6.1.5

Simple Harmonic Systems

Test yourself

SHM Experiment (Mass-Spring System)

A mass suspended on the end of a spring is also an excellent example of SHM.

Equipment

• Suspend a mass, m, from a spring as shown in the diagram.

Method

• The formula for the period, T, is $T=2\pi \sqrt{\frac{m}{k}}$ where k is the spring constant of the spring.
• Find the length of time for 10 complete oscillations, using the pointer against the vertical scale.

Method 2

• Repeat this and find the average time for 10 oscillations. Divide that time by 10 to find the period, T.
• The mass should be varied and the period remeasured for each value of m.
• Goggles/safety specs should be worn because there is a spring under tension.

Getting quality data

• The fiducial mark should be horizontal. Some form of pointer could be attached to the suspended mass.
• You can use a small section of string between the clamp and the mass-hanger to reduce the effect of any side to side motion given to the mass in releasing it.

Obtaining quality data 2

• You should check the mass, m, with a balance.
• The extension should not be so large that the elastic limit of the spring is reached. The spring should not be compressed so much that it no longer provides a restoring force.
• Keep the pointer close to but not touching the scale.
• Observe the pointer from eye level. Do this to reduce parallax errors.

Analysis

• You should plot a graph of T² against m to verify the equation.
• You would expect to plot a straight line through the origin with a gradient $=\frac{4\pi^2}{k}$.
• The straight line might miss the origin because the effective mass of the spring has not been taken into account.
• Plotted or given error bars would help to provide an estimate of the uncertainty in g by finding the worst acceptable line of fit.

SHM Experiment (Simple Pendulum System)

The simple pendulum is a classic experiment to demonstrate SHM.

Equipment

• Suspend a pendulum bob by a thread from a clamp.
• The thread should be squashed between two small blocks of wood or a bung/cork split in two. This makes the suspension point easy to define.

Method

• Time for 10 oscillations. Repeat and get an average.
• Divide this average time by 10 to find the period.
• Repeat for different lengths of pendulum thread.

Obtaining quality data

• Use a ruler to measure the length of the string from the suspension point to the centre of the bob.
• A micrometer or calipers might be useful to measure the diameter of the bob.
• Pull the bob to one side (position A) so that the thread makes a small angle with the vertical (use a protractor to measure this angle to keep it consistent and small. The maximum angle should be 10 degrees).

Obtaining quality data 2

• Start and then later stop the timer when the bob passes in front of a fiducial mark placed behind the thread (i.e. position B).
• This can be a card with a vertical line drawn on, or the clamp stand.
• This mark needs to be at the equilibrium position. This is where the bob is travelling fastest, so means there is the smallest error in starting and stopping the timer.

Analysis

• The equation for this is supposed to be $T = 2\pi \sqrt{\frac{l}{g}}$. So you should plot a graph of T² against l to verify the equation.
• You would expect to see a straight line through the origin with gradient $= \frac{4\pi^2}{g}$.
• You can find a value of g from the gradient ($g = \frac{4\pi^2}{gradient}$).
• Plotted or given error bars would help to provide an estimate of the uncertainty in g by finding the worst acceptable line of fit.