Simple Harmonic Oscillator

Purpose:

To explore the dynamics of a simple harmonic oscillator (SHO), using a simple pendulum.

Description:

This lab begins with the derivation of the differential equation which models the displacement s of a pendulum from equilibrium. Assuming a small displacement the model for the pendulum with an initial displacement of s₀ and an initial velocity of 0 is s'' = -²s,
s(0) = s₀,
s'(0) = 0. Students need to verify that s(t) = s₀ cos(t) is the solution of this initial value problem.

For the lab students construct a simple pendulum consisting of a string and a ball. They give it an initial displacement to set it into motion. Students use the MBL equipment to collected data giving the displacement of the pendulum for 6 to 8 cycles. They use this data to:

• Compute the average period of the pendulum.
• Calculate the acceleration due to gravity g.
• Plot velocity vs. time and position vs. time and discuss the relationship between the graphs.
• Plot the phase portrait for the pendulum.

Materials Needed:

• MBL equipment with distance probe.
• 1 meter rod with clamp.
• 1 metal bar clamp
• 1 ball attached to a string

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