With your CDs/DVDs you obtained some introductory materials.  For this experiment you will use some of the rubber bands, the die ('die' is the singular of 'dice'; you have one die, which was originally packed as one of ten dice), some paper clips, the straw, the push pin and , if they are included with your initial package, the two short bolts.  You will also use at least one of the paper rulers from the preceding experiment Measuring Distortion of Paper Rulers , and you will use the TIMER program.

Note that this experiment is designed to prepare you for later, more precise experiments using rotating objects and rubber bands.  There is significant uncertainty in the measurements you will be taking.  You should try to get reasonably accurate data on this experiment but there is no need to be overly precise or meticulous in your data measurements or your graphing. 

Note however that, while it is not overly precise, it is important to analysis your data correctly, according to the instructions given here.

This experiment is designed to take about an hour.  The average time reported by students is closer to 90 minutes, with reported times varying from 30 minutes to 3 hours or in rare cases more. 

You need to get some data here but don't worry if it isn't high-quality data.  The main thing is to get some data and be sure you know what it is telling you.

You will in later experiments obtain and analyze more precise data using better-behaved systems.

Rotating Straw Experiment

Spin the straw and time it

The straw has a hole drilled in it, near its center point.  If you trim the straw at the short flexible segment then the hole should be approximately in the middle of the remaining section of the straw.  The hole should be within a millimeter or so of the center of the trimmed straw.  A little extra trimming on one end or the other might be necessary to make it so.

The die has a hole drilled in or near the middle of one of its faces. 

• Place the die on a level tabletop with this hole facing upward. 
• Place the straw over the die so you can look down through the hole in the straw to the hole in the die. 
• The push pin will extend through the hole on one side of the straw and out the hole on the other side.  The part of the pin which is sticking out can be placed in the hole in the die, so that if you give it a spin the straw will make a few rotations around the pin.
• Spin the straw (not too fast, so you can count its revolutions) and see how many times it goes around before stopping.

If you have trouble getting the straw to spin, as an alternative you can place the pin beneath the straw, pointing upward, and spin the straw on the pin.  You might need to somehow stabilize the pin to keep it from tipping over, but some students report better results with this arrangement.

Now repeat the spin but this time use the TIMER to determine how long it takes to come to rest after being spun, and through how many revolutions it travels.  You can hold onto the clip with one hand and extend a finger of that hand to start the straw spinning, leaving your other hand free to operate the TIMER. 

A revolution consists of a 360-degree rotation of the straw about the axis.  You should easily be able to count half-revolutions and then estimate the additional number of degrees, to come up with the rotation within an error of plus or minus 15 degrees or so.  That's all the precision required here, so there is no need to bother with a protractor.

In the box below report in the first line the time in seconds and the number of degrees of rotation from the time you released the straw to the instant it came to rest.  Use comma-delimited format.  Starting in the second line give a brief description of what you did and how you made your measurements, and be sure to indicate whether you used a trimmed or untrimmed straw.

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Put weights in the ends of the straw and repeat

Now insert the two bolts into the ends of the straw.  If they won't work or if you don't have the bolts yet, slide a large paper clip or two onto each end.  If necessary elevate the die a bit to keep the clips from dragging (for example you could set it on top of an inverted drinking glass or mug).  Spin the straw.  If it is unbalanced you can move one bolt (or paper clip) in or out a little, or if necessary trim whichever end needs it, a little at a time, until you achieve good balance.  Then repeat the above exercise.

In the box below report in the first line the time in seconds and the number of degrees of rotation from the time you released the straw to the instant it came to rest.  Use comma-delimited format.  In the second line give the length of your straw and the units in which you measured the length.  Starting in the third line give a brief description of what you did and how you made your measurements.

currently no notes are associated with this requested response

Time at least a few 180-degree intervals and find midpoint clock times for intervals

Repeat one more time.  This time click the TIMER every time an end of the straw passes a selected point, so that you will have a timing for every 180 degrees of rotation.  From the data you obtain determine the average velocity of the straw, in degrees per second (this quantity is actually called 'angular velocity' because it is measured in units of angle per unit of time), for each 180 degree rotation.

Also calculate the clock time at the midpoint of each timed interval.  Recall that 'clock time' is the time on a running clock. 

