WEEK 1 HOMEWORK

Assignment 1 due Wednesday June 6th.

1. Read the text: Chapter 1.

2. Read Appendix I -WP Activity Guide

3. Read Appendix II -WP Activity Guide

5. Do 1-1, 1-2, 1-3 and 1-4 from Unit 1 problems below.

6, Do Unit 2 Homework that follows Unit 2 in the Activity Guide.

 

It is important that you read and understand Appendix I and II since it will help you analyze data throughout the semester. 

UNIT 1 PROBLEMS
 

1-1) Analysis of data obtained with a freely falling object

Ten measurements of the time that it takes a ball to fall two meters are given in the table below: 

Trial #

Time(s)

1

0.58

2

0.56

3

0.52

4

0.55

5

0.57

6

0.60

7

0.56

8

0.62

9

0.58

10

0.54

a)  Calculate the mean time of fall. __________

b)  Calculate the standard deviation of the measurements using Excel and explain its significance.
(i.e. what does it tell you about the data?)

d)  Calculate the standard deviation of the mean and explain its significance.
(i.e. what does it tell you about the data?)

f)  Give the result for time of fall measurements with the appropriate number of significant
figures and the appropriate amount of uncertainty.

g)  The time of fall measured by a much more precise method gets a result of 0.64s.
What type of error can explain the discrepancy. Would taking more data give you a more reliable result
i.e. closer to the precise result given above? Why or why not?


1-2)  You are working in your spare time for a consumer organization that is checking on
whether of not the small house brand cereal boxes sold the local supermarket each have
the number of grams of cereal listed on the box.  The market manager claims that their
food processors actually "over fills" the boxes so that consumers are not cheated, but
several members of your organization are skeptical.  You purchase six boxes of each of
three types of cereal, Tasteeos (rated at 21g), Honey Nut Tasteeos (rated at 25g), and
Raisin Bran (rated at 35g).  The packaging for all three types of cereal is identical and
consists of a cardboard box and an inner plastic wrapper.  Using an electronic balance
you determine that the mass a box and wrapper without the cereal is 17.7g.    Next you
find the total masses for each of the boxes.   Your results are shown in the following table.



Note: These data are based on Food Lion Brand cereal purchased in Carlisle, PA on
Sept. 1, 1996.

(a) Using the method described in Appendix C of the Workshop Physics Activity Guide,
find the average  and standard deviation  by hand for the mass of the cereal (excluding
the packaging) in the boxes of Tasteeos.   Notes: (1) N is not 12 in this case. (2) No
credit will be given unless you show your calculations!
(b) Use a spreadsheet to find the average, Standard Deviation ( ssd or SD), and
Standard Deviation of the Mean (SDM)  for the mass of the cereal (excluding the
packaging) in the boxes of Tasteeos.  Do this for the Honey Nut Tasteeos and the
Raisin Bran.  Just report the results.  No need to hand in the spreadsheet printout.
(c) Do any of the boxes contain a mass of cereal that is under the rated mass? Are the
boxes overfilled or does the supermarket appear to be cheating the customers? 
Discuss
(c) Although you notice that the volume of cereal is the same in all eighteen boxes of
cereal the masses of the Honey Nut Tasteeos are greater than those of the Tasteeos
and the Raisin Bran boxes have the greatest mass.  Why might the same "amount" of
each type of cereal have a different mass.
(d) What type of cereal has the largest percentage uncertainty  in its mass?   Note: To
find the percentage uncertainty in this case you should divide the Standard Deviation
from the Mean (SDM) of a set of measurements by the average (or mean value) of that
set of measurements and multiply by 100.  (See page A-32 of Appendix F of the
Workshop Physics Activity Guide.)


1-3)  Suzanna and Miguelo calculated the averages and standard deviations for several
types of measurements. They got into a rote mood and copied all the digits that
appeared on their scientific calculator.  Please boil these down to  the "correct"
number of significant figures and state how many significant figures is being reported
for each best estimate.

