Simple Harmonic Motion
Equipment
- LabQuest 2
- Vernier motion detector
- Lab stand with arm
- Springs with varying values of spring constant
- Masses ranging 50 - 250 grams
- A useful Google Sheets data table
Remarks on Hooke's Law. Note that we are working in the vertical direction, so we will use the expression
$$ F = k(y-y_0) $$Notes on simple harmonic motion (SHM)
$$ y(t) = A \sin (\omega t + \phi) + y_{offset} $$
Determining spring constant
- Hang spring from lab stand arm.
- Hang added mass and determine equilibrium position of the hanging mass.
- Add 4 additional masses and, for each one, find the new equilibrium position.
Position, Velocity and Acceleration of an SHM system
- Place the motion detector, face up, on the table below the hanging mass. Connect the motion detector to the LabQuest.
- Pull down on the hanging mass by a few centimeters and let go. Start recording data on the LabQuest.
- You should see record \( y(t)\) and \( v(t)\) plots. Repeat the trial until you get a nice, smooth plot for both.
Energy in an SHM system
- Do one trial in which you measure the initial displacement (amount you pulled down) of the hanging mass.
Determining spring constant
- Make a plot of equilibrium position of the hanging mass (y-axis) versus the total hanging mass.
- Determine the spring constant k from this plot.
Position, Velocity and Acceleration of an SHM system
- Use the \( y(t)\) vs. t plot to find the maximum displacement (Amplitude, A), maximum velocity (slope at the zero-crossing) and period of the oscillating motion.
- Use the \( v(t)\) vs. t plot to find the maximum velocity and compare to your value from the \( y(t)\) plot. Find the maximum acceleration (slope at zero-crossing of this graph).
- Sketch the \( y(t)\) vs. t and \( v(t)\) vs. t graphs plus your best guess at an \( a(t)\) vs. t plot.
- Determine \( \omega \) and \( \phi \) for the system and write the equation of motion in the form of Equation 2.
Energy in an SHM system
- Calculate the maximum kinetic energy (\(KE_{Max}\)) in the system
- Calculate the maximum spring potential energy (\(PE_{Max, spring}\)) in the system
- Consider the role of gravitational potential energy (\(PE_{grav}\)) in the system
- Compare total mechanical energy for the points when \(KE\) is maximum and when \(PE_{spring}\) is maximum. Is \(TE\) conserved?
Answer the following questions in your lab report:
- Why or why not?
- What could you say about it ?
- Assume that you will do the experiment one more time. Why or why not?
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Value is from NOAA for Latitude: 40.91597332879679, Longitude: -73.12491620370486, MSL Height: 41.0 Predicted gravity: 980269 +/- 2 milligals where a "gal" is a centimeter per second squared