Class 090831

Answer given on the change in slope from flat to on-side domino to on-end domino:

• The first interval is a change of 2.5x the second interval is a change of 2x and the overall change from flat to
  on its end is a 5x greater slope.

Comment:  You have given the ratio of the slopes.  The question asked for the 'difference', and since the ratios are different, your answer is not invalid.

However the term 'difference' can also be used to indicate a subtraction, which is the interpretation needed to answer a rate-of-change question.

The question would have been clearer had I asked for 'the change in the slope'.

The average rate of change of position with respect to clock time on the first, the third, the fifth, the seventh
and the ninth 1-foot ramps.
 

1 foot / .75 secs

.75 sec is a reasonable estimate for the time spent on the first ramp, and 1 foot is the displacement while on that ramp.  So this appears to be a reasonable result.

3 feet/ 1.86 secs 

The ball travels only 1 foot on the third ramp, and almost certainly spends less than 1.86 seconds on that ramp.  There is no 3-foot interval, or 1.86 second time interval, associated with the motion along the third ramp.

This is would be a reasonable result for the average velocity of the ball for the first 3-foot interval.

Some preliminary details unrelated to the physics of the course:

Use ic at the beginning of the title of anything you submit.

Remember:

• me smart ... you smart ... computer dumb

The computer will be looking for 'ic' at the beginning of your title.  It can't interpret your meaning if you indicate in any other way that you are in the classroom section.

 

The forms used to submit work are text forms.  If you do your work in a word processor as opposed to a text editor it's fine (you might want to use the editing features of a word processor), but only the text will come through on the form.

If you use a word processor, you should 'launder' the document by either saving it as a text file and opening it in the text editor, or by simply copy-pasting it into a text editor.  Then copy-paste the information file from the text editor into the form.

At least one student has told me that the form won't allow the entire document to be copied, saying it's too long.  That student is using a browser provided by his/her ISP, and I expect that's the problem.  If you encounter a similar problem I recommend that you install a standard browser (Internet Explorer, Firefox, Google, etc.) and use it for these forms.

Submit data using correct syntax: 

When submitting data, be sure you submit it in using the correct syntax.  More about this later in these notes.

Given Instruction:

Each line should consist the length of the pendulum being observed, the 1-minute count and the scale factor by which your reported length should be multiplied to get the length of the pendulum in centimeters. The three requested quantities should be separated by commas. Each line should therefore consist of three numbers, separated by commas. Don't include any words, units, or other information in any of these lines.

You may report as many lines as possible. Use the space below.

The first pendulum was the length of my middle finger to my wrist, the count was 70 oscillations .
The second pendulum was doubled the first and the count was 50 oscillations.
The third pendulum was doubled the second and the count was 35 oscillations.

Student's data report:

Your data report (start on the next line):

70, 3.7, 19
50, 1.3, 38
35, .46, 76

Instructor Comment: 

It appears that this student obtained excellent data, if I interpret the data correctly.

That being said, can I really be trusted to interpret the data, or should the data be given in a specified order with its meanings clearly defined?

I'm pretty good at reading and interpreting data, but I can't always read your mind.  The only safe thing to do is give the data in the specified order, according to the instruction

'Each line should consist the length of the pendulum being observed, the 1-minute count and the scale factor by which your reported length should be multiplied to get the length of the pendulum in centimeters'

Again, the data are so good that I can tell what they mean, which is very much to the credit of the student who reported them.

However if the data are interpreted according to instructions, each line would consist of the length, the 1-minute count and the scale factor.  So we would expect the lengths of the three pendulums to be 70, 50 and 35.  The 1-minute counts are 3.7, 1.3 and .46.  And to get the lengths of the pendulums in centimeters we would have to multiply the first given length by 19, the second by 38 and the third by 76 (indicating that the three pendulums were measured according to three different scales).

We know better.  In fact it doesn't take us long to figure out that the first number is probably the count, and the last number the length in centimeters.  We're left a bit in the dark about the meaning of the second number, and I confess that I still don't know what it means.

