A system will undergo periodic motion provided a restoring force acts in a direction opposite to displacement towards a stable equilibrium point. When the restoring force is directly proportional to the displacement then the system undergoes simple harmonic motion (SHM). The resulting equation of motion is sinusoidal and it can be shown that the period of motion is independent of the amplitude. In this lab you will investigate various 4 different oscillatory systems and use your knowledge of SHM to understand their physical properties.

First determine the spring constant k of each of the three springs you are given (plot a graph of applied force vs extension and find the slope). Then attach a mass m to one of the springs and suspend the system from the retort stand. Set the system into oscillatory motion and determine the period T. Using the equation for the period of motion T,

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find the spring constant k and compare this measurement with your earlier result. Next attach two springs together with the mass at one end and set this system into oscillatory motion. Measure the period and use the equation above to find the effective spring constant for the combined system. How does this relate to the spring constants of the individual springs? Can you postulate what the effective spring constant for the three springs will be? Check your hypothesis by oscillating a system of three springs.

Pull the pendulum back to a large angle (almost 180 degrees) and using the Logger Pro software obtain a plot of angular position vs time for the system. Due to damping you should see that the amplitude of oscillations decreases with time. From this plot make a table of period and corresponding amplitude (you can define the period as the time between consecutive maxima and the amplitude as the average height of those two maxima - it may work better to average the period and amplitude over several maxima). Plot period vs amplitude and fit a smooth curve through the data. Comment on the nature of the graph when the amplitude is large. Is period independent of amplitude? How does the graph behave when the amplitude is small? Is the period ever constant?

Choose one of the geometric shapes available as a physical pendulum. Choose a suitable axis of rotation and allow the shape to oscillate with a small amplitude. Measure the period of oscillations and use the relation

to determine the moment of inertia about the axis of rotation. Calculate the moment of inertia based on the geometry of the object you chose (you will probably need to use the Parallel Axis Theorem since your axis will not be at the centre of mass of the object. If you are lucky you may even need to do an integral!)

Using the motion detector and Logger Pro record the oscillations of the block spring system which is attached to the ceiling. Observe that the amplitude decreases slowly over time. Attached a piece of paper to the oscillating masses and record the oscillations again. Describe what you observe. Obtain an amplitude vs time graph from this plot and fit an exponential function to it. Hence determine the damping coefficient for the system.