### The Conservation of Mechanical Energy

Purpose:

The objective of this experiment is to test the idea of Conservation of Mechanical Energy in a situation in which we can assume that all non-conservative forces (friction) can be neglected.

## Equipment:

Air track, Blower, Glider and an Aluminum Flag. Photogatetimer, Micrometer, Triple-Beam Balance and a Riser Block.

## Introduction:

The Law of Conservation of Mechanical Energy states that the sum of potential and kinetic energy is constant when no work is being done by non-conservative forces.

One such situation is when the only force doing work is the force of gravity. The potential energy *U*is then just the gravitational potential energy *U _{g}*:

where *y* is the height above some arbitrary reference point. The kinetic energy *K* is:

The simplest possible situation is that of an object in free fall, if air resistance can be neglected. In this experiment you will test the Law of Conservation of Mechanical Energy in a slightly more complicated situation, a glider going down an inclined air-track.

## Pre-lab:

Before coming to the laboratory, use your data from the “Free Fall” experiment to verify the Conservation of Mechanical Energy in that case.

For each time for which you had calculated the velocity of the falling object, calculate the Potential, Kinetic and Total Energy of the object. (3 new columns in your spreadsheet) The mass of the object was 0.14 kg.

Using Excel^{®}, plot these three energies versus time on the same graph. You should find that the potential energy is decreasing, the kinetic energy is increasing, but the total energy remains constant. For the **total energy data** add a trend-line with equation (no R^{2}) and perform a regression to get the uncertainties in your equation.

**** **Turn in the table and graph when you come to the laboratory.** ****

## The experiment:

You will determine if mechanical energy is conserved for a glider sliding down an inclined air track. You will need to determine both the potential and the kinetic energy of the glider at several points along the track.

Level the air track as best you can, using the glider as a level.

The experiment will require you to determine the kinetic energy of the glider at various points along the inclined air track. This means you will need the velocity (*v*) at each of those points. One way to determine this is to measure the time it takes a small metal flag attached to the glider to pass through a photogate beam. If the flag width is *w* and the time to pass through the beam is Δ*t*, then the average speed of the glider at that point is. However, past experiments have shown that the effective width *w* of the flag is somewhat different from the actual width. That is, due to the width of the photogate beam and the electronics that detects the beam, the timer may “see” the flag as having an effective width that is different from its actual width. It is not unusual for the effective width and the actual width to differ by two or three millimeters. By moving the glider slowly past the photogate by hand, determine the effective width *w* of the flag and the uncertainty in the effective width. Record its width as .

Incline the track by placing a riser block under the single foot of the track. Use a riser block of thickness between 25 and 50 mm. You will need to determine the gravitational potential energy of the glider at various points along the inclined air track. This means you will need to know the glider’s height above some reference level at each of those points. Choose a point at the BOTTOM of the track for your reference point. One way to determine this is to use the height of the riser block with its uncertainty, the distance between the air track feet with its uncertainty and similar triangles to get *y* at each point along the incline.

Choose a starting point for the glider near the top of the track from which you will release the glider. Place a photogate timer at some distance down the track from the starting point. You will need 15 different locations along the track, so think about where you will put the photogate so as to space these out conveniently. Set the timer to measure the time it takes for the glider to pass through the photogate. Then release the glider and note the time measurement. Organize your data in a table, with proper title, column headings, and units. Take care to start the glider from rest at the same point each time. Then move the photogate timer and repeat. Continue until you have all the necessary data.

Determine the mass of the glider, and estimate the uncertainty in the mass as well.

## Analyzing your data:

UsingExcel^{®}, calculate the average time, velocity, kinetic energy, potential energy, and total energy for each location you selected along the inclined air track.

Graph the three energies versus position along the air track on the same set of axes. Include a trend line and equation but not the R^{2} value for the total energy data. Also perform a regression on the total energy data to get the uncertainties in the total energy equation.

Is the total mechanical energy constant? Would you conclude from your results that the total mechanical energy was conserved in this case?

Further Analysis of Data – Taking a closer look at your result:

Manually, select the limits of the y-axis to show the total energy in as much detail as possible.

You almost certainly will find that your total energy appears to have systematically either increased or decreased by a small amount as the glider went down the incline. Assuming you have not simply measured incorrectly or miscalculated, such a trend in the total energy is probably due to a small error in the determination of one or more of your parameters, riser block height, flag width, etc. This may overestimate the kinetic energy compared to the potential energy, or the other way around.

Try to eliminate any apparent variations in the total energy by adjusting one or more of these constants, within your estimated uncertainties.

Here you will be able to make use of one of the most important aspects of using a spreadsheet for your calculation. If you wanted to see how your results would change if you adjust the value for the width of the flag, you would have to change all your calculated velocities . If, on the other hand, you had entered the value of *w* into a cell in your spreadsheet, say cell C14, and then calculated velocities as $C$14/(cell with t value), you could then have Excel^{®} re-calculate everything by just changing the number in cell C14.

Copy your original data to a new page on the spreadsheet. On the new page, change your spreadsheet so that all your parameters, *w*, *m*, and *g*, are entered into their own cells and have Excel^{®} do all calculations by reference to these cells.

Now if, by making adjustments to these parameters (within their uncertainties), you can change the sign of the slope for the total energy, you would conclude that your results were consistent with Conservation of Mechanical Energy. Be sure to perform a regression analysis of your total energy for your optimal graph to get the uncertainties in the equation. You will probably note that changing the mass has no effect since both Potential and Kinetic Energy depend on the mass in the same way.

Finally, clearly state whether you think your results either are or are not consistent with the Conservation of Mechanical Energy, based on your graph, and why you think so.

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