course PHY 232
Class 090826xxxx
The idea of 'clock time'.: We imagine a clock which, once it starts running, continues running. The clock can measure the passage of time in any units (e.g., seconds, minutes, milliseconds, hours, days, years, centuries). Times measured in one unit can of course be converted to different units.
A pendulum could constitute a 'clock', which runs in units of 'cycles' or 'half-cycles'.
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?
Trial 1 Trial 2 Trial 3
Release of marble 13 8 8
Marble reaches end of ramp 18 16 13
Marble hit’s the floor 20 18 15
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?
Trial 1 Trial 2 Trial 3
Interval from release to end of ramp 13 to 18 8 to 16 8 to 13
Interval from end of ramp to floor 18 to 20 16 to 18 13 to 15
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?
The clock times in this question would be the dates which the measurements of the person were taken (January 1, May 1 and October 1)
What are the changes in clock times between measurements?
The first clock time was January 1 and the second clock time was May 1. The difference in these two clock times is 4 months. Four months could then be converted in to the number of days in the interval by adding up the total number of days in January, February, March and April. This calculation would total to 119 days, unless the year that this information was taken during was a leap year then the amount of days in this interval would equal 120. From this information the number of hours, minutes or even seconds contained in this interval can be calculated.
The interval between the second clock time (May 1) and the third clock time (October 1) could be figured by counting the number of months contained in this timespan which would be 5 months. Also as detailed in the previous paragraph it can be determined how many days are contained in this interval by adding up the number of days contained in each of the months contained in this interval. In this interval there would be no need to worry about the possibility of a leap year since leap years do not affect any of the months contained in this interval. The total number of days in this interval is 153 days.
Also calculations could be made using the entite intrerval of clock time in which all the measurments were taken. This interval contains 9 months or 272 days (273 days in the case of a leap year)
At what average rate did the child's height change with respect to clock time between Jan 1 and May 1?
For this question you would divide the change in height of the child over the given period (2 cm) by the change in clock time over which the height change occurred (4 months). This equation would give you the solution that the childs height increases one-half cm per month.
At what average rate did the child's height change with respect to clock time between May 1 and October 1?
For this as with the previous question you would take the change in height over the interval of clock time (3 cm) divided by the change in clock time of the interval (5 months). This equation would give you the solution of an increase of 0.6 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= 0.03333 cm
The A quantity? Height of the ramp
The B quantity? Distance of ramp
What is the change in the A quantity? 1 cm
What is the change of the B quantity: 30 cm
What is the slope of your ramp when supported by a domino lying 'on its side'? 2.5/30= 0.08333 cm
What is the A quantity? Height of the ramp
What is the B quantity? Distance of the ramp
What is the change in the A quantity? 2.5 cm
What is the change in the B quantity? 30 cm
What is the slope of your ramp when supported by a domino lying 'on its end'?
5/30= 0.16667 cm
What is the A quantity? Height of the ramp
What is the B quantity? Distance of the ramp
What is the change in the A quantity? 5 cm
What is the change in the B quantity? 30 cm
How much does the slope of the ramp change when you change the domino from flat to on-its-side to on-its-end?
From flat to on its side: From 1/30 to 2.5/30 is an overall increase of 1.5/30
From on its side to on its end: From 2.5/30 to 5/30 is an overall increase of 2.5/30
By how much does the landing position of the marble change as you move from the first slope to the second to the third?
Domino Position Landing position (distance from table leg)
Flat 12 cm
On its side 15 cm
On it end 22 cm
From the first slope (flat domino) to the second slope (on its side domino) the landing position of the marble increases by a distance of 3 cm from the table leg. From the second slope (on its side domino) to the third slope (on its end domino the landing position of the marble increases by 7 cm from the table leg.
What is the average rate of change of landing position with respect to ramp slope, between the first and second slope?
Between the first slope and the second slope the average change in landing position in regard to ramp slope would be figured by dividing the change in landing position between slope 1 and 2 (7cm) by the change in slope between one and two (1.5/30). This would give you the average change which is 140.
What is the average rate of change of landing position with respect to ramp slope, between the second and third slope?
The average rate of change between these two trials would be figured in the same manner as the average rate between the 1st and 2nd from the previous question. Change in landing position (7cm) divided by the change in slope (2.5./30) resulting in an answer of 84.000033.
