energy conversion 1

Energy conversion 1 Commentary

For part of this experiment you will use the calibrated rubber band you used in the preceding experiment 'Force vs. Displacement 1', as well as the results you noted for that experiment.

For this experiment you will need to use at least one rubber band in such a way as to make it useless for subsequent experiments. DO NOT USE ONE OF YOUR CALIBRATED RUBBER BANDS. Also note that you will use four of the thin rubber bands in a subsequent experiment, so DO NOT USE THOSE RUBBER BANDS HERE.
If your kit has extra rubber bands in addition to these, you may use one of them.

You are going to use a non-calibrated rubber band to bind three of your dominoes into a block. You should have plenty, but if you don't have extra rubber bands, you could use some of the thread that came with your kit, but rubber bands are easier to use.

The idea of binding the dominoes is very simple. Just set one domino on a tabletop so that it lies on one of its long edges. Then set another right next to it, so the faces of the two dominoes (the flat sides with the dots) are touching. Set a third domino in the same way, so you have a 'block' of three dominoes.
Bind the three dominoes together into a 'block' using a rubber band or several loops of thread, wrapping horizontally around the middle of the 'block', oriented in such a way that the block remains in contact with the table. The figure below shows three dominoes bound in this manner, resting on a tabletop.

Now place a piece of paper flat on the table, and place the block at one end of the paper, lying on the paper.

Give the block a little push, hard enough that it slides about half the length of the paper.
Give it a harder push, so that it slides about the length of the paper, but not quite.
Give it a push that's hard enough to send it past the other end of the paper.

You might need to slide the block a little further than the length of one sheet, so add a second sheet of paper:

Place another piece of paper end-to-end with your first sheet.
Tuck the edge of one sheet slightly under the other, so that if the block slides across the first sheet it can slide smoothly onto the second.

You are going to use a calibrated rubber band to accelerate the blocks and make them slide across the table.

Tie two pieces of thread to the rubber bands holding the blocks, at the two ends of the block, so that if you wanted you could pull the block along with the threads. One thread should be a couple feet long--a little longer than your two consecutive sheets of paper. The other thread needs to be only long enough that you can grasp it and pull the block back against a small resistance.
At the free end of the longer thread, tie a hook made from a paper clip.
Use the rubber band you used in the preceding experiment (the 'first rubber band' from your kit, the one for which you obtained the average force * distance results). Hook that rubber band to the hook at the free end of the longer thread.
Make another hook, and put it through the other end of the rubber band loop, so that when you pull on this hook the rubber band stretches slightly, the string becomes taut and the block slides across the tabletop.

You will need something to which to attach the last hook:

Now place on the tabletop some object, heavy enough and of appropriate shape, so that the last hook can in one way or another be fixed to that object, and the object is heavy enough to remain in place if the rubber band is stretched within its limits. That is, the object should be massive enough to remain stationary if a few Newtons of force is applied. Any rigid object weighing, or being weighted by, about 5-10 pounds ought to be sufficient. [ Technically we should put it as follows: the object should be heavy enough so the resulting static frictional force between it and the tabletop exceeds the force of your pull. ]
Your goal is to end up with a moderately massive object, to which the last hook is tied or attached, with the rubber band extending from the hook to another hook, a thread from that hook to the block (with a shorter thread trailing from the other end of the block)
With a slight tension in the system the block should be a few centimeters from the 'far' edge of the paper which is furthest from the massive object.
If the block is pulled back a little ways (not so much that the rubber band exceeds its maximum tolerated length) the rubber band will stretch but the last hook will remain in place, and if the block is then released the rubber band will snap back and pull the block across the tabletop until the rubber band goes slack and the block then coasts to rest.
The figure below shows the block resting on the paper, with the thread running from a hook to the rubber band at the far end, which is in turn hooked to the base of a computer monitor.

The rubber band is ready to be stretched between two hooks.

Consult your results from the Rubber Band Calibration experiment and determine the rubber band length required to support the weight of two dominoes. Pulling by the shorter piece of thread (the 'tail' of thread), pull the block back until the rubber band reaches this length (e.g., if you the rubber band supports two dominoes when its length is 7.9 cm, you would pull back until the rubber band is 7.9 cm long). On the paper mark the position of the center of the block (there might well be a mark at the center of the domino; if not, make one as near the center of the block as possible, and mark the paper accordingly). Release the thread and see whether or not the block moves. If it does, mark the position where it comes to rest as follows:

Make a mark on the paper at the center mark of the domino. Make your mark by drawing a short line segment, perhaps 3 mm long, starting from the center mark and running perpendicular to the length of the block.
Make another mark about twice the length of the first, along the edge of the block to that your two marks make sort of a 'long' T shape. The 'top' of the T indicates the direction of the resting domino, that 'tip' of the T marks the position of its center.
In the first figure below the lowest two marks represent the positions of the center of the dominoes at initial point and at the pullback point. The T-shaped mark is partially obscured by the domino.

