SHM – Types of Pendulums AP Physics

Pendulum Simple Physical/Compound  Oscillates due to gravity  Mass of pendulum bob is irrelevant  Oscillates due to gravity  Mass of rigid body contributes to the period of oscillation  Rotational Inertia present ω – angular frequency of SHM

SHM Angular Frequency  Mass on Spring  Simple Pendulum  Compound/Physical Pendulum d is distance from pivot to center of mass

Pendulum + Hooke’s Law Consider the free body diagram for a pendulum. Here we have the weight and tension. Even though the weight isn’t at an angle let’s draw an axis along the tension.   mgcos  mgsin  For a mass on a spring, the restoring force is found with Hooke’s Law.

Applying Newton’s Second Law to SHM  Acceleration is the second derivative of position.  Angular acceleration is the second derivative of angular position.  The angular frequency of SHM can be derived from Newton’s second law.

SHM equations to know These equations are derived with SHM. Angular frequency can be determined by applying Newton’s second law. Be careful: ‘x’ is simply a variable for a distance, it could represent length of a pendulum, the stretch/compression of a spring, or radial distance.  Summing Torques  Summing Forces  Linear to Angular:

SHM and position function  The position of an object undergoing SHM can be found using the following function:  X max is the maximum displacement from equilibrium  ω is the angular frequency  ɸ is the phase shift  Equals zero when max/min starts at t = 0  What about velocity and acceleration?

Velocity and Acceleration of SHM  dx/dt  dv/dt If position is positive, then velocity and acceleration are negative.