships between the variations of displacement, force, acceleration and velocity. In particular, there is a phase difference of π/2 between displacement and velocity, and between velocity and acceleration.

### Up next

### Broomstick pendulum, sinusoidal motion

**Demonstration**

To show that simple harmonic motion (S.H.M.) is sinusoidal.

Apparatus and Materials

- Broomstick pendulum
- Paper, roll of
- Paint brush
- Cathode ray oscilloscope, or PC-based digital oscilloscope
- Power supply, 12 V AC

Health & Safety and Technical Notes

Read our standard health & safety guidance

Set up the massive broomstick pendulum as in the diagram (left). Attach to the lower end of the broomstick a paint brush dipped in ink. This will produce an inked line as a roll of paper passes beneath it.

Alternatively, you could make the broomstick pendulum produce a fine line of sand (diagram right). Attach a plastic cup to the broomstick, using a drawing-pin. Fill the cup with fine dry sand. In the bottom of the cup is a hole (2-3 mm diameter) through which the dry sand flows onto the paper, as it is pulled across the floor.

An oscilloscope with a long persistence screen is an advantage but not essential.

Procedure

- Set the pendulum swinging and pull the paper steadily across the floor, under the pendulum, so that the brush makes a sinusoidal trace on the paper. (Pull the paper in a direction at right angles to the plane of oscillation.)
- Pull the paper faster, to show the sine wave spreads out more, in the same way as the trace on an oscilloscope does when you increase the timebase.
- Use the oscilloscope to demonstrate the waveform of the alternating voltage, first with the timebase switched off and then with the spot running across at constant speed. Compare it with the pendulum trace.
- Increase the speed of the timebase, drawing attention to the way that the waveform changes.

Teaching Notes

- Describe the line left by the oscillator as a
time trace

. The main points of this demonstration are that: - A simple harmonic motion produces a time trace with a sinusoidal shape
- Both the broomstick pendulum and the voltages from AC power supplies are simple harmonic motions
- The shape of the wave produced by a simple harmonic motion will depend on the timebase setting (speed that the paper is pulled)
- As the broomstick pendulum produces a time trace, point out
- The changing displacement of the pen/cup
- The amplitude of the oscillation (which gradually gets smaller)
- Once the time trace is complete, point out that the period (periodic time) remains constant. The frequency (inverse of periodic time) is also constant.
- You could go on to discuss the fact that isochronous oscillators make good clocks.

### Up next

### S.H.M. on a rope

**Class practical**

To show that the wave train on a rope has a sinusoidal shape.

Apparatus and Materials

- Rope, 6 m (or rubber tube)

Health & Safety and Technical Notes

Read our standard health & safety guidance

It is best if the rope is long enough for the wave to die out considerably by the time it reaches the far end, so that reflections do not cause standing waves. One should then see a continuous train of waves travelling along the rope.

Procedure

- Stretch a rope on the floor.
- Oscillate one end of it at about 4 Hz by hand.

Teaching Notes

- The oscillation that drives the motion of the rope is a S.H.M.
- This simple experiment can be used to link the experiment with and to link S.H.M. with wave behaviour.

### Up next

### Musical frequencies shown on a C.R.O.

**Class practical**

To show the similarities between S.H.M. and other oscillating systems.

Apparatus and Materials

*For each pair of students*

- Microphone
- Tuning fork
- Small rubber hammer or large rubber bung
- Power supply, low voltage, AC
- Bicycle dynamo
- Simple AC generator

*For a several related optional demonstrations...*

See the experiment The motor as a dynamo.

Health & Safety and Technical Notes

Read our standard health & safety guidance

Students unfamiliar with oscilloscopes may need to refer to the Apparatus note:

Procedure

- Connect the microphone to the input terminals of the oscilloscope. Set the oscilloscope to maximum gain. Set the time-base to a middle value.
- Hit the tuning fork gently with a small rubber hammer or strike it on a large rubber bung. [Striking the tuning fork on the bench will damage both the bench and the tuning fork.]
- Touch the base of the vibrating tuning fork to the microphone case. Adjust the time-base control to give a good display.
- Try singing and whistling across the microphone.

Teaching Notes

- Students should previously have seen the time trace produced by an oscillating pendulum in the experiment:
- Students may be able to conclude, from the fact that the waveform in step 3 s a sine wave, that the oscillation of the tuning fork is a S.H.M. However, there are two intervening processes: first the sound wave is converted into an equivalent electrical signal, then that signal is amplified and displayed on the oscilloscope screen. Those who find this analysis difficult to follow may prefer a direct demonstration of the tuning fork motion, in the experiment:
- Singing and whistling are likely to produce harmonics, so the waveform will not be so
clean

, though there will be discernable main frequency. - You could go on to show that several devices produce a similar trace on the oscilloscope: the alternating voltage from a transformer (AC power supply), a bicycle generator, and simple AC generator. In each case, the driver follows the pattern of a S.H.M.

