Question

Finding the Spring Constant

We can describe an oscillating mass in terms of its position, velocity, and acceleration as a function of time. We can also describe the system from an energy perspective. In this experiment, you will measure the position and velocity as a function of time for an oscillating mass and spring system, and from those data, plot the kinetic and potential energies of the system.

Energy is present in three forms for the mass and spring system. The mass, m, with velocity, v, can have kinetic energy KE

The spring can hold elastic potential energy, or PEelastic. We calculate PEelastic by using

where k is the spring constant and y is the extension or compression of the spring measured from the equilibrium position.

The mass and spring system also has gravitational potential energy (PEgravitational = mgy), but we do not have to include the gravitational potential energy term if we measure the spring length from the hanging equilibrium position. We can then concentrate on the exchange of energy between kinetic energy and elastic potential energy.

If there are no other forces experienced by the system, then the principle of conservation of energy tells us that the sum ΔKE + ΔPEelastic = 0, which we can test experimentally.

objectives

• Examine the energies involved in simple harmonic motion.
• Test the principle of conservation of energy.

Materials

computer

Vernier computer interface

Logger Pro

Vernier Motion Detector

wire basket

ring stand

slotted mass set, 50 g to 300 g in 50 g steps

slotted mass hanger

200 g hooked mass

spring, 15 N/m

twist ties

predicting the Motion

1. Sketch a graph of height vs. time for the mass on the spring as it oscillates up and down through one cycle.
2. Mark on the graph the times where the mass moves the fastest and, therefore, has the greatest kinetic energy.
3. Also mark the times when it moves most slowly and has the least kinetic energy.
4. From your graph of height vs. time, sketch velocity vs. time.

Procedure

Part I Preliminary data collection

1. Set up the experiment.

• 1. Mount the 200 g mass and spring, as shown in Figure 1. Securely fasten the 200 g mass to the spring, and the spring to the rod, using twist ties so the mass cannot fall.

Figure 2

• 1. Connect the Motion Detector to a digital (DIG) port of the interface. Set the Motion Detector sensitivity switch to Ball/Walk.
  2. Position the Motion Detector directly below the hanging mass, taking care that no extraneous objects could send reflections back to the detector. Protect the Motion Detector by placing a wire basket over the detector. The mass should be about 30 cm above the detector when it is at rest. Using amplitudes of 5 cm or less will then keep the mass outside of the 15 cm minimum distance of the Motion Detector.

1. Open the file “17a Energy in SHM” from the Physics with Vernier folder.
2. Start the mass moving up and down by lifting it 5 cm and then releasing it. Take care that the mass is not swinging from side to side. Click to record position and velocity data. Print your graphs and compare to your predictions. Comment on any differences.

Part II Determining spring constant

To calculate the spring potential energy, it is necessary to measure the spring constant, k. Hooke’s law states that the spring force is proportional to its extension from equilibrium, or F = –kx. You can apply a known force to the spring, to be balanced in magnitude by the spring force, by hanging a range of weights from the spring. The Motion Detector can then be used to measure the equilibrium position. You will plot the weight vs. position to find the spring constant, k.

1. Open the experiment file “17b Energy in SHM.” Logger Pro is now set up to plot the applied weight vs. position.
2. Click to begin data collection. Hang a 50 g mass from the spring and allow the mass to hang motionless. Click and enter 0.49, the weight of the mass in newtons (N). Press ENTER to complete the entry. Now hang 100, 150, 200, 250, and 300 g from the spring, recording the position and entering the weights in newtons. It is important that the length of the mass not change as the value of the mass is changed. When you are finished, click to end data collection.
3. Click Linear Fit, , to fit a straight line to your data.
4. What are the units of the slope?
5. What does the slope measure?
6. Record the slope value in the data table.

DATA TABLE

N/m

Questions

1. What is the spring force?

2. What does the spring constant describe.

3. Check your prediction. If you look at the velocity graph can you tell when the kinetic energy is the greatest?

4. If you where on the moon do you think you would get a different value for the spring constant?

Answer 1