You will make a similar mark for the final position for each trial of the experiment, and from these marks you will later be able to tell where the center mark ended up for each trial, and the approximate orientation of the block at the end of each trial.

Based on this first mark, how far, in cm, did the block travel after being released, and through approximately how many degrees did it rotate before coming to rest?
If the block didn't move, your answers to both of these questions will be 0.

Answer in comma-delimited format in the first line below. Give a brief explanation of the meaning of your numbers starting in the second line.

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Tape the paper to the tabletop, or otherwise secure it so that it doesn't move during subsequent trials.

Repeat the previous instruction until you have completed five trials with the rubber band at same length as before.

Report your results in the same format as before, in 5 lines. Starting in the sixth line give a brief description of the meaning of your numbers and how they were obtained:

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Now, without making any marks, pull back a bit further and release.

Make sure the length of the rubber band doesn't exceed its original length by more than 30%, with within that restriction what rubber band length will cause the block to slide a total of 5 cm, then 10 cm, then 15 cm.
You don't need to measure anything with great precision, and you don't need to record more than one trial for each sliding distance, but for the trials you record:

The block should rotate as little as possible, through no more than about 30 degrees of total rotation, and

it should slide the whole distance, without skipping or bouncing along.

You can adjust the position of the rubber band that holds the block together, the angle at which you hold the 'tail', etc., to eliminate skipping and bouncing, and keep rotation to a minimum.

Indicate in the first comma-delimited line the rubber band lengths that resulted in 5 cm, 10 cm and 15 cm slides. If some of these distances were not possible within the 30% restriction on the stretch of the rubber band, indicate this in the second line. Starting in the third line give a brief description of the meaning of these numbers.

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Now record 5 trials, but this time with the rubber band tension equal to that observed (in the preceding experiment) when supporting 4 dominoes. Mark and report only trials in which the block rotated through less than 30 degrees, and in which the block remained in sliding contact with the paper throughout (i.e., if the block bounced or rolled, either discard your measurements or don't measure them in the first place).

Report your distance and rotation in the same format as before, in 5 lines. Briefly describe what your results mean, starting in the sixth line:

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Repeat with the rubber band tension equal to that observed when supporting 6 dominoes and report in the same format below, with a brief description starting in the sixth line:

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Repeat with the rubber band tension equal to that observed when supporting 8 dominoes and report in the same format below, including a brief description starting in the sixth line:

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Repeat with the rubber band tension equal to that observed when supporting 10 dominoes and report in the same format below, including your brief description as before:

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In the preceding experiment you calculated the energy associated with each of the stretches used in this experiment.

The question we wish to answer here is how that energy is related to the resulting sliding distance.

For each set of 5 trials, find the mean and standard deviation of the 5 distances. You may use the data analysis program or any other means you might prefer.
In the box below, report in five comma-delimited lines, one for each set of trials, the length of the rubber band, the number of dominoes supported at this length, the mean and the standard deviation of the sliding distance in cm, and the energy associated with the stretch.
You might choose to report energy here in Joules, in ergs, in Newton * cm or in Newton * mm. Any of these choices is acceptable.
Starting in the sixth line specify the units of your reported energy and a brief description of how your results were obtained. Be sure to give a good description of how you obtained the energy associated with each stretch:

The force of gravity on the suspended dominoes corresponds to the force exerted by the rubber band on the block at its instant of release.

The force exerted by the rubber band immediately starts decreasing, and reaches zero when the rubber band goes slack.

Between release and this point an average force, which is about half the initial force, is exerted through a distance of at most a couple of centimeters.

After this point the rubber band does no more work on the block.

The rubber band energies are calculated using its force * displacement graph, as in the preceding exercise entitles 'Force vs. Displacement 1'.

A reasonable approximation to the potential energy of a rubber band at a given length is the 'linear' approximation between the maximum unstretched length and the given length.

potential energy = average force * change in length = (force at given length + 0) / 2 * (stretched length - max unstretched length).

For example a rubber band that starts exerting force at length 7.5 cm and exerts a force of 1.2 N at a length of 8.2 cm exerts an average force of about (1.2 N + 0 N) / 2 over the distance interval (8.2 cm - 7.5 cm) = .7 cm, so that its potential energy is about

approximate PE = ave force * distance = .6 N * .7 cm = .42 N cm

This 'straight-line' energy estimate could be refined by more accurately estimating the area under the corresponding interval on the force vs. length graph.

Sketch a graph of sliding distance vs. energy, as reported in the preceding box.

Fit the best possible straight line to your graph, and give in the first comma-delimited line the slope and vertical intercept of your line.
In the second line specify the units of the slope and the vertical intercept.
Starting in the third line describe how closely your data points cluster about the line, and whether the data points seem to indicate a straight-line relationship or whether they appear to indicate some sort of curvature.
If curvature is indicated, describe whether the curvature appears to indicate upward concavity (for this increasing graph, increasing at an increasing rate) or downward concavity (for this increasing graph, increasing at a decreasing rate).

