# 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.

**1** 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.

**2** 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 web site for a supplier of Fractional horsepower motors.

#### Procedure

**a** 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.

**b** 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.

**c** 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

**1 **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.

**2** 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 of *phase* 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.

**3** 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.

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