Energy Lab Physics

Energy Conservation

Objective: Investigation of energy conservation in oscillating motion.

Theory: If motion is driven by forces, forces are driven by energy. Energy is never created or destroyed; it is only transformed into other forms of energy. If we take an

isolated system, one that is not being influenced from outside, and keep track of all the

various kinds of energy that are involved in its motion, we should find that the sum

total of all the different kinds of energy add up to a constant, the energy of the system.

In the case of an oscillation of a mass on a spring under gravity, there are three kinds of

energy interchanging in the motion. One is the kinetic energy associated with any

matter in motion. Another is the potential energy due to gravity, and the third is the

potential energy due to the spring.

The kinetic energy of an object at any given time may be calculated by the formula

𝐾𝐸 = 1

2 𝑚𝑣2,

where m is the mass of the object and v is its velocity.

The potential energy of an object due to gravity at a given time may be expressed as

𝑃𝐸𝑔 = 𝑚𝑔ℎ,

where m is the mass of the object, g is the acceleration due to gravity (9.80 𝑚 𝑠2⁄

), and h

is the height of the object at that time. Note that the point from which we measure the

height is arbitrary; we track the relative change of the potential rather than its absolute

level.

The potential energy of a spring being stretched is

𝑃𝐸𝑠 = 1

2 𝑘𝑥2,

where x is the distance the spring is being stretched, and k is the spring constant of the

spring in question. The spring constant of a spring depends on both its geometry and its

material; the stiffer the spring, the higher the spring constant. The force required to

stretch a spring x meters is F = kx, so the spring constant has units of Newtons over

meters.

Although there are other types of energy present - such as heat and the energy present

in the matter itself - they do not vary much over the course of this experiment. If the

kinetic energy and the potential energies due to gravity and the spring are the only

significantly varying types of energy in our closed system, they should sum up to a

constant energy of the system. As the potential due to gravity oscillates with the height

of the mass, the energy in the spring is oscillating as well, reaching its highest when the

mass, and therefore the energy due to gravity, is at its lowest. Meanwhile, the kinetic

energy oscillates with the velocity of the mass, which is at its lowest at both the top and

bottom of the mass's oscillation, as it slows and turns around.

However, a truly closed system is much more difficult to achieve in practice than in

theory. In laboratory conditions, for example, the atmosphere interacts with all motions.

There may be friction in the system, dissipating our energy into heat. Additional

motions may be present in the mass, the energy of which is not included in our analysis.

In most real-world conditions, any motion will be damped over time by the

environment the mass is interacting with, and the energy of the system will slowly

decrease as it is redistributed out of the system. Since the rate of decay of energy from

the system usually depends on the amount of energy currently contained in the system,

it is often well described by an exponential decay curve. The time constant of that curve,

the coefficient of time in the equation, is a parameter that controls how quickly the

energy decays.

Apparatus: Force probe, motion sensor, LabQuest Mini, spring, post, cross bar, clamp, 200 g mass to attach to the spring, and a 1.000 kg or 0.500 kg mass to calibrate the force

probe.

Procedure: This lab has three parts: determining the spring constant, measuring energy, and observing the decay of a system with a high air drag coefficient.

1. If needed, switch your force probe to the ±10 N scale.

2. Attach your LabQuest Mini to the computer and the force probe and motion

sensor to the LabQuest Mini.

3. Insert a post in the hole in the table and a small crossbar as close to the top as

possible. You will need room for the mass to hang down on your spring from the

force probe attached to the bar, while staying at least 25 cm from the motion

detector. If there is not enough height to do it on the table, you will need to put

the motion detector on the floor, in which case you want the crossbar to be as

low as possible. Put the motion detector into wide-angle mode (the man

bouncing on the giant basketball) and directly under the mass.

4. Run Logger Pro and calibrate the force probe. For the first reading, the sensor

should be vertical on the crossbar with the hook hanging down; set that force to 0

N. Then use a calibration mass for the second reading.

Part 1: Determine spring elastic constant

5. Click on the clock icon in the menu bar of Logger Pro; set the duration of

measurement to 1.0 s and the samples/sec to 30.

6. Hang the spring from the force probe and the 200 g mass from the spring, with

the motion detector directly below the mass. (An easy way to ensure that the

mass is directly above the motion detector's sensor is to first hang the mass over

the force sensor's hook on a string, lower it down to the motion detector, and

adjust the position of the detector.) Collect some data with the mass oscillating to

ensure the detector is tracking the mass properly, and adjust positions if

necessary. Note that the light fixtures and vent gratings in the ceiling can

sometimes cause problems, if they are in the area scanned by the detector.

7. Hold the mass up by the hook just enough so that the spring is not stretching,

and click menu -> experiment -> zero. Our position measurement will now track

how far the spring is stretched out. Since our motion sensor is pointed in the

opposite direction from this stretching, its values for stretching will be reversed;

and so we will need to correct the sign. The sensor is pointing up in the direction

of greater height, however, so we can use height measurements as is.)

8. Now let the mass hang down at a steady equilibrium with the spring stretched

by the gravity on the mass; once it has settled, record the measured stretch

distance and the force that is stretching it in your worksheet.