Now we're all pretty smart, and we can often out what the data probably mean.  However I'm going to use the computer to collect the data you report.  The computer is very dumb.  It will only follow the precise instructions it's given.  When the data are not reported in the given order and with the given syntax, the computer is not going to know the difference, and when the data are all collected the results will be useless. 

It wouldn't be difficult for me to program the computer to reorder the data according to certain criteria, but besides taking my time away from things that might benefit students, that places a layer of interpretation between the reported data and the collected data.  That layer of interpretation can distort the data in ways that aren't always easy to predict.

Another student's data report:

19.15 135.33 1
38.3 97.33 2
76.6 70 4

The commas are missing.  Me smart ... you smart ... computer dumb.  We all know where one number ends and the next begins, but the computer will be told to look for commas to separate the numbers.  The computer won't be able to separate one number from another.  The information given here will confuse the collected data.

If we have some experience with pendulums we will also recognize that your first number in each line is probably length in cm and the second is probably the number of half-cycles counted.  To the numbers are reported in the specified order.

The third number is a problem, and was for most students, so we'll be discussing that.

The lesson here:  Follow the instructions precisely and report your data in the place and with the syntax defined by the instructions.

Another student's data report:

7.75, 59, 2.54
15.5, 45, 2.54
31, 32, 2.54

This is a good and completely correct data report.

The first pendulum has length 7.75, in units that aren't specified here.  The 1-minute count is 59 cycles.  We multiply the first number by 2.54 to get the length of the pendulum in centimeters, and we get 7.75 * 2.54 = 20, approx..  So the student is saying that a pendulum of length about 20 cm has a frequency of 59 cycles/minute.

One student's excellent explanation of reported data:

Clock time was measured with a clock timer (watch) which was started at the top of the extreme. To ensure the most accurate measurements of the pendulum lengths - the initial measurement (wrist to tip of middle finger) was measured as that. The second length was measuring using a needle nose plier and looping the string back to almost the middle of the center of the washer (not quite the middle because the string traveling around the pliers added length). The third length was measured this same method but using the second length to loop instead of the first length. The first pendulum was held between fingertips and the other two pendulum were help off the end of a desk with needle nose pliers and a rubberband around the handle (to reduce friction at the top of the pendulum). The 1/2 cycles were simply counted by hitting the Enter key in an Excel cell each time the pendulum reached an extreme. Uncertainty levels would be based on +/- 1/4 cycles accordingly to the pendulum - since the 1/2 cycles were counted of each pendulum. Other factors such as air resistance and friction from the string rubbing across the fingertips in the first pendulum also played a roll of uncertainty but no where near as in depth as clock time resulting from unsure counts that could be accuratly (physically) measured. The measurements may be off by +/- 2 mm due to trying to use an aluminum measuring tap to measure to lengths of not only the pendulum but of my wrist to middle finger tip.

This report also includes an excellent suggestion for counting cycles or half-cycles of the pendulum.

Galileo Experiment

Use the fastest pendulum you can reliably count. 

Count your pendulum for a minute.

The instructor will operate the experiment.  You just watch and time.

Time the ball as it moves down 10 feet of ramp, using a pendulum.

Repeat for 9 feet, then for 6 feet, then for 4 feet.

Graph position vs. clock time for the ball rolling down the ramp, with clock time in units of half-cycles. 

• Sketch the graph pretty carefully and sketch a smooth curve that you believe represents the actual position vs. clock time behavior of the ball. 
• The speed of the ball is constantly changing, so your graph will not contain any straight lines. 
• There is some uncertainty in your timing, so while your curve will probably come close to your data points, it shouldn't be expected to actually pass through any of them. 
• In other words, your curve should represent the actual behavior of the ball, as best you can infer it from the data points, but you shouldn't go out of your way to make your curve actually go through any of the points.

Based on your graph estimate the following:

• The time required to travel down the first one-foot ramp.
• The distance that would be traveled traveled during each half-cycle of your pendulum (i.e., the distance from the start to the end of the first half-cycle, the distance from start to end of the second half-cycle, etc.).
• The average rate of change of position with respect to clock time on the first, the third, the fifth, the seventh and the ninth 1-foot ramps.

