A mass m is attached to a spring of constant k, with the system arranged vertically as shown in Figure 1. All motion is vertical, so the coordinate system is chosen with up as positive. Two possible locations for the origin of the coordinates are shown, one where the spring is relaxed (coordinate s), and the other where the mass is in equilibrium (coordinate y). The right hand drawing shows an arbitrary position where the mass is moving with some speed v.
1. Find the magnitude of the stretch of the spring, |seq,| in terms of m, g, and k when the system is in equilibrium.
2. Write down the total mechanical energy, E, of the system when the mass, m, is a distance s vertically above the relaxed position of the spring, and is moving with a speed v.
3. Change variables by writing s in terms of y, the distance from equilibrium, and |seq,|. Substitute into your expression for mechanical energy and show that E = 1/2 m v2 + 1/2 k y2 + constant,. where the constant is some combination of m, k, and g.
Conservation of mechanical energy states Einitial = Efinal , so adding a constant to the energy makes no difference. Therefore we will define mechanical energy for this system as
E = 1/2 m v2 + 1/2 k y2 where y = 0 and E = 0 at equilibrium.
You will measure position and force, and from these calculate velocity and acceleration and eventually kinetic, potential, and total mechanical energies. Write several hypotheses about these quantities that you will be able to test. Submit your hypotheses before next class.
Real springs have mass, msp,distributed along the entire length of the spring. This mass must be accounted for somehow by an "effective mass of the spring." For a spring on a horizontal surface the effective mass would be (msp/3). It may different for a spring hung vertically.
The basic experimental arrangement is shown below.
1. Set the slide switch on the force probe to 10 N. Place the motion sensor on the floor. Attach the force sensor to In 1, and the motion sensor to Dig/Sonic 1.
2. Open the experiment Mechanical Energy.
3. Calibrate the force sensor.
4. Orient the spring with the wide end up, and tape masses to the mass hanger for a total of about 250 g. The mass should be about 5 or 10 cm below the bench top at equilibrium. Zero the sensors appropriately.
Check your set-up by lifting the mass straight up about 10 cm and releasing it. Avoid side-to-side motion. What do you expect to see for position as a function of time? Collect data--do you see what you expected? If not adjust sensors until you get what you expect.
5. Due to a hardware glitch you will need to adjust the zero manually. Return the system to equilibrium and zero all sensors. Collect data with the system at rest, and you may find that the position is not zero. Use Analyze:Statistics to determine the average position of the mass. Click on the oval Labpro icon to bring up the Sensors Window, and click on the motion sensor. On the pop up menu click Set Offset. And a box will pop up with an initial value of the offset. Subtract the average position from the offset and enter as the new offset and click OK. Collect data again and the position should have an average very close to zero. Close the Sensors window. (E.g. if the average position is 0.0123 and the initial offset is -0.612345, the new offset is -0.624645.) Each time you zero the motion detector you will need to repeat this.
Use Analyze Data Examine to look at the highest and lowest distance the mass reaches. How should the values compare for good data? If your data are not good, make appropriate adjustments and take new data.
Be absolutely certain that you have the equipment properly zeroed or your results will be bogus!
6. Start the mass in oscillation and let the mass oscillate for about a minute (why?), then Collect data again. Repeat the experiment until you have good data. Before you proceed, ask an instructor if you have good data.
7. If you want to plot more than one measurement on the vertical axis, click the name, click More on the pop up menu, and check the boxes for the items that you wish to plot.
Compare the Force and Distance graphs, and give a verbal description of their relation.
Compare the Force and Acceleration graphs, and give a verbal description of their relation.
8. Determine what graphs to plot to find the values of k and m (effective mass.) If you make a linear fit to data, be sure to record the slope and intercept, and to write the equation in proper terms. (The computer will always write y = mx + b, but you must write the variables to match what is plotted, F for force, etc.)
9. Program formulas for the kinetic and potential energies. K = 1/2 m v2 and U = 1/2 k y2 .
To program a new column for kinetic energy: From Data: New Calculated Columns enter values for the name (e.g. Kinetic Energy), short name (e.g. K), and units (J). In the Equation box enter the formula for kinetic energy. You should know the effective mass from part 8 (suppose you found 0.999 kg), and can get the velocity from the Variables pull down menu. The equation might look like
0.5 * 0.999 * "velocity"^2
In a similar fashion define the potential energy, U, and the total mechanical energy, E = K + U.
If you need to change a definition, select the column from Data:Column Options and the editing widow will appear.
10. Make graphs that allow you to discuss conservation of energy. Examples of graphs might be any one of the energies versus time (or position, or velocity, or acceleration), one energy versus another energy, etc. You need to try different graphs and decide what they tell you about energy conservation.
In your report be sure to discuss the graphs. For example from a graph of energies versus time you might discuss what is the value of potential energy when the kinetic energy is a maximum, a minimum, or zero. You should also discuss some of your hypotheses and decide what the data tells you about them.