# Oscillations and SHM

Student Learning Objectives
Lessons / Lecture Notes
Important Equations
Example Problems
Pencasts
Applets and Animations
Videos

Student Learning Objectives

• To understand the physics and mathematics of oscillations.
• To describe how the frequency of oscillation depends on physical properties of the system.
• To describe oscillatory motion with graphs and equations, and use these descriptions to solve problems of oscillatory motion.
• To understand and use energy conservation in oscillatory systems.
• To understand the basic ideas of damping and resonance.

Lessons / Lecture Notes

PY105 Notes from Boston University (algebra-based):

Introductory physics notes from University of Winnipeg (algebra-based):

HyperPhysics (calculus-based)

PHY2048 notes from Florida Atlantic University (calculus-based):

General Physics II notes from ETSU (calculus-based)

Important Equations

Example Problems

Problem 1
(a) A spring stretches by 0.015 m when a 1.75 kg object is suspended from its end. How much mass should be attached to the spring so that its frequency of vibration is f = 3.0 Hz?

(b) An oscillating block-spring system has a mechanical energy of 1.00 J, an amplitude of 10.0 cm, and a maximum speed of 1.20 m/s. Find the spring constant, the mass of the block, and the frequency of oscillation. (Solutions)

Problem 2
A 0.45 kg mass is attached to a spring with a force constant of 26.0 N/m and released from rest a distance of 3.25 cm from the equilibrium position of the spring. (a) What is the period of the mass? (b) Use conservation of energy to find the speed of the mass when it is halfway to the equilibrium position. (c) What is the maximum speed of the mass? (d) What is the magnitude of the maximum acceleration of the mass? (Solutions)

Pencasts

SHM
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Applets and Animations
 Simple Harmonic Motion I Demonstrating that one component of uniform circular motion is simple harmonic motion. Simple Harmonic Motion II Illustrating and comparing Simple Harmonic Motion for a spring-mass system and for a oscillating hollow cylinder. Masses and Springs A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring. Spring Mass SHM This applets plots the displacement, velocity, and acceleration of a mass connected to a vertical spring. This applet is excellent at showing the relationship between SHM and circular motion. Pendulum Lab Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It's easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of g on planet X. Notice the anharmonic behavior at large amplitude. Simple Pendulum The Simple Pendulum model displays the dynamics of a simple pendulum.  The pendulum is initially displaced from equilibrium and the pendulum bob has zero initial velocity. Pendulum Energy The Pendulm Energy Model shows a pendulum and associated energy bar charts. Users can change the initial starting point of the pendulum. Damped SHM The damping factor may be controlled with a slider. The maximum available damping factor of 100 corresponds to critical damping. Damped Oscillations The applets shows a mass connected to a horizontal spring on a surface with friction. The user can control the  degree of damping. Driven SHM A harmonic oscillator driven by a harmonic force. The frequency and damping factor of the oscillator may be varied. Damped Driven Simple Harmonic Oscillator The Damped Driven Simple Harmonic Oscillator model displays the dynamics of a ball attached to an ideal spring with a damping force and a sinusoidal driving force.  The spring is initially unstretched and the ball has zero initial velocity. Forced Oscillations (Resonance) This applet plots the motion of a driven simple harmonic oscillator. The user can control the frequency of the vibrator, the spring constant, the degree of damping, and the mass. Couple Oscillations and Normal Modes The Coupled Oscillators and Normal Modes model displays the motion of coupled oscillators, two masses connected by three springs. Spring Pendulum The Spring Pendulum model displays the model of a hollow mass that moves along a rigid rod that is also connected to a spring.  The mass, therefore, undergoes a combination of spring and pendulum oscillations.

Videos