Trapezoids

Sketch a y vs. x coordinate system.  The y axis is vertical (up and down the page), the x axis horizontal (left and right across the page).

You have a 'graph trapezoid' on your desk. 

• A 'graph trapezoid' has the property that one of its sides is perpendicular to two other sides.

Orient your trapezoid so that its base rests somewhere on the x axis.  (The base is the side which is perpendicular to two other sides; not every trapezoid has a base in this sense, but a 'graph trapezoid' does).

Estimate the two 'graph altitudes' of your trapezoid, its 'altitude' and its 'base'.  You can use any unit with which you are comfortable to make your estimate (e.g., centimeters, inches, feet, kilometers, nanometers, pounds, liters, gallons, kilograms, slugs, cubic feet, miles per hour; whatever you think works best for you is fine, though length units are probably most appropriate to this exercise).  (The 'graph altitudes' are the sides which are parallel to the vertical axis when the base rests on the horizontal axis).

Make on fold in the trapezoid so that if you tear the paper along the fold, the two pieces can be reassembled to make a rectangle.

Answer the two questions below, and in your answers explain your reasoning by giving the estimated dimensions, and a complete description of what you did.  Your explanation show how you proceeded from your estimates to your results.

• What is the 'graph slope' associated with your trapezoid?
• What would be the dimensions of this rectangle?

Definitions of average velocity and average acceleration:

These are the central definitions for the first part of your course.  Everything you do in analyzing motion should come back to these definitions:

• The average velocity of an object on an interval is its average rate of change of position with respect to clock time on that interval.
• The average acceleration of an object on an interval is its average rate of change of position with respect to clock time on that interval.

Analyzing the motion of the Lego racer:

We estimated that the Lego racer traveled 60 cm in 1.5 seconds to rest as it traveled in the direction opposite our chosen positive direction, then 30 cm in 1.2 seconds to rest as it traveled in our chosen positive direction. 

Applying the definition of average velocity to the second motion:

• By the definition, we are finding average rate of change of position with respect to clock time.
• The A quantity is the position of the racer.
• The B quantity is the clock time.
• The average rate is by definition of average rate equal to (change in A) / (change in B).
• Having identified the A and B quantities we find that

average velocity = average rate of change of position with respect to clock time = (change in position) / (change in clock time).

• According to our information, the change in position is +30 cm and the change in clock time is +1.2 seconds.
• Thus our average velocity is

average velocity = average rate of change of position with respect to clock time = (change in position) / (change in clock time) = (+ 30 cm) / (+1.2 s) = +25 cm / sec.

Find the average velocity for the first motion, using similar steps to connect your result with the definition of average velocity.

If we sketch a graph of velocity vs. clock time for the second motion:

• we know that the velocity ended up at zero
• we know that the cart was moving in the positive direction as it slowed to rest
• if we assume that the graph is a straight line, we conclude that the line decreases toward a point on the horizontal axis during the 1.2 second interval (you should have a sketch of the graph in your notes)
• we know that the average velocity is 25 cm / s; since the final velocity is zero we conclude that the initial velocity is greater than 25 cm / s; and since we expect the average velocity to occur at the middle of the time interval we conclude that the initial velocity was 50 cm/s

Our graph therefore forms a trapezoid with base 1.2 seconds, and altitudes 50 cm/s and 0 cm/s (in this case the trapezoid is in fact a triangle).  We could find the trapezoid's associated slope and area. 

• The slope is rise / run.  The rise is the change in velocity.  Velocity changes from 50 cm/s to 0, so the change in velocity is

change in velocity = final velocity - initial velocity = 0 cm/s - 50 cm/s = - 50 cm/s.

• The run is 1.2 seconds.  So the slope is

slope = rise / run = - 50 cm/s / (1.2 s) = -42 (cm / s) / (s) = -42 (cm / s) * ( 1 / s) = -42 cm / s^2.

• Since the rise represents change in velocity and the run represents change in clock time, our calculation gives us (change in velocity) / (change in clock time).

