3 The Simple Pendulum Not all periodic motion is simple harmonic motion. Only a system whose equation of motion has the form will exhibit simple harmonic motion with angular frequency a. The period of the motion (the time required for one complete cycle) is T = 2x/w. One feature of simple harmonic motion is that the angular frequency (and hence the period) is independent of the amplitude of the motion. As an example, consider the simple harmonic oscillator – a mass on a spring. Applying Newton's Second Law to this system gives ma = F We see that x(t) has an equation of motion that matches the form of simple harmonic motion with a = /k/m. Now, let's apply a similar analysis to a simple pendulum -– a mass on a string. Consider a simple pendulum that consists of mass M at the end of a very light, rigid rod of length L. If the pendulum is displaced from its vertical equilibrium position by angle 0, the restoring force is -Mg sin 0. (Part of the weight of the mass is cancelled by the tension in the rod. Only the uncompensated component of the weight is a restoring force.) The "position" of the pendulum is the length of the circular arc connecting the current position of the mass to its equi- librium position, s = Le. Applying Newton's Second Law to this system gives --Mg sin0 Ma = F ML (a) Will this result in simple harmonic motion? Why or why not? If so, what is the angular frequency as of the motion? For small angles, sin 0 ru 0. (The error is less than 1 percent for 0 < 0.2447, which is about 14°.) Using this approxima- tion, the equation of motion for the simple pendulum becomes ML d²e -Mg® Ma = F (b) For small angles, is the pendulum's motion approximately simple harmonic motion? Why or why not? If so, what is the angular frequency w of the motion? Now let's apply your result for the period of the motion. (c) Suppose you constructed two identical simple pendulums with masses of 1 kg attached to light, rigid rods of length 1m. You then contract with SpaceX to set up one of the pendulums on the surface of the moon. If each pendulum is pulled back to the same initial angle and released from rest, will the pendulum on the moon run faster than, slower than, or at the same rate as the one on the earth? What is Tmoon/Tearth? (d) What mass and length should you use for the lunar pendulum to match the period of the earth pendulum?

3 The Simple Pendulum Not all periodic motion is simple harmonic motion. Only a system whose equation of motion has the form will exhibit simple harmonic motion with angular frequency a. The period of the motion (the time required for one complete cycle) is T = 2x/w. One feature of simple harmonic motion is that the angular frequency (and hence the period) is independent of the amplitude of the motion. As an example, consider the simple harmonic oscillator – a mass on a spring. Applying Newton's Second Law to this system gives ma = F We see that x(t) has an equation of motion that matches the form of simple harmonic motion with a = /k/m. Now, let's apply a similar analysis to a simple pendulum -– a mass on a string. Consider a simple pendulum that consists of mass M at the end of a very light, rigid rod of length L. If the pendulum is displaced from its vertical equilibrium position by angle 0, the restoring force is -Mg sin 0. (Part of the weight of the mass is cancelled by the tension in the rod. Only the uncompensated component of the weight is a restoring force.) The "position" of the pendulum is the length of the circular arc connecting the current position of the mass to its equi- librium position, s = Le. Applying Newton's Second Law to this system gives --Mg sin0 Ma = F ML (a) Will this result in simple harmonic motion? Why or why not? If so, what is the angular frequency as of the motion? For small angles, sin 0 ru 0. (The error is less than 1 percent for 0 < 0.2447, which is about 14°.) Using this approxima- tion, the equation of motion for the simple pendulum becomes ML d²e -Mg® Ma = F (b) For small angles, is the pendulum's motion approximately simple harmonic motion? Why or why not? If so, what is the angular frequency w of the motion? Now let's apply your result for the period of the motion. (c) Suppose you constructed two identical simple pendulums with masses of 1 kg attached to light, rigid rods of length 1m. You then contract with SpaceX to set up one of the pendulums on the surface of the moon. If each pendulum is pulled back to the same initial angle and released from rest, will the pendulum on the moon run faster than, slower than, or at the same rate as the one on the earth? What is Tmoon/Tearth? (d) What mass and length should you use for the lunar pendulum to match the period of the earth pendulum?