Simple Harmonic Motion

Part 1: Mass-Spring System

Equipment

• Spring: The oscillating element.
• 50g masses with holder: Allows for mass adjustment up to 500g.
• Stand and clamp: To securely hold the spring.
• Pin and Blu-Tack: Used as a fiducial marker at the equilibrium position.
• Metre ruler: For accurate positioning.
• Stopwatch: To time oscillations.

Graphs and Calculations

1. Graph of T² vs. m:

• Plot T² (y-axis) against mass m (x-axis). Draw a line of best fit.
• The gradient of this graph is 4π²/k, where k is the spring constant.

2. Equation of Motion:

• The period of a mass-spring system in SHM is given by:

$$T = 2\pi \sqrt{\frac{m}{k}} \Rightarrow T^2 = \frac{4\pi^2}{k} m$$

• Using the gradient of the graph, calculate k for the spring.

Improvements and Notes

• Vertical Oscillation: Ensure the spring oscillates vertically; any horizontal motion can affect timing accuracy.
• Use of Fiducial Marker: The marker should be placed at the centre of oscillation to reduce timing errors.
• Data Logger: Using a motion tracker or data logger can improve timing accuracy by removing human reaction time errors.

Part 2: Simple Pendulum

Equipment

• Pendulum bob on 2m string: For generating oscillations.
• Stand and clamp: To secure the pendulum.
• Pin and Blu-Tack: Fiducial marker at equilibrium position.
• Metre ruler: To measure string length.
• Stopwatch: To time oscillations.
• Two wooden blocks: To support the pendulum setup.

Method

1. Setup:

• Set up the pendulum with a string length L of 1.5m (distance from the suspension point to the bob's centre of mass). Place the fiducial marker at the equilibrium position.

2. Initiate Oscillations:

• Displace the pendulum by a small angle (less than 15°) and release. Ensure the motion is in a straight line.

3. Measure Oscillation Time:

• Start timing when the bob passes the fiducial marker. Measure the time for 10 oscillations and record as T₁₀.
• Calculate the period T for one oscillation by dividing T₁₀ by 10.

4. Decrease Length:

• Shorten L by 0.100m increments, measuring T for each length down to 0.500m.

5. Repeat for Accuracy:

• Repeat each measurement twice more to obtain mean values.

Graphs and Calculations

1. Graph of T² vs. L:

• Plot T² (y-axis) against pendulum length L (x-axis) and draw a line of best fit.
• The gradient of this graph is 4π²/g, where g is the acceleration due to gravity.

2. Equation of Motion:

• The period of a pendulum in SHM is:

$$T = 2\pi \sqrt{\frac{L}{g}} \Rightarrow T^2 = \frac{4\pi^2}{g} L$$

• Calculate g from the gradient.

Safety

• Low Risk: There are minimal safety concerns, but ensure the pendulum has enough space to swing without obstruction.

Improvements and Notes

• Small Bob: Use a small pendulum bob to make length measurements easier.
• Length Accuracy: Measure from the centre of mass of the bob for accurate L.
• Data Logging: As with the mass-spring system, a motion tracker can improve timing accuracy by eliminating manual timing errors.