In this demonstration, a mass hangs from a spring. A motion sensor is placed on the floor under the spring and it will measure the distance from the sensor to the bottom of the mass. Initially the mass is 80 cm above the sensor.
1. Sketch a graph showing how the position of the mass changes with time when it is pulled down 10 cm from its equilibrium position and released.
2. How does your graph compare to your neighbors?
3. What are the important characteristics of your graph that you used to compare your graph to your neighbor's graph?
Now observe what happens when the mass is pulled down.
Interactive Mathematics Lecture Demonstration
4. How does the actual graph compare to your prediction?
A general equation to describe this type of graph is
y = A cos(x - C)) + D or y = A sin (x - C)) + D
6. For each variable in the equation, state in your own words either the physical meaning of the variable or what part of the graph is related to that particular variable for this experiment.
y
x
A
B
C
D
7. Indicate on the following graph any of the above variables that can be directly determined by looking at the graph. For instance what part of the graph is represented by A?
Interactive Mathematics Lecture Demonstration
To find the parameters for a sinusoidal function, it is convenient to use the equation in the form y = A cos(x - C)) + D.
In this form the parameters can be found as follows:
To find A, take half the difference between the largest and smallest values of y.
The parameter B represents the number of cycles the sinusoidal function makes during the natural period of the cosine function. To find B, you first need to find the period of the waveform, which is the time it takes the waveform to repeat itself. The period is most easily found by measuring the time between two successive peaks or valleys. Then
, the parameter B is given the name angular frequency and the symbol is commonly used instead of B.
The parameter C represents the horizontal shift (or phase shift) of the cosine function. Since the cosine function has a maximum value when its argument is zero, the time when your data first reaches a maximum will serve as an estimate of C. [If you are using the sine function, y = A sin(x - C)) + D, C = x-coordinate of maximum – T/4.] The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive
8. From the following graph, determine the
Amplitude, A
Period, T
Angular frequency,
Phase shift, C
Zero offset, D
Interactive Mathematics Lecture Demonstration
Appendix 1. Laboratory Measured Data of Mass-Spring Oscillation
Time Distance Time Distance Time Distance (sec) (m) (sec) (m) (sec) (m)
Instructor Notes:
Learning Outcomes:
Upon completion of this module the students should be able to:
Find an equation for an object in simple harmonic motion.
Equipment: mass, spring, stand to hold the spring
3. frequency of oscillation, amplitude, sinusoidal shape
5. cosine of sine function
6. y – height of mass above the floor x – time
A – amplitude of oscillation B – is related to the period
C – related to how much the sine wave has been shifted horizontally D – initial height of the mass
7.
8. To measure the system parameters, you can either have the students estimate the parameters from the graph, or you can give them the measured data values listed in Appendix 1.
amplitude: m
Period: T = Peak2 time – Peak1 time = 1.825 – 0.775 = 1.05 sec
Angular frequency, Phase shift, C = 0.775 sec
Interactive Mathematics Lecture Demonstration
Zero shift,
