course PHY 201
10:48 PMxxxx
Sunday, August 30, 2009 " "Quick Exercise:
Using the short pendulum, set up an incline so the marble will roll off the incline and fall to the floor. Release the pendulum and start counting. Then release the ball at the top of the incline, and count its half-cycles until it reaches the floor. Note the count at the instant of release, at the instant it reaches the end of the ramp and at the instant it strikes the floor.
Your counts are the 'clock times' for this clock.
You observed three 'clock times' with this clock. What were they?
My Response:
5
9
10
(half-cycles)
Your three 'clock times' define two 'time intervals', one that lasted from release until the ball reached the end of the ramp, and another from the end of the ramp to the floor.
What were the two time intervals?
My Response:
4 half-cycles
1 half-cycle
When you analyze situations involving a clock, you will need to take care to distinguish between clock times and time intervals.
* Use an adjective whenever you use the term 'time': When you refer to a clock time or a time interval, use the term 'clock time' or the term 'time interval', rather than just the term 'time'.
Definition of Average Rate of Change:
The average rate of change of A with respect to B is defined to be
* average rate = change in A / change in B
Examples:
A child's height is 100 cm on Jan 1, 102 cm on May 1 of the same year, 105 cm on October 1 of the same year.
* What was the clock time at each measurement?
My Response:
Jan 1, May 1, and Oct 1
* What are the changes in clock times between measurements?
My Response:
4 months, 5 months
* At what average rate did the child's height change with respect to clock time between Jan 1 and May 1?
My Response:
2cm / 4mo = 0.5 cm/mo
* At what average rate did the child's height change with respect to clock time between May 1 and October 1?
My Response:
3cm / 5mo = 0.6 cm/mo
To answer a question related to an average rate of change on an interval, always answer the following questions:
* What is the A quantity?
* What is the B quantity?
* What is the change in the A quantity for the interval?
* What is the change in the B quantity for the interval?
* What therefore is the average rate of change of A with respect to B?
Answer these questions for the above example.
My Response:
The A quantity is height in cm.
The B quantity is time in months.
The change in A quantity is 2cm for the first interval and 3cm for the second interval.
The change in B quantity is 4mo for the first interval and 5mo for the second interval.
For a marble rolling down a ramp, off the edge and falling to the floor:
* What is the slope of your ramp when supported by a 'flat' domino?
My Response:
slope = domino thickness / ramp length = 1cm / 30cm = 0.0333
* What is the slope of your ramp when supported by a domino lying 'on its side'?
My Response:
slope = 2.5cm / 30cm = 0.0833
* What is the slope of your ramp when supported by a domino lying 'on its end'?
My Response:
slope = 5cm / 30cm = 0.1667
* How much does the slope of the ramp change when you change the domino from flat to on-its-side to on-its-end?
My Response:
0.1667 - 0.0333 = 0.6337
there are three slopes and therefore two changes here
* By how much does the landing position of the marble change as you move from the first slope to the second to the third?
My Response:
0 cm to 8cm to 11cm
Intervals: 8cm and 3cm
* What is the average rate of change of landing position with respect to ramp slope, between the first and second slope?
My Response:
ave rate = change in landing position / change in slope
= (8cm - 0cm) / (0.0833 - 0.0333)
= 8cm / 0.05
= 160 cm / half-cycle
* What is the average rate of change of landing position with respect to ramp slope, between the second and third slope?
My Response:
ave rate = (11cm - 8cm) / (0.1667 - 0.0833)
= 3cm / 0.0834
= 35.9712
For the same marble on the same ramp:
* How long does it take the ball to roll down the incline with the domino lying 'flat'?
My Response:
I'm not sure we timed the marble but I will use the data from the previous assignment:
8 half-cycles
* How long does it take the ball to roll down the incline with the domino lying 'on its side'?
My Reponse:
4.25 half-cycles
* How long does it take the ball to roll down the incline with the domino lying 'on its end'?
