course Phy 201
Aug, 29, 1:36
Quick Exercise:xxxx
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; 2,6,6.5
Trial 2; 4,8,8.5
Trial 3; 9,13,13.5
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?
2, .5
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? .5cm,.6cm
What are the changes in clock times between measurements? .1cm
At what average rate did the child's height change with respect to clock time between Jan 1 and May 1? .5cm
At what average rate did the child's height change with respect to clock time between May 1 and October 1? .6cm
To answer a question related to an average rate of change on an interval, always answer the following questions:
What is the A quantity? Is the child height
What is the B quantity? The months that went past.
What is the change in the A quantity for the interval? 100,102,105
these are the A quantities, not the changes in the A quantity
What is the change in the B quantity for the interval? 4months, 5months
What therefore is the average rate of change of A with respect to B? .5cm, .6cm
The numbers are right, but the B quantity is measured in months. So the change in the B quantity is in months. The average rate will therefore be in cm / month, not cm.
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? .5
There are two intervals here, and you need to find the change for each interval.
Neither of the changes will be .5.
You should indicate how you got this result.
By how much does the landing position of the marble change as you move from the first slope to the second to the third? By.1
what is the unit of this change?
What is the average rate of change of landing position with respect to ramp slope, between the first and second slope? .4
How did you get this result and what are its units?
What is the average rate of change of landing position with respect to ramp slope, between the second and third slope? .2
How did you get this result and what are its units?
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?
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? It represent the change in A
The altitudes represent the A quantity. The change in the altitude represents the change in the A quantity.
The A quantity also has a name: speed.
It also has unit: miles / hour.
What is the rise between the two points of this graph? 50/60=.83
What you give is the ratio between the 'graph altitudes', which is often an important quantity.
However the rise is the difference between the 'graph altitudes', not the ratio. The rise is 60 - 50 = 10.
In fact the rise has a meaning--it is the change in speed. It also has units: mile / hour.
So the rise is 60 mph - 50 mph = 10 mph, and it represent the change in speed.
What is the run between these points? 2/7=.28
The run is the difference, not the ratio. Easily corrected:
The run is 7 - 2 = 5.
However the B quantity has a name and units. Its name is 'clock time', and its unit are hours.
So the run is 7 hrs - 2 hrs = 5 hrs, and it represents the change in the B quantity, which is the change in clock time.
What therefore is the slope associated with this graph trapezoid? .83/.28=2.96
You divided your rise by your run, and this is the correct thing to do.
However rise and run are defined by differences, not ratios. This will be easy for you to correct.
What does the slope mean? how step the graph is
The slope is calculated as
rise / run = change in speed / change in clock time = 10 mph / (5 hrs) = 2 mph / hr.
What does the base of the graph represent? miles
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? How much coverage the points have
We will do more work in class Monday on areas and equal-area graphs, and I'll defer comment on this part until then.
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? It represent the change in A
What is the rise between the two points of this graph? 1000/1200=.83
What is the run between these points? 16/40=.4
What therefore is the slope associated with this graph trapezoid? .83/.4=2.0
What does the slope mean? how step the graph is
What is the name of the A quantity, and what are its units?
What is the B quantity, and what are its units?
What is the change in the A quantity, what are its units, and what name would you give the change in the A quantity?
What is the change in the quantity, what are its units, and what name would you give the change in the B quantity?
What number do you get when you divide the change in the A quantity by the change in the B quantity?
What are the units of the quantity you get when you do this division?
What therefore is the numerical value of the slope, and what are the units of the slope?
What is the meaning of the slope for this situation?
What does the base of the graph represent? Dollars per month
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? How much coverage the points have
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).
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 change in A
What does the base of the graph represent? The change in B
What is the area of the graph? 120
What does the area of the graph represent?
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 change in A
What does the base of the graph represent? The change in B
What is the area of the graph?36000
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? The altitudes of the graph repersent the Y value and also the change in A.
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? .15
What does the slope mean? The average rate of change of A with respect to B is average rate=change in A/change in B.
What does the base of the graph represent? The base repersents the X value also the change in B
What are the dimensions of the equal-area graph rectangle?
What is the area of the graph? 6000
What does the area of the graph represent? How much coverage the rectangle has.
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 altitude of the graph repersents the y value and also the change in A.
What is the rise between the two points of this graph? 12/10=1.2
What is the run between these points? 5/20=.25
What therefore is the slope associated with this graph trapezoid? 1.2/.25=4.8
What does the slope mean? The average rate of change of A with respect to B is average rate=change in A/change in B.
What does the base of the graph represent? The base reprsents the X value also the change in B
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? How much area is in the two points
Explain how you construct a 'graph rectangle' from a 'graph trapezoid'.
-You cut the top off of the trapezoid which makes a small triangle, then you flip the triangle over so that the long flat side is at the top of the rectangle.
Explain how to find the area of a 'graph trapezoid'.
-To find the area of a graph trapezoid width times length.
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?
-Yes they could verify that you walked at a 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?
-Yes I did increase but not a constant rate of what I noticed. The other person could not judge that I was increasing at a constant but did note that I did increase but nothing they could measure.
Describe the motion of the dice on the ends of the strap, as you see them from your perspective."
You're doing well here. Though like everyone else in the class you have some errors, you are doing much of this right, and you are using an intelligent approach to these problems.
Review my notes, and think about how you would change your answers to some of the questions. Most of the questions will be assigned again, and I expect you'll be in very good shape after we review these ideas on Monday.