• The second column of the TIMER represents clock times; the third column represents time intervals.  Several trials are typically included in TIMER output.  However in the process of analysis it is more convenient to think of a different clock for each trial.
• The running clock for a given trial (e.g., a given spin of the strap) is generally assumed to read t = 0 at the initial instant. 
• The initial instant for a given trial would usually be the instant of the first 'click' of that trial.
• Clock times can be found by successively adding up the time intervals.  If you have time intervals of, say, 3 s, 5 s, 9 s and 15 s, then if the clock is started at t = 0 the clock times of the corresponding events would be 3 s, 8 s, 17 s and 32 s.
• Clock times can also be found by subtracting the TIMER's clock time for the first 'click' of a trial from the clock time of each subsequent 'click'.  For example the TIMER might show clock times of 63 s, 66 s, 71 s, 80 s and 95 s during a trial.  This means that the second 'click' occurred 66 s - 63 s = 3 s after the initial click; the third click was 71 s - 63 s = 8 s after the initial click; the fourth and fifth clicks would have occurred 80s - 63 s = 17 s and 95 s - 63 s = 32 s after the initial 'click'.  So the corresponding clock times would have been t = 0 (corresponding to TIMER clock time 63 s), then 3 s, 8 s, 17 s and 32 s.
• The midpoint clock times would be the clock times in the middle of the intervals. In the preceding example the first interval runs from clock time t = 0 to t = 3 s, so the midpoint clock time is 1.5 s. The second interval runs from t = 3 s to t = 8 s, so the midpoint clock times is the midpoint of this interval, t = 5.5 sec. The midpoints of the remaining two intervals would be t = 12.5 sec and t = 24.5 sec.
• A clock time is generally designated by t, and if t is used for the variable then it refers to clock time. A time interval is generally referred to by `dt; a time interval is a change in clock time. A midpoint clock time might be referred to as t_mid.

Copy the relevant part of the TIMER output into the box below. 

Starting in the second line after your TIMER output, give a table of average velocity vs. midpoint clock time (each line should include the midpoint clock time, then the average velocity for one time interval).

Starting in the line below your table, explain how you used your data to calculate your average velocities and the midpoint clock times. 

A common error here is to reverse the columns of the table.  Remember the convention that

• y vs. x has x in the first column.

A graph of vAve vs midpt clock time would have the midpt clock time in the first column.

Another common error is to confuse time intervals with clock times:

• For example if clock times are 2, 5, 15 and 17 seconds, then the time intervals are respectively 3, 10 and 2 seconds.  The clock starts at the beginning of the observation and continues running through the end, and clock times are the times showing on the clock.

• Conversely, if your time intervals are respectively 3, 10 and 2 seconds, then if your initial clock time is 2 seconds, the next clock time must be 3 sec + 2 sec = 5 sec; the next would be 5 sec + 10 sec = 15 sec; and the third would be 15 sec + 2 sec = 17 sec.  Thus, if your clock was at 2 seconds when the first interval began, it would be at 5 sec, then 15 sec, then 17 sec at the beginnings of the next three intervals.

The clock times depend on where your clock was at the beginning.  The initial clock time doesn't necessarily have to be 2 seconds, at it was in the preceding example.  You might just as well assume a clock that started at 0 seconds.  The same time intervals as before. Successive intervals of 3, 10 and 2 seconds would then end at clock times 0 sec + 3 sec = 3 sec; 3 sec + 10 sec = 13 sec; 13 sec = 2 sec = 15 sec.   Thus, if your clock was at 0 seconds on the first 'click', it would be at 3 sec, then 13 sec, then 15 sec on each of the next three 'clicks'.

Still another common error is to report clock times rather than midpoint clock times:

• The midpoint clock time for a time interval is halfway between the clock time at the beginning and the clock time at the end of the interval.

• Continuing the earlier example, if the clock times are 2, 5, 15 and 17 seconds, then the midpoint clock times are 3.5 seconds (halfway between clock times 2 and 5 seconds), 10 seconds (halfway between 5 and 15 seconds) and 16 seconds (halfway between 15 and 17 seconds).

A common error in reporting midpoint clock times is to report half of each time interval.  In this example time intervals are 3, 10 and 2 seconds, and someone making this error would report 1.5, 5 and 1 second.

If your clock times or midpoint clock times do not increase from one interval to the next, you are making an error:

• One characteristic of clock times is that they do not decrease during a trial. 

• Each interval in a trial occurs later than the interval that precedes it. 

• It should be clear that midpoint clock times therefore cannot decrease, whereas time intervals can either increase or decrease.

The average velocity on an interval is the average rate of change of position with respect to clock time on that interval. 

• For example, if an object's positions at clock times 2, 5, 15 and 17 seconds are 10, 20, 50 and 60 cm, then

On the first interval clock time changes by 5 sec - 2 sec = 3 sec, and position by 20 cm - 10 cm = 10 cm, so the average rate of change of position with respect to clock time on this interval is (change in position) / (change in clock time) = 10 cm / (3 sec) = 3.3 cm/s, approx..

On the second interval clock time changes by 15 sec - 5 sec = 10 sec, and position by 50 cm - 20 cm = 30 cm, so the average rate of change of position with respect to clock time on this interval is (change in position) / (change in clock time) = 30 cm / (10 sec) = 3.0 cm/s, approx..

On the third interval a similar calculation would indicate an average rate of 5 cm/s.

