Harmonic Motion

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Simple harmonic motion (dynamics and energy relationships)

Students should understand simple harmonic motion, so they can:

(1)  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.

(2)  Write down an appropriate expression for displacement of the form A sin ωt or
A cos ωt  to describe the motion.  

(3)  Find an expression for velocity as a function of time.

(4)  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.

(5)  State and apply the relation between frequency and period.

(6)  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. 

(7)  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.

Mass on a spring

Students should be able to apply their knowledge of simple harmonic motion to the case of a mass on a spring, so they can: 

(1)  Derive the expression for the period of oscillation of a mass on a spring.

(2)  Apply the expression for the period of oscillation of a mass on a spring.

(3)  Analyze problems in which a mass hangs from a spring and oscillates vertically.

(4)  Analyze problems in which a mass attached to a spring oscillates horizontally.

Pendulum and other oscillations

Students should be able to apply their knowledge of simple harmonic motion to the case of a pendulum, so they can:  

(1)  Derive the expression for the period of a simple pendulum.  

(2)  Apply the expression for the period of a simple pendulum. 

(3)  State what approximation must be made in deriving the period.
 

 

 

Simple Harmonic Motion

SHM is exemplified by a block oscillating at the end of a spring (ignoring other forces for now).

Hooke's Law says that the Force exerted by a spring is proportional to the amount of displacement (stretching or compression) of the spring, and a proportionality constant called the spring constant (k). 

Fs = -kx

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  k indicates the “stiffness” of the spring

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  the negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement.

Click here to view a movie of a block oscillating on a spring.

The Potential Energy stored in a spring is determined by

Example #1

A 12 cm-long spring has a force constant (k) of 400 N/m.  How much force is required to stretch the spring to a length of 14 cm?

Example #2

A block of mass m = 1.5 kg oscillates on a spring, whose force constant k is 500 N/m.  The amplitude of the oscillations is 4.0 cm.  Calculate the maximum speed of the block.

Example #3

A block of mass m = 2.0 kg is attached to an ideal spring of force constant k = 500 N/m.  The amplitude of the resulting oscillations is 8.0 cm.  Determine the total energy of the oscillator and the speed of the block when it’s 4.0 cm from equilibrium.

Example #4

A block of mass m = 2.0 kg is attached to an ideal spring of force constant k = 500 N/m.  The block is at rest at its equilibrium position.  An impulsive force acts on the block, giving it an initial speed of 2.0 m/s.  Find the amplitude of the resulting oscillations.

Kinematics of SHM

An oscillating object repeats its motion (cycle) in a certain time.  This time is called the period (T) of the oscillation.  The number of cycles occurring in a certain period of time is called the frequency (f) of oscillations.  One cycle per second is called one Hertz (Hz).

Example #5

A block oscillating on the end of a spring moves from its position of maximum spring extension to maximum spring compression in 0.25 s.  Determine the period and frequency of this motion.

Example #6

A student observing an oscillating block counts 45.5 cycles of oscillation in one minute.  Determine its frequency (in Hertz) and period (in seconds).

Other properties of springs

Imagine a thick spring (high k) oscillating with a small mass attached.  It should have small, rapid oscillations.  A thin spring (low k) with a large mass attached should have large, slow oscillations.  In other words, the rate of oscillation depends on the mass attached to the spring, and the spring constant.

Comparison of circular and oscillating motion

Since 2p is associated with circular motion, it relates to oscillating motion as well.   Click here to see a movie illustrating simultaneous circular motion and oscillatory motion.

Example #7

A block of mass m = 2.0 kg is attached to a spring whose force constant, k, is 300 N/m.  Calculate the frequency and period of the oscillations of this spring-block system.

Example #8

A block is attached to a spring and set into oscillatory motion, and its frequency is measured.  If this block were removed and replaced by a second block with 1/4 the mass of the first block, how would the frequency of the oscillations compare to that of the first block?

Example #9

A block of mass m = 1.5 kg is attached to a vertical spring of force constant k = 300 N/m.  After the block comes to rest, it is pulled down a distance of 2.0 cm and released.

A) What is the frequency of the resulting oscillations?

B) What are the minimum and maximum amounts of stretch of the spring during the oscillations of the block?

Simple Pendulums

A simple pendulum consists of a mass m attached to a massless rod that swings without friction about a vertical equilibrium position.

Simple Pendulum Equations

If qmax is relatively small, then the downward component of gravity will approximately equal gravity itself, producing a fairly constant restoring force.  Therefore,

Example #10

A simple pendulum has a period of 1 s on Earth.  What would its period be on the Moon (where g is one-sixth of its value here)?

Internet Simulation Links

Conservation of Energy in a Spring-Mass System

Pendulum Motion

Internet Simulation Links

Damped Harmonic Motion

by Professor Mark Sutherland, University of Toronto.

http://thorin.adnc.net/~topquark
/fun/JAVA/dho/dho.html

Control the initial amplitude and damping coefficient for a mass attached to a spring. Displacement, velocity and energy versus time are plotted.

Professor Sutherland has a collection of physics applets and tutorials at fun@learning.physics.

The Simple Harmonic Oscillator

by Dan Russell, at Kettering University in Flint, MI .

http://www.kettering.edu
/~drussell/Demos/SHO
/mass.html

One of many excellent animations at Russell's Acoustics and Vibration Animations. This page shows a derivation of the Simple Harmonic Oscillator (SHO) including an animated gif showing the simple harmonic motion of three undamped mass-spring systems.

Waves and Sound

from the ExploreScience site of Raman Pfaff at the University of Michigan, Ann Arbor.

http://www.explorescience.com
/activities/Activity_page.cfm?
ActivityID=22

In order to use this site you must download and install the Macromedia Shockwave Plug-in on your computer.

This activity allows you to compare a simple pendulum and a simple harmonic oscillator. You can control the mass on each as well as the lengh and spring constant. The really fun part comes with controling the acceleration due to gravity and making it different than Earth's.

Spring Pendulum

by Walter Fendt, Augsburg, Germany.

http://www.walter-fendt.de
/ph14e/
springpendulum.htm

Control the mass, spring constant and amplitude of a mass hung on a spring. Displacement, velocity, acceleration force or energy vs time can be plotted in real time. The applet allows you to experiment and determine the relationship between these variables and the period.

Herr Fendt has a collection of other physics applets on his site.