My Response:
4 half-cycles
* For each interval, what is the average rate of change of the time required to roll down the incline with respect to ramp slope?
My Response:
ave rate = change in clock time / change in slope
= 3.75 / 0.05
= 75
ave rate = 0.25 / 0.0834
= 2.998
* For each interval, what is the average rate of change of the ball's position with respect to clock time as it rolls down the ramp?
My Response:
ave rate = change in ball position / clock time
= 30cm / 3.75
= 8 cm / s
ave rate = 30cm / 0.25
= 120 cm / s
Do the following, as best you can. We've had limited discussion of graphs so if you don't do well, it's OK. We'll have further discussion in our next class. However do the best you can.
* Make a graph of marble position vs. clock time as it rolls down an incline of length 30 cm, starting from rest, in 3 seconds.
* Make a graph of marble velocity vs. clock time as it rolls down an incline of length 30 cm, starting from rest, in 3 seconds.
* For your marble rolling down the three inclines, graph position vs. clock time for each incline.
* For your marble rolling down the three inclines, graph velocity vs. clock time for each incline.
Describe the four graphs you have constructed (again do your best; we will soon develop some language for describing graphs).
My Response:
The graph of marble position vs. clock time is linear from a point (0, 30), decreasing at a constant rate of 8cm/s until reaching a point at (3, 14). (maybe I've got this one backwards)
The graph of marble velocity vs. clock time would begin at a point (0, 0), increasing at an increasing rate until it reaches a point near (3, x), where x is the final velocity at the end of the ramp (would this be 8cm/s?)
The graph of position vs/ clock time has a slight curve, which decreases at a decreasing rate.
The graph of velocity vs clock time
A 'graph rectangle' is a rectangle, one of whose sides is part of the horizontal axis.
* The quantity which is represented by the length of the side which is part of the horizontal axis is the 'base' of the graph rectangle.
* The quantity represented by the length of either of the sides perpendicular to the 'base' is the 'altitude' of the graph rectangle.
* The 'area' of the graph rectangle is the product of the quantity represented by its 'base' and the quantity represented by its 'altitude'.
On a graph of speed in miles / hour vs. clock time in hours, we find a graph rectangle with base 3 and altitude 40.
* What does the altitude of the graph represent?
My Response:
The altitute is the speed quantity.
* What does the base of the graph represent?
My Response:
The base is the clock time quantity.
* What is the area of the graph?
My Response:
area = base * altitude
= 3 * 40
= 120
* What does the area of the graph represent?
My Response:
The area of the graph represents the distance traveled in 3 hours at 40 mi/h.
On a graph of income stream in dollars per month vs. clock time in months, we find a graph rectangle with base 36 and altitude 1000.
* What does the altitude of the graph represent?
My Response:
The altitude is the dollars per month quantity.
* What does the base of the graph represent?
My Response:
The base is the clock time quantity.
* What is the area of the graph?
My Response:
area = 36 * 1000
= 36 000
* What does the area of the graph represent?
My Response:
The area of the graph represents the income earned in 36 months.
On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30.
* What does the altitude of the graph represent?
My Response:
The altitude is the force quantity.
* What does the base of the graph represent?
My Response:
The base is the position quantity.
* What is the area of the graph?
My Response:
area = 200 * 30
= 6 000
* What does the area of the graph mean?
My Response:
The area of the graph is the total force over 200 feet.
On a graph of density in grams / centimeter vs. position in centimeters, we find a graph rectangle with base 16 and altitude 50.
* What does the altitude of the graph represent?
My Response:
The altitude is the density quantity.
* What does the base of the graph represent?
My Response:
The base is the postion quantity.
* What is the area of the graph?
My Response:
area = 16 * 50
= 800
* What does the area of the graph mean?
My Response:
The area of the graph is density over 16 cm.
A 'graph trapezoid' is defined by two points on a graph, as follows:
* The 'left altitude' is the line segment parallel to the y axis, running from the leftmost of the two points to the horizontal axis.
