4. Simple Harmonic Motion I.

The following is a simulation of a mass on a spring. The graph shows the y location of the mass. The force acting on the mass in this case is called a Hooke's Law force: F = -ky where k is called the spring constant, in N/m indicating the stiffness of the spring and y is the location of the mass from some equilibrium position.

Note: Animations may take a few seconds to load.

Questions:

4.1. Click on the 'set parameter' button, left click the mouse over the red mass on the spring and drag it to a starting location. Click 'play' to see the motion and the graph of the mass location. How is the graph you see here similar to the graph you saw for the motion of the red circle in the transverse wave case?

4.2. Determine the period, frequency and angular frequency of this motion from the values on the graph (Hint: Frequency in Hz is the inverse of period and angular frequency is 2pf).

4.3. Click the 'show velocity' box to see graphs of both position and velocity. Where is the mass when the velocity is a maximum? Where is the mass when the velocity becomes zero?

4.4. Try different spring constant values, k, between 0.5 N/m to 5.0 N/m, releasing the mass at the same point each time. What is the relationship between spring constant and frequency? Note that you can right-click on the graph to create a copy at any time to save the results of several different cases.

4.5. For the same spring constant try releasing the ball at several different amplitudes. What is the relationship between amplitude and frequency?

In the previous simulation (3. Transverse Waves) the red circle moved up and down as the result of a transverse wave traveling horizontally along the string of particles. The equation for the motion of the entire string is

. If we were to assume that the red particle was located at x = 0 the equation describing the motion of just the red circle would be

where A is the maximum amplitude, w is the angular velocity and f is the phase. This is also the formula for simple harmonic motion which describes the location of a mass on a spring as a function of time. In other words the motion of each point on a transverse wave is exactly the same as if each of those points was undergoing simple harmonic motion but with a slightly different phase from its neighbor.

4.6. If

is the location of the mass on the spring and the time derivative of location is velocity, then the velocity of the mass is given by

. Click the 'show velocity' button and run a simulation with an initial amplitude of 6.0 m and a spring constant of 2. From the graph find the angular frequency and calculate the maximum amplitude v_max = Aw. How does this number compare with the maximum value on the velocity graph; are they the same?

Credits.

Go To: IUS Physics Top Page.
Contact Dr. K. Forinash, for comments/suggestions/corrections.