AP Physics 1: Simple Harmonic Motion

Introduction to Simple Harmonic Motion (SHM)

• Simple Harmonic Motion (SHM) describes repetitive back-and-forth motion.
• Examples include springs and pendulums.
• Textbook reference: Physics Fundamentals Ch. 15: Rotation.
• Standards: No NGSS standards addressed; Massachusetts Curriculum Frameworks (2006) - 4.1.
• AP Physics 1 Learning Objectives:
  • Predict properties determining SHM motion (3.B.3.1).
  • Design plans to collect data for oscillatory motion (3.B.3.2).
  • Analyze data for relationships in oscillatory motion (3.B.3.3).
  • Explain oscillatory behavior with restoring force evidence (3.B.3.4).
  • Use object model to analyze system behavior (5.B.2.1).
  • Describe systems with internal potential energy (5.B.3.1, 5.B.3.2, 5.B.3.3).
  • Calculate changes in kinetic and potential energy (5.B.4.2).

Simple Harmonic Motion

• SHM involves regular, periodic oscillation.
• Acceleration is opposite to displacement, slowing and reversing motion.
• Ideal systems without friction continue SHM indefinitely.
• Displacement vs. time graphs resemble sine or cosine functions.

Examples of Simple Harmonic Motion

• Springs: Spring force accelerates it back to equilibrium.
• Pendulums: Gravity accelerates it back to equilibrium.
• Waves: Medium oscillates like a duck on water.
• Uniform Circular Motion: Vertical position oscillates between +r and -r.

Kinematics of Simple Harmonic Motion

• Position as a function of time is a sine or cosine function.
• General equations:
  • Position: ( x(t) = A\cos(\omega t + \phi) )
  • Velocity: ( v(t) = -A\omega\sin(\omega t + \phi) )
  • Acceleration: ( a(t) = -A\omega^2\cos(\omega t + \phi) )
• Amplitude is the maximum displacement.
• Phase angle is the starting position relative to equilibrium.
• Frequency ( f ) is related to angular frequency ( \omega ) by ( \omega = 2\pi f ).

Springs

• Spring Force: Hooke's Law: ( F_s = -kx )
  • ( F_s ): Spring force, ( k ): Spring constant, ( x ): Displacement.
• Potential Energy: ( U = \frac{1}{2}kx^2 )
• Period: ( T = 2\pi\sqrt{\frac{m}{k}} )
• Frequency: ( f = \frac{1}{T} )

Sample Problems:

1. Force needed to compress a spring: ( F = kx )
2. Potential energy stored: ( U = \frac{1}{2}kx^2 )
3. Period of oscillation: ( T = 2\pi\sqrt{\frac{m}{k}} )

Pendulums

• Forces: Constant gravity force and tension force.
• Period: ( T = 2\pi\sqrt{\frac{l}{g}} ) for small angles.
• Potential energy is gravitational potential energy at maximum displacement.

Sample Problems:

1. Period of oscillation using length and gravity.
2. Determine pendulum length for a given frequency.

These notes cover the fundamental concepts of Simple Harmonic Motion, focusing on the behavior of springs and pendulums, as well as the related kinematic equations and sample problems for practical understanding.