### Up next

### Vibrating tuning fork and rotating mirror

**Demonstration**

It is worth the effort of setting up this demonstration to project the image of the oscillating tuning fork.

Apparatus and Materials

- Small piece of mirror (5 mm x 2 mm)
- Tuning fork (256 Hz or more)
- Small piece of a thin plane mirror, or aluminized Melinex polyester film (gauge 25) (see Technical notes)
- Motor, suitable to drive the rotating mirror
- Contact adhesive (for example, Evo-stik)
- Compact light source
- Converging lens (+ 7D)
- Screen
- Power supply, low voltage, variable

Health & Safety and Technical Notes

Put a guard around the mirror so it cannot fly off and hit observers.

Read our standard health & safety guidance

A tuning fork with a mirror attached to it does not make a beam of light oscillate enough to show clearly. The motion can be made visible by attaching the mirror to the tuning fork arm. The mirror is then driven by the fork with much the same motion, but larger, and it will deflect a small beam of light with considerable motion. The rotating mirror sweeps the light across a screen.

To set up the apparatus: Carefully secure the small piece of mirror to one limb of the tuning fork [alternatively, secure a small strip of aluminized Melinex to the fork, to act a mirror.]

Firmly clamp the fork in the vertical position.

Using the lens, converge light from the lamp filament to an image on the screen about 2 m away, and then place the tuning fork mirror in this beam about a metre from the lens.

Mount a single plane mirror, so that it can rotate about a vertical axis when driven by the fractional horsepower motor, as near as possible to the fork where it can intercept the reflected beam. The image is received on a screen about a metre away.

Alignment is tricky and needs some playing around with: persevere, possibly setting up one mirror at a time.

Procedure

- Drive the motor at about 600 rotations per minute. The light spot will be seen to travel across the screen.
- Pluck the fork vigorously and the light spot will trace a sinusoidal wave form. You may need to adjust the frequency of the rotating mirror so that the trace is repeated in the same position on each rotation.

Teaching Notes

- The experiment shows that a vibrating tuning fork oscillates with a S.H.M. Students should previously have seen the time trace produced by an oscillating pendulum
- This experiment is similar to... ...but demonstrates the nature of a tuning fork’s motion directly. Though the other one is much easier to set up, students may find this experiment easier to understand. The two experiments could be demonstrated one after the other.

### Up next

### Lath with a load

**Demonstration**

A demonstration experiment that makes a time trace of the oscillation and illustrates damping.

Apparatus and Materials

- Either metre rule or long lath
- G-clamps, large, 2
- clean smooth paper (e.g. roll of wallpaper or computer print-out paper)
- Felt-tip pen or brush with ink
- Rubber bands, 2
- Masses, 0.5kg or 1 kg, 2

Health & Safety and Technical Notes

Read our standard health & safety guidance

Clamp the metre rule so that it oscillates in a horizontal plane. The diagram below shows one possible way of obtaining the time trace.

The lath is clamped firmly at one end to a pair of table legs, about 0.1 m above the floor. In order to slow its rate of oscillation, suitable masses can be attached to its free end with rubber bands or string.

As shown, the paper rests over several layers of soft paper on a wooden board running on steel rollers. This allows the pen to write smoothly despite bumpy movement of the moving board, and is needed if the decay of the oscillations is to be anything like exponential.

Procedure

- To obtain a time trace, deflect the lath or metre rule slightly to one side and release, while at the same time slowly advancing the paper sheet placed underneath it. The pen on the end of the lath will write on the moving paper. To get a clear time trace, you will need to adjust both the oscillation period and also the rate at which the paper moves.
- One way of moving the board and paper at a steady rate is to pull it along with a string wrapped round a shaft that is turned by a motor.

Teaching Notes

- The first thing to point out is the sinusoidal shape of the trace produced on the moving paper. The lath or metre rule has a constant period of oscillation. Changing the rate at which the paper moves has the same effect as changing the time-base on an oscilloscope when studying an electrical oscillation.
- If appropriate, ask students if they can recognize the (exponential) shape of the envelope of the oscillation curve from earlier work.

*This experiment has yet to undergo a health and safety check.*

### Up next

### Straight line graphs

## Drawing straight line graphs

Once you have plotted the points of a graph, checked for any anomalies and decided that the best fit will be a straight line:

- To select the best fit straight line, take a weighted average of your measurements giving less weight to points that seem out of line with the rest.
- Use a ruler to draw the line.

## Interpreting straight line graphs

Proportionality:
A straight line through the origin represents direct proportionality between the two variables plotted, *y = mx*. If the plotted points (expressing your experimental results) lie close to such a line, then they show the behaviour of your experiment is close to that proportionality.