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'? 5 half cycles of the pendulum
How long does it take the ball to roll down the incline with the domino lying 'on its side'? 3 half cycles of the pendulum
How long does it take the ball to roll down the incline with the domino lying 'on its end'? 2 half cycles of the pendulum
For each interval, what is the average rate of change of the time required to roll down the incline with respect to ramp slope?
From flat to on its side: 2 half cycles/ (1.5/30)= 40
From on side to on end: 1 half cycles/ (2.5/30)= 12
From flat to on its end: 3 half cycles/ (4/30)= 22.5
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?
Change in ball position/ change in time it takes to roll down ramp
From flat to on it side: 30cm / 2 half cycles= 15 cm per half cycle
From on side to on end: 30 cm/ 1 half cycle= 30 cm per half cycle
From flat to on its end: 30 cm/ 3 half cycles= 10 cm per half cycle
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? Speed in miles/hour
What does the base of the graph represent? Clock time in hours
What is the area of the graph? 120 miles
What does the area of the graph represent? Miles traveled in the 3 hours traveling at 40 miles per hour
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? Dollars per month
What does the base of the graph represent? Clock time in months
What is the area of the graph? 36,000
What does the area of the graph represent? Amount of income in the given amount of 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? Force in pounds
What does the base of the graph represent? Position in feet
What is the area of the graph? 6,000
What does the area of the graph mean? the total anount of force in pounds at any distance up to and including 30 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? Density in grams/centimeter
What does the base of the graph represent? Position in centimeters
What is the area of the graph? 800
What does the area of the graph mean? number of grams present
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? Speed in miles/ hour
What is the rise between the two points of this graph? 10 mph
What is the run between these points? 5 clock hours
What therefore is the slope associated with this graph trapezoid? 2 miles/ hour
What does the slope mean? that there was an increase of an average of 2 miles/hour for 5 hours to achieve the change in speed of 50 miles/hour to 60 miles/hour
What does the base of the graph represent? Clock time in hours
What are the dimensions of the equal-area graph rectangle? Base:5 Height: 55
What is the area of the graph? 275
What does the area of the graph represent? The number of miles traveled in the given amount of clock time
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? Income stream in dollars per month
What is the rise between the two points of this graph? 200
What is the run between these points? 24
What therefore is the slope associated with this graph trapezoid? 8.333333
What does the slope mean? the average rate of change of the income stream in dollars per month
What does the base of the graph represent? Clock time in months
What are the dimensions of the equal-area graph rectangle? Base: 24 Altitude: 1100
What is the area of the graph? 26,400
What does the area of the graph represent? The steam of income taken in over the course of the entire clock time represented on this graph
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? Force in pounds
What is the rise between the two points of this graph? 30
What is the run between these points? 200
What therefore is the slope associated with this graph trapezoid? 0.15
What does the slope mean?
What does the base of the graph represent? Position in feet
What are the dimensions of the equal-area graph rectangle? Base:200 Altitude:30
What is the area of the graph? 6,000
What does the area of the graph represent? The total amount of force at all distances included on the graph
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? Density in grams/centimeter
What is the rise between the two points of this graph? -2
What is the run between these points? 15
What therefore is the slope associated with this graph trapezoid? -0.13333333
What does the slope mean? that as the density in grams per centimeter increases the position in centimeters decreases.
What does the base of the graph represent? Position in centimeters
What are the dimensions of the equal-area graph rectangle? Base 7.5 altitude: 11
What is the area of the graph? 82.5
What does the area of the graph represent? The density in grams per cm at all positions covered by this graph
Explain how you construct a 'graph rectangle' from a 'graph trapezoid'.
To construct a graph rectangle from a graph trapazoid you first find all the areas where one side is not equal to the same side on the opposite side of the trapazoid and move each side in or our in proportion to the other until they line up. Once you have done this with all the sides you will have a graph rectangle. This ensures that opposing sides of the newly formed rectangle are of the same length.
Explain how to find the area of a 'graph trapezoid'.
Figure what the demensions of the sides would be if you were to take the graph trapezoid and make it into a graph rectangle. Then take those dimensions and figure the area.
Good work. I'll be sending commentary on the last several problems, which are re-assigned for Wednesday, after Wednesday's class. Your answers to those problems are mostly correct and you don't need to redo them.