This is the form of an average rate of change.  Recalling the definition of average rate of change, we see that velocity is the A quantity, clock time the B quantity, so that this is the average rate of change of velocity with respect to clock time.

This is the definition of acceleration.

The slope of this graph represents the acceleration of the car. 

• Note that our reasoning requires that the v vs. t graph be a straight line.  Otherwise we could not have concluded that the initial velocity is 50 cm/s.

What is the slope of the graph of the first motion (the distance was 30 cm and required 1.5 seconds)?

• Answer by duplicating the reasoning used above.

What is the area of the graph trapezoid corresponding to the first motion, and what does this area represent?

• Answer by identifying all the quantities you use to find the area, and as best you can reason out the meaning of your result.  Reason in detail similar to that used above, though the reasoning process will be different for the question of area.

Homework:

Count the pendulum:

• Take a pendulum home and give it an accurate count. 
• Submit using form at Pendulum Counts Report. 
• This form is self-titled so you can't add 'ic' to the title.  This is OK, since this particular form is used only for classroom students.
• If you submitted the data using the wrong syntax, submit it again.  You need only submit the data, but if you wish to revise anything else you are welcome to do so.

Report your results from today's class using the Submit Work Form.  Answer the questions posed in the notes above.

Take another look at Chapter 1 in your text, and see what you pick up this time that you missed before.  Submit questions if you have thenm.

You should have viewed the key systems at the link given below.  If you haven't, you need to do this.

http://vhcc2.vhcc.edu/ph1fall9/frames_pages/introduction_to_key_systems.htm

Do the preliminary question-answer exercise:

• Go to the Submit Work Form.  Fill out the identifying information (access code, email address, etc.), just type a simple 'hello' message in the last box, and submit.  The instructor will respond by posting a copy to your access site, along with a short comment.  Posting will typically occur on the evening of the day you submit the form, or on the following evening.

On a graph of income stream in dollars per month vs. clock time in months, we find the two points (16, 1000) and (40, 1200). 

• What do the altitudes of the graph represent?
• What is the rise between the two points of this graph?
• What is the run between these points?
• What therefore is the slope associated with this graph trapezoid?
• What does the slope mean?
• What does the base of the graph represent?
• What are the dimensions of the equal-area graph rectangle?
• What is the area of the graph?
• What does the area of the graph represent?

On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30. 

• What do the altitudes of the graph represent?
• What is the rise between the two points of this graph?
• What is the run between these points?
• What therefore is the slope associated with this graph trapezoid?
• What does the slope mean?
• What does the base of the graph represent?
• What are the dimensions of the equal-area graph rectangle?
• What is the area of the graph?
• What does the area of the graph represent?

On a graph of density in grams / centimeter vs. position in centimeters, we find the points (5, 12) and (20, 10).

• What do the altitudes of the graph represent?
• What is the rise between the two points of this graph?
• What is the run between these points?
• What therefore is the slope associated with this graph trapezoid?
• What does the slope mean?
• What does the base of the graph represent?
• What are the dimensions of the equal-area graph rectangle?
• What is the area of the graph?
• What does the area of the graph represent?

Explain how you construct a 'graph rectangle' from a 'graph trapezoid'.

Explain how to find the area of a 'graph trapezoid'.

Ongoing question:  What is the smallest possible percent difference you think you could detect, using the pendulum, in the times required for the ball to travel down two ramps?

Drop a coin simultaneous with the release of a quarter-cycle long pendulum.  Find the minimum height at which the pendulum clearly strikes the wall first, and the maximum height at which the coin clearly strikes the floor first.

Walk down the sidewalk at constant velocity while someone times you with a pendulum of appropriate length.  Can they verify that you walked at constant velocity?

Walk down the sidewalk, increasing your velocity gradually while someone times you with a pendulum of appropriate length.  According to their results, did you speed up at a constant, an increasing or a decreasing rate?  According to your perceptions, did you speed up at a constant, an increasing or a decreasing rate?

Describe the motion of the dice on the ends of the strap, as you see them from your perspective.