Simple Harmonic Motion Model

The EJS simple harmonic motion Model shows a mass m situated at the end of 2 springs of length l = 2.0 m of negligible massThe motion is restricted to one dimension, the horizontal. (We choose a coordinate system in the plane with origin at centre of the mass-spring system and with the X axis along the direction of the spring). The floor is assumed to be frictionless.

Four Plots vs t shows
1 displacement (in m) versus time (in s).
2 velocity (in m/s) versus time (in s).
3 acceleration (in m/s^2) versus time (in s).
4 energies (in J) versus time (in s).

Three Plots vs X shows

5 velocity (in m/s) versus displacement (in m).
6 acceleration (in m/s^2) versus displacement (in m).
7 energies (in J) versusdisplacement (in m).

Users can examine and change the model if they have Ejs installed.

free oscillations

A simple harmonic oscillator is an oscillator that is neither driven nor damped. Its motion is periodic— repeating itself in a sinusoidal fashion with constant amplitude, A. Simple harmonic motion SHM can serve as a mathematical model of a variety of motions, such as a mass on a spring.

For simplicity, we assume that the reaction of the springs to a displacement dx from the equilibrium point follows Hooke's Law, F(dx) = -k dx , where k is a constant which depends on the physical characteristics of the spring.

This, applying Newton's Second Law, leads us to the second order differential equation

d2x / dt2 = -k/m (x-l),

where x is the horizontal position of the mass from the from the origin centre of the springs.
This is similar to what is commonly describe in SHM as
a = - ?2x
a acceleration
w omega is angular velocity of SHM
x displacement of object in SHM from the equilibrium position

Exercises: Designed for http://www.seab.gov.sg/aLevel/20102011Syllabus/9646_2011.pdf
Oscillations
Content
• Simple harmonic motion

1. Run the simulation with b = 0 (no damping) and X driver = 0 ( no driver amplitude). Explore the various sliders to make sense of the sliders. Describe the motion of these free oscillations with reference to acceleration and displacement. Describe and relate to other examples of simple free oscillations.
2. Investigate the relationship of the displacement, velocity and acceleration versus time by exploring the Plot vs t checkbox to reveal the graphical display of the experimental view of the setup. Describe, with graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion.
3. Explore the terms amplitude, period, frequency, angular frequency and phase difference in the virtual laboratory by looking for the hints in the virtual lab. Play with the sliders and make sense of these terms used commonly in SHM.
4. Explore and record the period, T in terms of both frequency, f and angular frequency, ?. Select the 'expert' checkbox and look for the values of f and ? in relations to T.
5. The equation a = –?2x is the defining equation of simple harmonic motion. Select the Plot vs X checkbox and record down the graph. Why is the equation is correct? Explain the negative sign and meaning of ? in terms of k and m.
6. The equation  v = vocos? t can be used to describe the graph of v versus t (select checkbox Plot vs t and check v) Why is the equation is correct? Under what conditions is the equation valid?
7. The equation  v = ±? Math.sqrt ( xo2 - x2 )  can be used to describe the graph of v versus x (select checkbox Plot vs x and check v) Why is the equation is correct? Under what conditions is the equation valid?
8. Explore degree of damping and the importance of critical damping by varying the slider of b. Design and record down how the values of b affects the graph of displacement vs time. Hint: The graph of energies vs time would be of interest in describing the effects of damping.
9. Explore the amplitude and frequency of the driving force (Fdriver) and it effects on the motion of the system.

recently change 07 June 2009
1. allow the mass to be drag and remember the x initial value.
2. resize the screen and the velocity vector and text to prevent the autoscale to cause the view to move to much.
April 2010
3 updated with slightly better GUI and color scheme.
4 remove plotperiod
5 added dots on all 7 graphs for better visualization of value of graphs.
6 update sliders and checkbox with associated background colors
09 February 2011