Linear relationships:
In many experiments the best straight line fails to go through the origin. In that case, there is a simple linear relationship, *y = mx + c*. Historically, one of the most far-reaching examples is the graph of pressure of gas in a flask (constant volume) against temperature. The intersect on the temperature axis gives an absolute zero of temperature, and an estimate of its value.

Identifying systematic errors:
In some experiments, all measurements of one quantity are wrong by a constant amount. This is called a ‘systematic error’. (For example, in a pendulum investigation of *T* against *l* all the lengths may be too small because you forgot to add the radius of the bob. Plotting T^{2} against *l* will still give a straight line if every value of *l* is too short by the radius but the line does not pass through the origin.) In such cases, the intersect can give valuable information.

Checking for constancy:
Consider the acceleration of a trolley. If you plot *s* against t^{2}, where *s* is the distance and *t* is the total time of travel from rest, then you hope to get a straight line through the origin. [A straight line through the origin shows that s = constant t^{2}]

In fact we know that *s* is proportional to t^{2} for any case of constant acceleration from rest. Simple mathematics lead from the statement that Δv / Δt = acceleration, giving s = 1/2at^{2} providing *a* is constant. [Δv = change of velocity, Δt = time taken.]

IF *a* is constant, THEN *s* = 1/2at^{2} because logic does that. So why might you plot the graph? To find out whether the trolley moved with constant acceleration.

### Up next

### Simple pendulum experiments

In different teaching schemes, class experiments with pendulums take several forms:

- experiments to demonstrate relationships or verify laws
- training in techniques of timing and observing
- scientific investigation
- an accurate measurement of
*g*

## Experiments to demonstrate relationships or verify laws

Many students welcome routine experiments with definite instructions and a clear outcome. This form of experiment, in which the answer is stated first, does not demonstrate real science practice but can show that physical phenomena behave predictably.

## Training in techniques of timing and observing

You may think that training in techniques is useful, but training does not transfer easily. It is unlikely to spread to other fields of science, unless you ensure that its potential value makes a strong impression.

## A scientific investigation

Even if students know that you are well aware of the answer

, that there is some formula

that describes how pendulums behave, they can do genuine scientific work if they regard the experiment as an investigation.

The experiment itself may be almost the same as verifying the formula

but the instructions need an essential twist. Ask students to find something out, not show that what you have already said is true. Say what should be measured (e.g. many swings should be timed) or what relationships should be investigated. Certainly do not say what results are to be expected.

## An accurate measurement of *g*

Advanced students may enjoy extracting the value of the gravitational field strength, *g* , by this indirect method.

### Up next

### Oscillations, waves and mathematical models

## Mechanical oscillators

Every mechanical oscillator, isochronous or not, has these features:

- it is displaced successively to one side then the other of an equilibrium position;
- it is accelerated towards the equilibrium position by a force; the force is related to its displacement in some way;
- it has inertia, which means that it continues through the equilibrium position, rather than coming to rest there;
- it stores energy both kinetically and gravitationally; at the extremes there energy stored kinetically is zero, at equilibrium the change in the energy stored gravitationally is zero
- there are resistive forces against which it must do work; as a result some energy is stored thermally.

## Oscillations can make waves

Every mechanical wave has its origin in an oscillating object. The particles in the medium themselves oscillate, in a similar way to the originating object, but a certain time later.

Electrons oscillating in a transmitting aerial give rise to fluctuating electric and magnetic fields which travel and cause electrons to oscillate in a receiving aerial.

## Isochronous oscillators

There are simple arguments for relating the steady time-keeping of an isochronous oscillator to the relationship between displacement and restoring force.

- The time is constant. If the amplitude is doubled, the average speed must be doubled. The double speed is acquired in the same time, so the average acceleration is doubled. Double the acceleration means double the force. So the average acceleration is proportional to the amplitude. This can be achieved by having a restoring force proportional to the displacement.
- If the restoring force is proportional to the displacement, twice the displacement gives twice as much force, giving twice as much acceleration. Thus the velocity gained in any given short time is doubled, and twice the distance is covered in this same time. But the displacement was doubled to start with. There is just twice as far to go, and double the distance will be covered in the same time as the original motion.

The motion of an oscillator for which the restoring force is inversely proportional to displacement from its equilibrium position would be called harmonic

. Simple harmonic motion is an idealized kind of motion. Real oscillators (atoms in a crystal or a molecule, car bodies on springs, buildings, and bridges) are likely only to approximate to this motion.

## Simple harmonic motion as a mathematical model

You might consider whether restoring force is likely to be accurately proportional to displacement, in one or two practical cases. Mathematical models can be built up, to be used in describing phenomena. Such models are often strictly limited to ideal cases, and represent real events more or less well. But such seeming inadequacy can be a strength, for the ideal model can be quite simple and can apply to many things. If necessary, more complicated mathematical models can be devised, to cope with damping, for example.