🔖 Topics

  • Simple harmonic motion

🎯 Objectives

  • Describe, calculate, and graph different properties of simple harmonic motion
  • Relate circular motion to simple harmonic motion

📋 Sequence

  • We have seen SHM before - mass-spring oscillator, pendulum
  • Stable vs. unstable equilibrium
  • Review - what is a spring constant?
  • Restoring forces
    • Non-linear restoring forces can be approximated as linear for small oscillations
    • \( \vec{F} = -k\vec{x} \)
  • Energy of a mass-spring oscillator
    • \( E = KE + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 \)
    • \( E_{total} = \frac{1}{2}kA^2 \)
    • Setting \( E_{total} = \frac{1}{2}mv^2 \) yields \( v_{max} = \sqrt{k/m} A \)
  • Acceleration of a mass-spring oscillator
    • Draw a FBD and sum the forces
    • \( F_x = -kx = ma_x \Rightarrow a_x = kx/m \)
    • Maximum acceleration can be found when we are at the maximum amplitude
  • SHM and Circular Motion
    • \( \theta(t) = \omega t \)
    • \( x(t) = A cos \theta = A cos(\omega t) \)
    • Radial acceleration: \( a_x = -a_r cos \theta = - \omega^2 A cos \omega t \)
    • \( a_x(t) = - \omega^2 x(t) \)
    • We showed before that \( a_x = kx/m \)
    • Thus, angular frequency is \( \omega^2 = k/m \)
    • Frequency: \( \omega = 2\pi f \)
    • Period: \( T = 1/f \)
    • Maximum speed \( v_{max} = \sqrt{k/m}A = \omega A \)
    • Maximum acceleration \( a_{max} = kA/m = \omega^2 A \)
    • You can see that \( v_{max}^2 = a_{max} A \)
  • Graphs of SHM
    • Derivatives: \( cos(x) \rightarrow -sin(x) \rightarrow -cos(x) \rightarrow sin(x) \rightarrow cos(x) \)
    • Position vs. Time => \( cos(\omega t) \)
    • Velocity vs. Time => \( -sin(\omega t) \)
    • Acceleration vs. Time => \( -cos(\omega t) \)
    • Kinetic Energy vs. Time
    • Potential Energy vs. Time
    • Momentum vs. Time
    • Phase

🖥️ Animations, Simulations, Activities

📝 Practice Problems

Tuning Fork: Each prong of a tuning fork moves back and forth precisely in simple harmonic motion. The distance the prong moves between its extrema is 2.24 mm. If the frequency of the tuning fork is 440 Hz, what are the maximum velocity and maximum acceleration of a prong?

Diaphragm of a Speaker: The diaphragm of a speaker has a mass of 50 grams and responds to a signal frequency of 2.0 kHz. It moves back and forth with an amplitude of \(1.8 \times 10^{-4}\) m at that frequency.

  • What is the period of the diaphragm’s motion?
  • What is the angular frequency of the diagram’s motion?
  • What is the maximum force acting on the diaphragm?
  • What is the total mechanical energy of the diaphragm?

Equation of Object on Spring: A 170 g object on a spring oscillates on a frictionless surface with a frequency of 3.00 Hz and an amplitude of 12.0 cm.

  • What is the spring constant of the spring?
  • If the object starts at its maximum amplitude at time t = 0, what equation describes its position as a function of time?

✅ Partial Solutions

N/A

📘 Connected Resources

  • Giambattista, Alan, et al. College Physics With an Integrated Approach to Forces and Kinematics. 5th ed., McGraw-Hill Education, 2020.