energy conversion 1

Phy 231

Your 'energy conversion 1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

Your optional message or comment:

How far and through what angle did the block displace on a single trial, with rubber band tension equal to the weight of two dominoes?

1, 0

The rubber band was stretched to 7.2cm but only moved the block 1 cm. I made sure the string connected the block in the middle so the block didn’t rotate as it moved. The second number is the number of degrees the block rotated.

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of two dominoes:

1,0

.5, 0

1, 0

.5, 0

The first number in each line is the number of cm the block moved when I stretched the rubber band to 7.2cm. The second number in each line is the number of degrees the block rotated. The string is right in the middle of the block, and I pulled the block straight so the block didn't rotate at all.

Rubber band lengths resulting in 5 cm, 10 cm and 15 cm slides:

7.5, 7.9, 8.5

These numbers are the length in cm that the rubber band was stretched for the block to slide 5, 10, and 15cm. The second numbers are the rotation in degrees of the block.

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of four dominoes:

8.2, 13

8.2, 7

8.7, 15

8.0, 5

8.0. 7

The first numbers in each line are the length the rubber band slid when I stretched the rubber band 7.8cm. The second numbers are the rotation in degrees of the block.

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of six dominoes:

12.5, 20

12.0, 17

11.8, 15

12.7, 22

12.4, 10

The first numbers in each line are the length the rubber band slid when I stretched the rubber band 8.1cm. The second numbers are the rotation in degrees of the block.

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of eight dominoes:

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of ten dominoes:

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:

7.2, 2, .8, .2739, .30

7.5, 4, 4.88, .2949, 3.7

7.8, 6, 8.22, .2864, 9.4

8.1, 8, 12.28, .3701, 18.7

8.2, 10, 12.52, .2588, 23.8

The energy in each line is in N*cm. I took the N force gravity exerts on the dominos from the previous experiment and multiplied this by the average length of the slide of the block.

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.

So it is not appropriate to multiply the initial force exerted by the rubber band, by the distance of the slide.

Slope and vertical intercept of straight-line approximation to sliding distance vs. energy, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature:

2, 0

Newtons, N*cm

according to your graph, if you have the vertical and horizontal coordinates on the appropriate axes (sliding distance on vertical, energy on horizontal), your slope would be about (20 cm) / (10 N * cm) = 2 / N or 2 N^-1.

My points are in a curvature that suggests the best fit line is increasing at an increasing rate.

Lengths of first and second rubber band for (first-band) tensions supporting 2, 4, 6, 8 and 10 dominoes:

5, 0

Newtons, N*cm

The points are clustered around a line that is increasing at an increasing rate.

Mean sliding distance and std dev for each set of 5 trials, using 2 rubber bands in series:

7.2, 7.0

7.5, 7.3

7.8, 7.7

8.1, 7.8

8.2, 7.8

Slope and vertical intercept of straight-line approximation to sliding distance vs. energy, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature:

.9, .2793

3.3, .2966

6.7, .2702

10.5, .2121

11.5, .2966

1-band sliding distance and 2-band sliding distance for each tension:

.8, .9

4.88, 3.3

8.22, 6.7

12.28, 10.5

12.52, 11.5

Slope and vertical intercept of straight-line approximation to 2-band sliding distance vs. 1-band sliding distance, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature:

1.2, 0

no units, cm

The points cluster around the best fit line very well. The line is almost perfectly straight. There is no curvature.

Discussion of two hypotheses: 1. The sliding distance is directly proportional to the amount of energy required to stretch the rubber band. 2. If two rubber bands are used the sliding distance is determined by the total amount of energy required to stretch them.

Since this graph is so linear, then this experiment supports the hypothesis that the sliding distance is directly proportional to the amount of energy required to stretch the rubber band.

How long did it take you to complete this experiment?

Optional additional comments and/or questions:

I performed this experiment last semester, put it in a word document then forgot to turn it in.

Good, except that you didn't calculate your rubber band energies correctly.

See my commentary below.

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.

****************

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:

****************

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.

****************

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:

****************

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:

****************

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:

****************

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:

****************

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:

****

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:

****

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).

****

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:

****

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).

****

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.

energy conversion 1

Your 'energy conversion 1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

** Your optional message or comment: **

** How far and through what angle did the block displace on a single trial, with rubber band tension equal to the weight of two dominoes? **

** 5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of two dominoes: **

** Rubber band lengths resulting in 5 cm, 10 cm and 15 cm slides: **

** 5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of four dominoes: **

** 5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of six dominoes: **

** 5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of eight dominoes: **

** 5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of ten dominoes: **

** 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: **

** Slope and vertical intercept of straight-line approximation to sliding distance vs. energy, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature: **

according to your graph, if you have the vertical and horizontal coordinates on the appropriate axes (sliding distance on vertical, energy on horizontal), your slope would be about (20 cm) / (10 N * cm) = 2 / N or 2 N^-1.

** Lengths of first and second rubber band for (first-band) tensions supporting 2, 4, 6, 8 and 10 dominoes: **

** Mean sliding distance and std dev for each set of 5 trials, using 2 rubber bands in series: **

** Slope and vertical intercept of straight-line approximation to sliding distance vs. energy, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature: **

** 1-band sliding distance and 2-band sliding distance for each tension: **

** Slope and vertical intercept of straight-line approximation to 2-band sliding distance vs. 1-band sliding distance, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature: **

** Discussion of two hypotheses: 1. The sliding distance is directly proportional to the amount of energy required to stretch the rubber band. 2. If two rubber bands are used the sliding distance is determined by the total amount of energy required to stretch them. **

** How long did it take you to complete this experiment? **

** Optional additional comments and/or questions: **

Good, except that you didn't calculate your rubber band energies correctly. See my commentary below.

Energy conversion 1 Commentary

****************

****************

****************

****************

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.

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.

****

****

****

****

****

****

Your optional message or comment:

How far and through what angle did the block displace on a single trial, with rubber band tension equal to the weight of two dominoes?

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of two dominoes:

Rubber band lengths resulting in 5 cm, 10 cm and 15 cm slides:

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of four dominoes:

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of six dominoes:

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of eight dominoes:

5 trials, distance in cm then rotation in degrees, with rubber band tension equal to the weight of ten dominoes:

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:

Slope and vertical intercept of straight-line approximation to sliding distance vs. energy, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature:

Lengths of first and second rubber band for (first-band) tensions supporting 2, 4, 6, 8 and 10 dominoes:

Mean sliding distance and std dev for each set of 5 trials, using 2 rubber bands in series:

Slope and vertical intercept of straight-line approximation to sliding distance vs. energy, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature:

1-band sliding distance and 2-band sliding distance for each tension:

Slope and vertical intercept of straight-line approximation to 2-band sliding distance vs. 1-band sliding distance, units of slope and vertical intercept, description of the graph and closeness to line, any indication of curvature:

Discussion of two hypotheses: 1. The sliding distance is directly proportional to the amount of energy required to stretch the rubber band. 2. If two rubber bands are used the sliding distance is determined by the total amount of energy required to stretch them.

How long did it take you to complete this experiment?

Optional additional comments and/or questions:

Good, except that you didn't calculate your rubber band energies correctly.

See my commentary below.

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

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

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.