SHM model for inquiry learning virtual laboratory

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. 

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 / dt_² = -k/m (x-l),

where x is the horizontal position of the mass from the from the origin centre of the springs.

Exercises:

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, ω. Select the 'expert' checkbox and look for the values of f and ω in relations to T.
   
5. The equation a = –ω²x 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 ω in terms of k and m.
6. The equation  v = v_ocosω 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 = ±ω Math.sqrt ( x_o² - 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.
   

• http://www.seab.gov.sg/SEAB/aLevel/syllabus/2010_GCE_A_Level_Syllabuses/9646_2010.pdf
  

Credits:

The simple harmonic motion Model was created by Loo Kang WEE, Francisco Esquembre, Fu-Kwun Hwang and Wolfgang Christian using the Easy Java Simulations (EJS) authoring and modeling tool.

You can examine and modify a compiled EJS model if you run the program by double clicking on the model's jar file.  Right-click within the running program and select "Open EJS Model" from the pop-up menu to copy the model's XML description into EJS.  You must, of course, have EJS installed on your computer.

Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.