course phy 201
8/31 around 10:20 am
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?
Clock Time when ball was released. Clock Time when ball got to end of ramp. Clock Time when ball hit floor.
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 ticks (half cycles) from top to bottom. ~ 1 tick between end to floor, this was harder to measure as accurately as the marble greatly increased in speed once it left the table
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, oct 1
What are the changes in clock times between measurements? 4 months and 5 months
At what average rate did the child's height change with respect to clock time between Jan 1 and May 1? .5 cm/month
At what average rate did the child's height change with respect to clock time between May 1 and October 1? .6 cm/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? Height
What is the B quantity? Clock time
What is the change in the A quantity for the interval? 5 cm
What is the change in the B quantity for the interval? 9 months
What therefore is the average rate of change of A with respect to B? 5cm / 9 months
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? 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.
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'.
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 using a sheet of graph paper as our measuring device, we found that between the 1st and 2nd slope we had a change of 10.75 graph blocks. The difference between the 2nd and 3rd slopes was 14.5 graph blocks
What is the average rate of change of landing position with respect to ramp slope, between the first and second slope? (10.75 0) / ((1/12) - (1/30)) = (10.75/ (1/20) = 215??? Im guessing that this is giving me what the change would be in landing spot with respect to the slope going all the way to 1. Not sure that accomplishes what I am looking for though.
Very good answer. Only the units are missing (see the annotated document I sent earlier today).
What is the average rate of change of landing position with respect to ramp slope, between the second and third slope? 14.5 / (1/12) = 174 same aside as last question not sure this is what im looking for.
For the following questions I was not aware we were supposed to measure time intervals for the 3 different slopes. We only measured landing spots. Therefore I do not have the required information to do these questions. Sorry.
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?
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).
For the preceeding questions I was not aware we were supposed to measure time intervals for the 3 different slopes. We only measured landing spots. Therefore I do not have the required information to do these questions. Sorry.
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? Amount of mph
What does the base of the graph represent? Amount of hours
What is the area of the graph? 120
What does the area of the graph represent? Total miles traveled
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? Amount of money made per month
What does the base of the graph represent? Months worked
What is the area of the graph? 36,000
What does the area of the graph represent? 36,000 dollars made
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? Position in feet
What does the base of the graph represent? Force in pounds
What is the area of the graph? 6,000
What does the area of the graph mean? A total of 6,000 pounds of pressure over a 200 foot area
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? Position in centimeters
What does the base of the graph represent? Density in grams / centimeter
What is the area of the graph? 800
What does the area of the graph mean? Density of 800 g/cm over 50 centimeters
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 mph
What is the rise between the two points of this graph? 10
What is the run between these points? 5
What therefore is the slope associated with this graph trapezoid? 2/1
What does the slope mean? Over the 5 hour span, speed went up 2 mph per hour.
What does the base of the graph represent? Clock time
What are the dimensions of the equal-area graph rectangle? 55 x 5
What is the area of the graph? 275
What does the area of the graph represent? Total miles traveled in 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? Money made 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? 25/3
What does the slope mean? For every 3 months worked the average amount of money made went up 25 dollars
What does the base of the graph represent? Clock time
What are the dimensions of the equal-area graph rectangle? 1100 x 24
What is the area of the graph? 26,400
What does the area of the graph represent? Over the 2 year span a total of $26,400 was made
On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30.
This question is answered above. The proper coordinates are not given to answer these questions
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? density
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? 2/15
What does the slope mean? Density goes up as position is moved back? (P.S. this question is worded in a way that I cant really apply it to a tangible idea of some real world scenario.
What does the base of the graph represent? Position in cm
What are the dimensions of the equal-area graph rectangle? 11 x 15
What is the area of the graph? 165
What does the area of the graph represent? Total pounds of pressure over a certain change in position ????
Explain how you construct a 'graph rectangle' from a 'graph trapezoid'. Add the 2 altitudes then divide by 2. this gives you the altitude for both sides to match up
Explain how to find the area of a 'graph trapezoid'. Take the altitude found above and multiply it by (x2-x1)
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? 25%
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.
Good work. See my notes, compare with the annotated document sent earlier today, and let me know if you have questions.