The graph will be increasing--more pullback means more energy as well as longer sliding distance.

Ideally the graph would be linear, but it is commonly reported that the graph increases at a decreasing rate (i.e., that it is concave down).

The 'rise' of the graph is typically around 20 cm, the 'run' typically around 1 N * cm. This would result in a graph slope of about 20 cm / (1 N * cm) = 20 / N or 20 N^-1.

Now repeat the entire procedure and analysis, but add a second rubber band to the system, in series with the first.

For each trial, stretch until the first rubber band is at the length corresponding to the specified number of dominoes, then measure the second rubber band and record this length with your results.
When graphing mean sliding distance vs. energy, assume for now that the second rubber band contributes an amount of energy equal to that of the first. You will therefore use double the energy you did previously.
When you have completed the entire procedure report your results in the boxes below, as indicated:

Report in comma-delimited format the length of the first rubber band when supporting the specified number of dominoes, and the length you measured in this experiment for second band. You will have a pair of lengths corresponding to two dominoes, four dominoes, ..., ten dominoes. Report in 5 lines:

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Report for each set of 5 trials your mean sliding distance and the corresponding standard deviation; you did five sets of 5 trials so you will report five lines of data, with two numbers in each line:

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Give the information from your graph:

Give in the first comma-delimited line the slope and vertical intercept of your line.
In the second line specify the units of the slope and the vertical intercept.
Starting in the third line describe how closely your data points cluster about the line, and whether the data points seem to indicate a straight-line relationship or whether they appear to indicate some sort of curvature.
If curvature is indicated, describe whether the curvature appears to indicate upward concavity (for this increasing graph, increasing at an increasing rate) or downward concavity (for this increasing graph, increasing at a decreasing rate).

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In the box below, report in the first line, in comma-delimited format, the sliding distance with 1 rubber band under 2-domino tension, then the sliding distance with 2 rubber bands under the same 2-domino tension.

The in the subsequent lines report the same information for 4-, 6-, 8- and 10-domino tensions.

You will have five lines with two numbers in each line:

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The preceding box will comprise a table of 2-rubber-band sliding distances vs. 1-rubber-band sliding distances.

Sketch a graph of this information, fit a straight line and determine its y-intercept, its slope, and other characteristics as specified:

Give in the first comma-delimited line the slope and vertical intercept of your line.
In the second line specify the units of the slope and the vertical intercept.
Starting in the third line describe how closely your data points cluster about the line, and whether the data points seem to indicate a straight-line relationship or whether they appear to indicate some sort of curvature.
If curvature is indicated, describe whether the curvature appears to indicate upward concavity (for this increasing graph, increasing at an increasing rate) or downward concavity (for this increasing graph, increasing at a decreasing rate).

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To what extent do you believe this experiment supports the following hypotheses:

The sliding distance is directly proportional to the amount of energy required to stretch the rubber band. If two rubber bands are used the sliding distance is determined by the total amount of energy required to stretch them.

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Additional Notes:

A student gives the following data; the data look very reasonable:

Rubber band length, the number of dominoes supported at this length, the mean and the
standard deviation of the sliding distance in cm, and the energy associated with the stretch, for each
set of 5 trials:

0,0 , 8cm 3 dominos 
.61,.0746 8.4cm 4 dominos 
.97,.09747 8.7cm 6 dominos 
1.17,.1204 8.9cm 8 dominos 
1.3,.03536 9.2cm 1o dominos

The student asks the following question:

I need you to explain to me how to get the Joules cause Im sure I didnt do it right so If you could
explain it I would appreciate it 

INSTRUCTOR RESPONSE:

Each domino supported requires an additional tension force of .2 Newtons.

When force is exerted in the direction of motion, as in the case of the rubber band propelling the domino block, the work done by the force is

`dW = average force * distance.

The average force exerted between the 8 domino length (where force is about 8 dominoes * .2 N / domino = 1.6 N) and the 10 domino length (where force is about 2.0 N) is very close to 1.8 N, the average of the minimum and maximum forces exerted on that interval.

The distance through which this force acts is the difference between the 8 domino length and the 10 domino length.

The standard unit of work in the Joule, which is a Newton * meter. So if your average force is expressed in Newtons and your distance in meters, the work will be in Joules.

The fundamental unit of a Newton is a kg * m / s^2; the fundamental unit of a Joule is a kg m^2 / s^2.

See also links to solutions/discussion of related cq problems, which have addressed this type of situation, and the preceding experiment on force vs. displacement.

For your data it appears that the rubber band contracts .3 cm between the 8 and 10 domino lengths, so that the work done in this contraction would be .3 cm * 1.8 N = .003 m * 1.8 N = .054 Joule.

Similar calculations could be made for the intervals between 6 and 8 dominoes, between 4 and 6 dominoes, etc..

If the rubber band contracts from the 10 domino length to its original length, the total work would be obtained by adding up the work for each intervening interval.