Introduction to SHM Objectives Describe simple examples of free oscillation. Define and use the terms used to describe simple harmonic motion (SHM). Select and use the equation period = 1/frequency Explain that in SHM, period is independent of amplitude.

Frequency and period for SHM Displacement Time This is the graph we would expect to get from an object oscillating as it gradually loses energy. We can model this with a “wig-wag”

First we need to establish some terminology before we go further into SHM so we can talk with confidence! Use the text book or internet to get a definition for “free and forced vibrations” making sure you can describe free vibrations. Now use a ruler or hack saw blade connected to the desk leg, with a paint brush secured to one end, to draw a trace of a free vibration as the ruler oscillates. Slowly pull the paper along underneath the ruler.

Your sheet should look like this when you are done The less friction between the brush and paper the longer it will take for the oscillations to decay because ………

Equilibrium position Equilibrium position – Where the pendulum or oscillating body would normally rest Displacement Displacement – distance from equilibrium position taking note of the direction Amplitude Amplitude – Maximum displacement from equilibrium position Cycle Cycle – complete range of motion from current position eg a full out and back swing Period Period – Time taken for one complete cycle/oscillation Frequency Frequency – Number of complete oscillations per second Displacement Time

A displacement–time graph for a frictionless pendulum or a real one in a “long case clock”.

A displacement–time graph for a pendulum with no driving mechanism.

S.H.M. using angle sensor

Free vibrations decaying over a long period of time.

Simple Harmonic Motion SHM In order to understand what SHM is you may find it easier to think about the motion of a swing or pendulum. Answer the following questions about a pendulum swinging. In which direction does it accelerate when left to drop? When will it have maximum speed? What makes it slow down and stop at the other side? In which direction must the acceleration be NOW? Does the acceleration change? When would it be a maximum and minimum?

So considering a pendulum the motion would have the following characteristics. a is the acceleration x is the displacement a a x x At this point x = 0 a = 0

Now complete the activity making notes regarding all motion. 1)Use a pendulum, firstly to observe it then to measure the quantities as described with the ticker timer. Try changing things concerning the motion like mass, how far you pull it back and length of string.. Attempt to plot graphs of the motion. You may be able to approximate the way the speed varied during the oscillations. Do this purely on observations and measurements, NOT previous knowledge.

Plotting displacement, velocity and acceleration for SHM. In order to look at the motion of a pendulum accurately we can use the set up shown below. You will need to repeat this 3 times in each direction and be quick with your calculations. Ticker timer By connecting the ticker timer to the pendulum we can accurately plot the displacement and Velocity against time for the first half of the cycle the repeated swings from both sides will build up the whole picture!

To do this you need to divide the tape into 5 dot sections, ie 5 dots and gaps as shown THIS IS 0.1 SECONDS!! The total length of each section gives the displacement in 0.1 SECS, so calculate the running total to find the displacement from the start position. Plot the results on a graph. Now you should be able to plot the velocity for each section as a function of the time after the release. The velocity can be calculated from the length and the fact that each section is 0.1s long or by drawing tangents to your displacement time graph. Plot this on a graph.

In order to plot the acceleration you will need to calculate the change in velocity from one section to another over the period of 0.1s and plot a graph of this with respect to time. You should now have graphs that look something like these. Displacement t t Velocity Acceleration t Make sure your graphs are printed off, annotated and inserted into your files! The Insight graphs generated by the angle sensor look like this.