(a) 5.666666667 ± 0.028314389 Volts
(b)   -18.25723 ± 2.31684 degrees celsius
(c) 0.0373749738 ± 0.005174523 meters
(d) 3546.78349 ± 188.549763 seconds (express this answer with and without scientific
notation)
(e) Explain why the result of rounding off in part (d) is ambiguous unless scientific
notation is used.

1-4) Wanda and Chris are studying projectile motion and they have derived a
theoretical equation that allows them to predict how far a projectile will move as
a function of its angle of launch, height above the floor, and initial speed.  Each
of them agrees that for the initial angle of launch, height above the floor and speed
of the ball that they measured, the theoretical distance for a launched ball ought
to be 4.58 meters.  The verify the theoretical equation, Wanda tapes the center
of a piece of paper at a horizontal distance of 4.58m from the point of launch
and shoots the projectile six times.  She puts a piece of carbon paper on top
of her other paper so that each time the projectile hits the paper it leaves a
smudge.  Chris thinks he can get less variation in his data by steadying the
table and adjusting the angle of launch more carefully.  He does these things,
puts down a clean paper, and launches the projectile four more times.



(a) Assuming that the picture of each piece of paper that follows is its actual size,
so distances can be measured in cm directly on the diagrams below: What is the average
distance of Wanda's six launches?  What is Chris's average
distance?   What is the standard deviation (οsd  or SD) and standard deviation of
the mean (SDM) of each set of measurements?

(b) Briefly discuss which of the two students has the least uncertainty
associated with his or her launches and is thus more precise.   Which one of
the students is the most accurate is the sense of having best estimate that
closely matches the theoretical calculation.  Which  one of the students
appears to have a systematic error associated with his or her launches?
What factors in the experimental technique used by each student might
cause one to be more precise and another more accurate?

 

 

Assignment 2 due Monday June 11th.

1. Read the text: CH 2 in your text and go over the examples carefully. Until now you have been assigned very little reading from the text. It is critical that you now begin to put what you have done in the lab and the discussion into a coherent framework. Motion is described verbally, in graphs and by mathematical equations, and you need to connect these three representations of motion. Go over the conceptual questions at the end of the chapter and make sure you understand and can answer them. You do not need to turn those in for grading.

2. Do Questions 1-1 to 1-5 given below. These are based on work done in Unit 3, the BB lab.

3. Do Unit 4 homework in the WP I manual.

4. Do Unit 5 homework in the WP I manual.

 

UNIT 3 QUESTIONS

1-1 (a) Make a data table showing the times and distances you recorded for the faster ball and sketch the data points as you did in Activity 1-2 of that lab, but use a large sheet of graph paper and most of the area on the sheet. Don't forget label axes give units and use a ruler for all straight lines.

(b) Draw a straight line that passes as close as possible to the data points you have graphed.

(c) Using methods you were taught in algebra, calculate the value of the slope, m, and find the value of the intercept, b, of the line you have sketched through the data. Give an appropriate number of significant figures in you results.

(d) Give the mathematical equation that describes the motion in terms of the position and time variables using the constant values obtained from your graph.

1-2) Using Excel create a mathematical model (an equation) of the motion for the faster bowling ball from the same data as in question 1. I will explain how to do this in class. However a step by step account is given below.

 (a) Open a new worksheet enter a title for your bowling ball graph and enter a title in cell A1. (b) Enter the Distance (m) and Time (s) headings and data for the faster ball from activity 2-2 in cells B3-B7 and C3-C7. Refer to your data table in Activity 2-2  (c) Enter the heading of a third column with theoretical distance (m) in D3. Place the symbol m (for slope) in the cell F4 and an estimated value, or 1, in G4. Place the symbol b (for y-intercept) in cell F5 and an estimated value, or 1, in G5. (f) Enter the appropriate equation for a straight line of the form Distance = m*Time + b in cell D4. Be sure to refer to cells G4 for slope and G5 for y-intercept as absolutes, i.e. use $G$4 and $G$5 when referring to them. (g) Click and drag the formula cell equation in cell D4 down to cell D7 to get the rest of the theoretical values ( get the small black cross that will appear at bottom right corner of D4 first.) (h) Create a graph of the data in the D-exp and D-theory columns as a function of the data in the Time column. Series 1 is time vs measured distance and series 2 is time vs. theoretical distance (i) Change the values in cells G4 and G5 until your theoretical line matches as closely as possible with your experimental data points in the graph window. (j) Hand in a printout of your modeling spreadsheet with the following written on it: 1. Your name, the date, and Problem 1-2 on it. 2. Report the average speed of the bowling ball that you calculated in Activity 2-2 compared to the slope of the "best" fitting graph for your mathematical model. 3. The equation that provides the "best" mathematical model for the motion you studied in the form Distance (m) = ( ? m/s) Time (s) + ( ? m) 4. Discuss the meaning of the slope of a graph of Distance vs. Time. What does it tell you about the motion of the bowling ball? 5. Compare the values of m and b that you obtained using hand calculations with those just obtained using computer modeling. Are they similar? They should be!