9. From the measurements taken in step 8, determine the spring constant and

record it in your worksheet.

Figure 1 An example of graphs when the mass is oscillating

10. Set the mass to oscillating; after its initial wobbles have settled into a steady

rhythm, collect data. Each graph should be sinusoid, as in Figure 1 above; if not,

try to figure out what is wrong and correct it before collecting data again. Once

suitable data is collected, copy the graphs of position and force into the

Conservation tab of the worksheet. Copy the time, force, position, and velocity

columns into the data cells of the worksheet, and also copy it somewhere you can

do calculations from it.

Figure 2 An example of a Force vs Position graph

11. The formula we used for the spring constant implies that if we graph force over

position, k will be the slope of that line. Click on the horizontal time axis of the

force graph in Logger Pro, and show distance on that axis instead. Then add a

linear fit to the data to find the slope. Notice since we have measured position

and force here in opposite directions, the slope will have the opposite sign of its

true value; the spring constant is actually positive. Show the equation and the R2

(or correlation) value for the fit, and include the graph in the Conservation tab.

12. Record the value of k obtained from the linear fit of this measurement.

Part 2. Evaluation of Energy Conservation

13. Now we are ready to begin our analysis of the energy of the system. Next to

your working data, enter a formula into a kinetic energy column to calculate it at

each moment of time using the current value of velocity. Note that we will not be

using significant figures for our energy calculations. Use numeric values for the

unchanging constants, and cell values for the variables. Label the column at the

top, including the units.

Table 1: Force, Motion and Energy of an Oscillating Mass

time (s) force (N) position

(m) velocity (m/s)

kinetic energy

(J)

elastic energy (J)

potential energy (J)

total energy

(J)

change in energy (J)

14. In a labeled elastic energy column, use a formula to calculate the energy stored in

the spring, using the position measurement at each time and the spring constant

as determined by linear fit.

15. In the next column, label and calculate the gravitational potential energy.

16. In the next, label and calculate from the previous columns the total energy.

17. In the last column, calculate the change in the total energy between each

measurement.

18. Make a graph of the kinetic, spring, gravity, and total energies, plotted together

against time. You should include the labels in your collected data, so they will be

used in the graph. You can collect the time data first, then hold the Ctrl key while

you collect the energy data. If your total energy does not seem to at least be

steadier than its components, check your calculations. After appropriately

labeling the graph and its axes, place it in the conservation tab of the worksheet.

19. Make a graph of the change in energy over time, and place it in the worksheet.

20. Question: Within the range of noise in our measurement, does the total energy

seem to be conserved during the oscillation? Looking at your graphs, do you see

any evidence that some other process may have been involved in the system,

interchanging energy with the others?

Part 3 Damping of Oscillations

1. Close Logger Pro, and start the EnergyLab file. This configures Logger Pro to do

our calculations and graph them for us on the fly. Go to Data->User Parameters

to set the spring constant and mass to the values you are using. Next, make sure

the mass is hanging still at equilibrium, and select experiment -> zero to zero the

position. Then set the oscillation going again and hit collect. Note how the total

energy oscillates between the various kinds of energy in turn. You may have to

try the oscillation a few times to get a measurement with relatively little noise.

2. Even though the measured energy of our system is roughly conserved in the

short term, it's not an ideal system. If there were no external interference, and no

degrees of freedom in the system except for the height of the mass, then the

energy of the oscillation would be conserved forever. But there are a number of

factors that will tend to dampen the oscillation of the spring, draining away some

of the energy from the system. Click on the clock symbol in the menu to change

the sampling duration to 30 seconds, and collect data. Try a few until you get one

with relatively little noise. The overall curve we see here is an exponential decay

of total energy; click on the f(x)= button and select a natural exponent fit on the

total mechanical energy.

3. In an exponential decay 𝑦 = 𝐴 𝑒−𝐶𝑡 + 𝐵, the coefficient of time in the exponent

controls the speed of the decay (it is sometimes called the time constant.) Record

the value of the time constant and put the graph in the damping tab of the

worksheet.

4. It may be that much of our energy is being lost in doing the work of moving air

around - air resistance may be damping our oscillation. To check this hypothesis,

let's tape an index card to the bottom of the mass to increase its air resistance.

Make sure it is tightly attached so it does not flap loosely.

5. Set the oscillation going again and collect data; get a few until you get one with

relatively little noise. Fit an exponential curve to the total energy, and place the

graph in the damping tab of the worksheet.

6. Question: How does this curve compare to the previous? Has increasing the air

resistance increased the rate of decay of energy from the system?

7. Record the time constant – the coefficient of time in the exponent of the fitted

curve.

8. Question: How does this time constant with the card compare with the time

constant without the card, and how is that related to the two rates of damping?

Report: Answer the questions in red in the Questions tab; write up a conclusion for a lab report, as described in the Lab Report Rubric, and include it in the Conclusion tab of

the worksheet. Be sure to mention at least 2 potential sources of error besides your

calculations or instruments, and to suggest ways of eliminating or accounting for them.