Lesson Plan - Pendulum Motion

d2 theta / dt2 = - g/L * sin(theta)

For a pendulum on a flexible rope, of course this would not work for angles greater than pi/2. However, one might consider a pendulum consisting of a very thin but stiff metal rod, with a massive pendulum bob at the end. For this pendulum, the above equation is valid for all angles.

This exercise attempts to let students explore questions which cannot be handled with simple pendulums, and to understand both the usefulness and the limitations of approximations in science.

This lesson could be presented in a variety of ways, either by using a Euler method solution in a spreadsheet (though be sure to be careful with the time step used, as this problem quickly becomes unstable with explicit integration methods), a symbolic math package such as Maple, or a modeling package such as Stella. We have presented a Java applet created from a Stella model. The pre configured solution has the advantage of letting the students see immediately what the effect of changing the length or the acceleration due to gravity, while using a spreadsheet or similar solution lets students tackle more difficult problems, such as the effect of a length that changes over time.

The exploration addresses the following common misconceptions among students of Physics (from C3P):

• The period of oscillation depends on the amplitude.
• The restoring force is constant at all points in the oscillation.
• The heavier a pendulum bob, the shorter its period.
• All pendulum motion is perfect simple harmonic motion, for any initial angle.
• Harmonic oscillators go forever.
• A pendulum accelerates through lowest point of its swing.
• Amplitude of oscillations is measured peak-to-peak.
• The acceleration is zero at the end points of the motion of a pendulum.

The exploration is designed to work with content in the following areas:

• Displacement
• Velocity and acceleration
• Vectors
• Rotational Motion
• Oscillations
• Simple Harmonic Motion

The exploration meets the following National Standards:  Science Content Standards: 9-12

• CONTENT STANDARD A:
• Abilities necessary to do scientific inquiry
• Understandings about scientific inquiry
• Fundamental abilities and concepts that underlie this standard include:
  • Identify questions and concepts that guide scientific investigations.
  • Design and conduct scientific investigations.
  • Use technology and mathematics to improve investigations and communications.
  • Formulate and revise scientific explanations and models using logic and evidence.
  • Recognize and analyze alternative explanations and models.
  • Communicate and defend a scientific argument.
  • Understandings about scientific inquiry
• CONTENT STANDARD B:
  • Motions and forces

Also be sure to check out the Pit and the Pendulum model.