Unit 6: Oscillations

Big Ideas:
1. Simple harmonic oscillations are defined by specific mathematical relationships.
2. Simple harmonic oscillations are executed by a variety of mechanical devices.
3. A simple pendulum executes approximate simple harmonic motion.

Learner Objectives (as published by the College Board):
1. Students should understand simple harmonic motion, so they can:
A. Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period, and frequency of the motion.
B. Write down an appropriate expression for displacement of the form Asinwt or Acoswt to describe the motion.
C. Find an expression for velocity as a function of time.
D. State the relations between acceleration, velocity, and displacement, and identify points in the motion where these quantities are zero or achieve their greatest positive and negative values.
E. State and apply the relation between frequency and period.
F. Recognize that a system that obeys a differential equation of the form d²x/dt² = ω²x must execute simple harmonic motion, and determine the frequency and period of such motion.
G. State how the total energy of an oscillating system depends on the amplitude of the motion, sketch or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic.
H. Calculate the kinetic and potential energies of an oscillating system as functions of time, sketch or identify graphs of these functions, and prove that the sum of kinetic and potential energy is constant.
I. Calculate the maximum displacement or velocity of a particle that moves in simple harmonic motion with specified initial position and velocity.
J. Develop a qualitative understanding of resonance so they can identify situations in which a system will resonate in response to a sinusoidal external force.

2. Students should be able to apply their knowledge of simple harmonic motion to the case of a mass on a spring, so they can:
A. Derive the expression for the period of oscillation of a mass on a spring.
B. Apply the expression for the period of oscillation of a mass on a spring.
C. Analyze problems in which a mass hangs from a spring and oscillates vertically.
D. Analyze problems in which a mass attached to a spring oscillates horizontally.
E. Determine the period of oscillation for systems involving series or parallel combinations of identical springs, or springs of differing lengths.

3. Students should be able to apply their knowledge of simple harmonic motion to the case of a pendulum, so they can:
A. Derive the expression for the period of a simple pendulum.
B. Apply the expression for the period of a simple pendulum.
C. State what approximation must be made in deriving the period.
D. Analyze the motion of a torsional pendulum or physical pendulum in order to determine the period of small oscillations.