1-3) The graphs below represent possible functional relationships between time, t, and the position, x, of an object.

 wpe3.jpg (42501 bytes)

In each case indicate which, if any, of the four conditions hold: 1. x is proportional to t  2. x is a linear function of t  3. x always increases when t increases 4.  x always decreases as t increases Note: It is possible in each graph that none the conditions hold, or only one holds, or two hold, or three hold at the same time.

1-4) On the same set of axes, x vs t, draw four lines that represent the motion of a bowling ball that is described by the equation   x = m*t + xo  for each of the following situations:

       a)  m is + and xo is zero.

       b)  m is the same as in a but xo is negative.

       c) m is greater than in b, but xo is the same as in a.

       d) m is negative and xo is the same as in b.

        Label each line a, b etc.

    b) Compare and explain the differences between the motion in a, b, c and d.

1-5) First a brief review: in Unit 3 you arrived at a mathematical description of the motion of a bowling ball and in the absence of friction - in practice the effect of friction on a pretty massive bowling ball is very small - the ball moves at constant velocity. That means it has constant speed and moves in a straight line - the effect of spin also being conveniently ignored. This motion was found to be described very well by an equation, with some uncertainty in measurement that showed up in the mean square error term that is also calculated in the curve fit process.

The equation you arrived at was of the form 

                                                            y = mx + b

Writing this equation instead with variables commonly used to describe motion, which are x and t, we get the equation  x = mt + b

What is the physical significance of the constants, for m and b in your experiment? To understand that we have to go back to how motion is described. The term velocity is used to describe the rate of change in position, or more specifically it is defined as:

                                                            v = (x – x0)/( t - t0)

where the subscript 0 refers to the initial time and position and the variables without a subscript to the final time and position.  It is usual to assume that the clock is started at t0, so that t0 = 0. Now the equation reduces to :

                                                            v = (x – x0)/ t

Rearranging, we get

                                                            x = v t +  x0

This corresponds exactly to the equation you obtained from the modeling exercise if the physical significance of m is the velocity, a constant for this motion. Also b must be x0, the position of the ball at t = 0.

Note: x is not a distance, it is a position, and x0 is the position at t = 0. However (x- x0) is a distance, or more accurately since it is a vector, (x- x0) is really the displacement in the x direction for the recorded motion.

Now lets use the equation in some exercises that connect equations to graphs and graphs to equations.

a. Draw the graphs for the following equations:

 b. Write equations for the following graphs. Estimate values for the constants.

 

c. For each of the graphs in b above, describe the motion, i.e. how you would move, to produce the position-time graph shown.

 

Due Thursday June 15th:

The Measurement Contest.

EITHER: Measure the height, from ground level, of the NE corner of Smith Hall (don't include the elevator shaft sitting on the roof). Remember this is an indirect measurement so don't make any direct measurements on the building itself.  You have been given a handout for this activity in class and I will be glad answer any questions you may have. Any equipment you need for measurement you can get from me of the lab tech. The reports are due at the beginning of class Thursday June 14th. 

                                                      OR

In case you you don't care for the first assignment, estimate the volume  flow of water in the Ohio River at Huntington. There will be no hints given with this assignment.