Education and Manpower Bureau
Using Datalogger in the Teaching of Physics

Simple Harmonic Motion

Experiment Profile:

1. Background

Simple Harmonic Motion is a form of periodic motion in which a point of body oscillates along a line about a central point in such a way that it ranges an equal distance on either side of the central point and is always proportional to its distance from it. One way of visualising SHM is to imagine a point rotating around a circle of radius r at a constant angular velocity w. If the distance from the centre of the circle to the projection of this point on a vertical diametre is y at time t, this projection of the point will move about the centre of the circle with Simple Harmonic Motion. A graph of y against t will be a sine wave, whose equation is y = rsinwt.

We will use the mass-spring system to illustrate the concept of Simple Harmonic Motion in this experiment. During SHM, the mass moves upwards and downwards, changing the length of the spring. Three forms of energy are involved in this motion - gravitational potential, translation kinetic and elastic potential. In this lab, you will examine the relationships between these three quantities throughout a single cycle of motion and test the conservation of mechanical energy.

2. Objective

 To study the characteristics of Simple Harmonic Motion. To investigate the motion of a mass-spring system. To test the conservation of mechanical energy in Simple Harmonic Motion.

3. Equipment List

 Datalogger interface connected to PC Motion sensor Masses and mass hanger Base and support rod with clamp Spring

 Experiment Collision and Conservation of Momentum Simple Harmonic Motion Acceleration due to Gravity Pushing and Pulling a Dynamics Cart Acceleration of a Dynamic Cart

 Procedure.

 1 Connect the datalogger interface to a PC with the software installed. 2 Connect the motion sensor to the appropriate channel of the interface. See Figure 1. 3 Record the total mass. 4 Position the mass so it hangs 60-70 cm above the motion sensor when it is in equilibrium. 5 Launch the software needed for measuring position and velocity during your experiment. 6 Pull the mass down approximately 10 cm (if the spring will allow this much extension) and release it, setting it in to SHM. 7 When the motion is smooth and straight up and down, begin data collection. 8 Stop the data collection after two or three cycles have elapsed. 9 Save the data file collected for further analysis. Repeat the procedure using a new hanging mass or a new spring.

Analysis:

1. As the mass moves up and down, what energies are involved? How did the mass get the original amount of each kind of energy?

2. What relationship gives the amount of elastic energy in the spring?

3. How can you tell that this is a Simple Harmonic Motion?

4. Compare the amounts of each kind of energy (in a qualitative way) at the three extremes of motion - highest, middle and lowest point.

5. Construct a spreadsheet that contains the values of the following quantities:
 * Constants: Mass, Spring Constant * Variables: Position (height), Velocity, Time * Calculated Values: Gravitational Potential Energy (Ug) Kinetic Energy (K) Elastic Potential Energy (Ue) Total Mechanical Energy (Et)

6. Record the position, velocity and time for at least 15 different positions during a single cycle of the motion.

7. Calculate Ug based on height above the lowest point of the motion.

8. Calculate Ue based on distance below the highest point of the motion.

9. Graph all four calculated values as functions of time.

10. What is your conclusion regarding the total mechanical energy during a cycle of motion?

Extensive:

If one considers that energy must be conserved, and therefore the total energy at each position must be the same, the lab can be re-configured to dynamically determine the spring constant k. What value of k would keep the total energy constant, and how does this agree/disagree with the value of k determined in a separate measurement?

 Acknowledgements We are particularly grateful to SKH Kei Hau Secondary School for their help in the production of this video.

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