* The 'right altitude' is the line segment parallel to the y axis, running from the rightmost of the two points to the horizontal axis.
* The 'slope segment' is the line segment between the two points.
* The 'base' is the part of the x axis between the two altitudes.
The 'graph slope' between two points is the slope of the 'slope segment' of the graph trapezoid defined by the two points.
On a graph of speed in miles / hour vs. clock time in hours, we find graph points (2, 50) and (7, 60)
* What do the altitudes of the graph represent?
My Response:
The altitudes are the speeds.
* What is the rise between the two points of this graph?
My Response:
The rise is 60 - 50 = 10
* What is the run between these points?
My Response:
The run is 7 - 2 = 5
* What therefore is the slope associated with this graph trapezoid?
My Response:
slope = rise / run
= 10 / 5
= 2
* What does the slope mean?
My Response:
The slope is the change in speed.
* What does the base of the graph represent?
My Response:
The base is the clock time quantity.
* What are the dimensions of the equal-area graph rectangle?
My Response:
altitude = (50 + 60) / 2 = 55
base = 7 - 2 = 5
* What is the area of the graph?
My Response:
area = a * [(b1 + b2) / 2]
= 5 * (110 / 2)
= 275
* What does the area of the graph represent?
My Response:
The area represents the average speed traveled over 5 hours.
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?
My Response:
The altitudes represent the income stream.
* What is the rise between the two points of this graph?
My Response:
The rise is 200.
* What is the run between these points?
My Response:
The run is 24.
* What therefore is the slope associated with this graph trapezoid?
My Response:
The slope is 200 / 24 = 8.33
* What does the slope mean?
My Response:
The slope is the change in income stream.
* What does the base of the graph represent?
My Response:
The base is clock time.
* What are the dimensions of the equal-area graph rectangle?
My Response:
altitude = (1000 + 1200) / 2 = 1100
base = 40 - 16 = 24
* What is the area of the graph?
My Reponse:
area = 1100 * 24
= 26 400
* What does the area of the graph represent?
My Response:
The area represents the average income stream over 24 months.
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?
My Response:
The altitudes represent force.
* What is the rise between the two points of this graph?
My Response:
The rise is 0.
* What is the run between these points?
My Response:
The run is 200.
* What therefore is the slope associated with this graph trapezoid?
My Response:
The slope is 0.
* What does the slope mean?
My Response:
The slope is the average force.
* What does the base of the graph represent?
My Response:
The base represents position.
* What are the dimensions of the equal-area graph rectangle?
My Response:
base 200
altitude 30
* What is the area of the graph?
My Response:
area = 200 * 30
= 6 000
* What does the area of the graph represent?
My Response:
The area represents force over 200 feet.
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?
My Response:
The altitudes represent density.
* What is the rise between the two points of this graph?
My Response:
The rise is 10 - 12 = -2
* What is the run between these points?
My Response:
The run is 20 - 5 = 15
* What therefore is the slope associated with this graph trapezoid?
My Response:
The slopse is rise / run = -2 / 15 = -0.133
* What does the slope mean?
My Response:
The slopse is the average change in density.
* What does the base of the graph represent?
My Response:
The base is position.
* What are the dimensions of the equal-area graph rectangle?
My Response:
altitude = (12 + 10) / 2 = 11
base = 15
* What is the area of the graph?
My Response:
area = 11 * 15
= 165
* What does the area of the graph represent?
My Response:
The area represents the average change in density vs position.
Explain how you construct a 'graph rectangle' from a 'graph trapezoid'.
My Response:
Average the altitudes to ""even out"" the trapezoid into a rectangle.
Explain how to find the area of a 'graph trapezoid'.
My Response:
To find the area of the graph trapezoid, use the constructed graph rectangle and multiply by the base.
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?
My Response:
Probably 10-15%."
Good work. Your approach is great and your answers are nearly all valid.
I'll comment more on some of the later problems in an email after class Wednesday.