A table of average summarizing position vs. clock time and average velocity vs. midpoint clock time, assuming the clock to be at t = 2 seconds at the initial instant, would thus read: 

clock time (sec)	position (cm)	midpt clock time (sec)	ave rate of change of position with respect to clock time (cm/s)
2	10
5	20	3.5	3.3
15	50	10	3.0
17	60	16	5.0

Were we to assume that the same events were observed by a clock that started at t = 0 sec rather than t = 2 sec our table would be very similar, with just a shift in clock times:

clock time (sec)	position (cm)	midpt clock time (sec)	ave rate of change of position with respect to clock time (cm/s)
0	10
3	20	3.5	3.3
13	50	10	3.0
15	60	16	5.0

 

What is your evidence that the straw is speeding up or slowing down?  Is there any way you can determine in a meaningful way the rate at which the straw is speeding up or slowing down?

Student comment:  The average velocities decrease as time progresses.

I could find the average deceleration by dividing the change in velocity be the change in time. However I doubt the rate is constant so I am not sure other than finding the deceleration between each interval.

Instructor response:  You could sketch a curve representing the trend of the v vs. t information, then see if some function does a good job of approximating that curve (e.g., if the graph is well approximated by a straight line, you could easily determine the equation of the best-fit line). However for this experiment, it turns out that friction varies in an unpredictable manner and you probably wouldn't get a meaningful result.

Measure the lengths of two opposing rubber bands

Now choose one of the thin rubber bands and one of the thicker rubber bands.  Make sure there are no obvious defects on the rubber band you choose; otherwise your choice should be random.

Bend three paperclips to form hooks.  Hook each rubber band to an end of one hook, and attach the other hooks to the free ends of the rubber band.  Pull gently on the end hooks until the rubber bands pretty much straighten out and take any data necessary to determine their lengths, as accurately as is reasonably possible with the paper rulers.

Now pull a little harder so the rubber bands stretch out a little.  Stretch them so that the distance between the end hooks increases by about 1 cm.  Take data sufficient to determine the lengths of the two rubber bands.

Repeat so that the distance between the end hooks increases by another centimeter, and again take data sufficient to determine the two lengths.

Repeat twice more, so that with your last set of measurements the hooks are 4 cm further apart than at the beginning.

In the box below report in the first line the lengths as determined by your first measurements, with the 1 cm stretch.  Report in comma-delimited form, with the length of the thicker rubber band first.  In the second, third and fourth lines make a similar report for the three additional stretches.  Starting in the fifth line, give a summary of how you made your measurements, your raw data (what you actually observed--what the actual readings were on the paper ruler) and how you used your raw data to determine the lengths.

The thin rubber band will stretch more than the thick one, so the changes in its length will be greater than those of the thick rubber band.  The report below is reasonable.  Note that the changes in the second-column figures are at least double those in the first column, consistent with the changes in length of the thin rubber band.

7.55,7.3
7.85,8.2
8.10,8.78
8.35,9.50

The report below is not reasonable.  Even with two thick rubber bands, it is very unlikely that the two lengths would change by exactly the same amount with every increase in stretch, and a thin rubber band would certainly exhibit greater changes in length than the thick.

6.5,5.5
7.5,6.5
8.5,7.5
9.5,8.5

Sketch a graph of length_thin vs. length_thick, where length_thin is the length of the thin rubber band and length_thick is the length of the thick rubber band.  Fit the best straight line you can to the data, using manual fitting methods (i.e., actually draw the line on the graph--don't use a graphing calculator or a spreadsheet to find the equation of the line, but measure everything as in the Fitting a Straight Line to Data activity).

In the box below give in the first line the slope and vertical intercept of your straight line.  Starting in the second line, discuss how well the straight line actually fit the data, whether the data seems to indicate curvature, and what the slope and vertical intercept mean in terms of your rubber band system:

If the length of the thin rubber band corresponds to the vertical axis, and the thin to the horizontal, then the rise of the graph between two points will correspond to the change in length of the thin rubber band, and the run to change in length of the thick.  The slope will therefore be greater than one. 

For example the data

7.55,7.3
7.85,8.2
8.10,8.78
8.35,9.50

would be associated with a slope between about 2.5 and 3.

Observe 2 rubber bands in series vs. a single rubber band

Flip a coin.  If it comes up Heads, add a paper clip and a second thin rubber band to the system so you have a chain of two thin rubber bands pulling on a single thick rubber band.  If it comes up Tails, instead add a second thick rubber band so you have a chain of two thick rubber bands pulling on a single thin rubber band.

Repeat the experiment using these two chains.  Observe the length of the 2-rubber-band chain vs. the length of the 1-rubber-band chain.  Report the slope of your graph in the box below below.  Starting in the second line, discuss

• how the slope of the this graph differs from that of your previous graph
• why the slope should differ
• how you would expect the slope to differ if the two thin rubber bands were identical
• to what extent your results support the hypothesis that the two thin rubber bands do not in fact behave in identical ways.

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