course PHY 201
8-30 2pm
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.xxxx
Your counts are the 'clock times' for this clock.
You observed three 'clock times' with this clock. What were they?
Instant of release- 4, instant the ball reaches the end of the ramp- 8, and the instant the ball strikes the floor- 9
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?
4, 1
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? Jan 1, May 1, October 1
What are the changes in clock times between measurements? About 4 months
At what average rate did the child's height change with respect to clock time between Jan 1 and May 1? .5 cm per month
At what average rate did the child's height change with respect to clock time between May 1 and October 1? .6 cm per month
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.
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? 1/30
What is the slope of your ramp when supported by a domino lying 'on its side'? 1/12
What is the slope of your ramp when supported by a domino lying 'on its end'? 1/6
How much does the slope of the ramp change when you change the domino from flat to on-its-side to on-its-end? .05, .083
By how much does the landing position of the marble change as you move from the first slope to the second to the third? 9.5 lines of paper or 6.9 cm, 10 lines of paper or 7.4 cm
What is the average rate of change of landing position with respect to ramp slope, between the first and second slope? 138 cm
What is the average rate of change of landing position with respect to ramp slope, between the second and third slope? 88.8 cm
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'?
How long does it take the ball to roll down the incline with the domino lying 'on its side'?
How long does it take the ball to roll down the incline with the domino lying 'on its end'?
For each interval, what is the average rate of change of the time required to roll down the incline with respect to ramp slope?
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?
I cannot do these calculations because we never timed the pendulum we used for this experiment (one of half and a quarter the length of our book) against 60 seconds, therefore I have no conversion factor to calculate time using the pendulums clock time. Further, we did not time the ball in each situation (with the changing domino), we merely measured the change in landing position.
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.
The graph is a line starting at (0 cm,0 sec) and ending at (30 cm, 3 seconds) that increases proportionally
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. The graph is a line starting at (0 cm/sec, 0 sec) and ending at (20 cm/sec, 3 sec) that increases proportionally.
For your marble rolling down the three inclines, graph position vs. clock time for each incline. The graphs are similar, but as the slope increases on the inclines, the total clock time decreases.
For your marble rolling down the three inclines, graph velocity vs. clock time for each incline. The graphs are similar, but as the slope increases on the inclines, so does the end velocity, but the total time decreases.
Describe the four graphs you have constructed (again do your best; we will soon develop some language for describing graphs).
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? The altitude of the graph, 40, represents 40 miles per hour
What does the base of the graph represent? The base of the graph, 3, represents the time interval of three hours
What is the area of the graph? The area of the graph is 120 miles
What does the area of the graph represent? The area of 120 miles represents the distance covered when traveling 40 miles per hour for 3 hours.
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? The altitude, 1000, represents $1000 per month
What does the base of the graph represent? The base, 36, represents the time interval of 36 months
What is the area of the graph? The area is 36,000
What does the area of the graph represent? The area represents $36,000 made over the course of 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? The altitude, 30, represents 30 lbs of force
What does the base of the graph represent? The base, 200, represents a change in position of 200 feet
What is the area of the graph? The area of the graph is 30*200 or 6000
What does the area of the graph mean? The area of 6000 represents 6000 foot pounds of foce
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? The altitude, 50, represents a density of 50 grams per centimeter
What does the base of the graph represent? The base, 16, represents a change in position of 16 centimeters
What is the area of the graph? The area is 16*50 or 800
What does the area of the graph mean? The area of 800 represents 800 grams
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? The altitudes of the graph represent initial speed in miles per hour and final speed in miles per hour
What is the rise between the two points of this graph? The rise between the two points is 10 miles per hour
What is the run between these points? The run between these two points is 5 hours
What therefore is the slope associated with this graph trapezoid? The slope is therefore 2
What does the slope mean? This slope represents an increase of speed of 2 miles every hour for the five hours traveled
What does the base of the graph represent? The base, 5, represents a time interval of 5 hours
What are the dimensions of the equal-area graph rectangle? The dimensions of the equal area graph are base 5 by altitude 55
What is the area of the graph? The area is therefore 275
What does the area of the graph represent? This area represents 275 miles covered while traveling first 50 miles per hour, then finally 60 miles per hour, for a total of 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? The altitudes represent an initial pay of $1000 per month followed by a final pay of $1200 per month
What is the rise between the two points of this graph? The rise is $200 dollars per month
What is the run between these points? The run is 40-16 which is 24 months
What therefore is the slope associated with this graph trapezoid? The slope is 200/24=8.33
What does the slope mean? The slope represents an increase of 8.3 dollars each month for 24 months
What does the base of the graph represent? The base, 24, represents a time interval of 24 months
What are the dimensions of the equal-area graph rectangle? The dimensions of the equal area graph are altitude 1100 by a base of 24
What is the area of the graph? The area of this graph is 26,400
What does the area of the graph represent? The area of 26400 represents $26,400 made over the course of 24 months, first making $1000 per month and finally making $1200 per month.
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? The altitude represents 30 lbs of force
What is the rise between the two points of this graph? No points given
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? The base, 200, represents a change in position of 200 feet
What are the dimensions of the equal-area graph rectangle? 30 by 200
What is the area of the graph? 6000
What does the area of the graph represent? This area represents 6000 foot pounds of force
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? The altitudes represent an initial density of 12 grams/cm and a final density of 10 g/cm
What is the rise between the two points of this graph? The rise is -2 grams/cm
What is the run between these points? The run is 15 cm
What therefore is the slope associated with this graph trapezoid? The slope is therefore -2/15, or -.133
What does the base of the graph represent? The base represents a change in position of 15 cm
What are the dimensions of the equal-area graph rectangle? The dimensions of the equal are graph are 11 by 15
What is the area of the graph? The area is 165
What does the area of the graph represent? The area represents 165 grams
Explain how you construct a 'graph rectangle' from a 'graph trapezoid'.
To construct a graph rectangle from a graph trapezoid, you find the midpoint on the line of slope of the trapezoid. Since this is halfway between both extremes, a graph rectangle of this height has the exact same area, and its height may be used instead.
Explain how to find the area of a 'graph trapezoid'.
Finding the area of a graph trapezoid is easiest done by making a graph rectangle of the graph trapezoid. First find the midpoint of the trapezoids line of slope, then use this as the altitude of the new graph rectangle. Using the same base, multiply base times new altitude to find area.
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.
Describe the motion of the dice on the ends of the strap, as you see them from your perspective.
Very nice work. See my notes.