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A. Error Propagation Exercises 1. Suppose you measure the diameter of a circle to be (12.34 ± 0.05) cm. Calculate the Show your work. (Note: It is recommended that you first write the formulas for C and A in terms of the diameter instead of the radius.) Circumference: C= x=(7.00±0.05) cm y=(9.20±0.05) cm Area: 2. Suppose x and y have the following measured values. Calculate A and B. State your final correctly rounded results with uncertainty. Show your work. allas E A = 2xy A = B= 4x³

A. Error Propagation Exercises 1. Suppose you measure the diameter of a circle to be (12.34 ± 0.05) cm. Calculate the Show your work. (Note: It is recommended that you first write the formulas for C and A in terms of the diameter instead of the radius.) Circumference: C= x=(7.00±0.05) cm y=(9.20±0.05) cm Area: 2. Suppose x and y have the following measured values. Calculate A and B. State your final correctly rounded results with uncertainty. Show your work. allas E A = 2xy A = B= 4x³

University Physics Volume 1
University Physics Volume 1
18th Edition
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:William Moebs, Samuel J. Ling, Jeff Sanny
Chapter1: Units And Measurement
Section: Chapter Questions
Problem 73P: (a) How many significant figures are in the numbers 99 and 100.? (b) If the uncertainty in each...
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Question number 1 and 2 can you show how to calculate to solve ? Here’s the hint for the formula
A. Error Propagation Exercises Calculate the 1. Suppose you measure the diameter of a circle to be (12.34 ± 0.05) cm. circumference and area of the circle. State your final correctly rounded results with uncertainty. Show your work. (Note: It is recommended that you first write the formulas for C and A in terms of the diameter instead of the radius.) Circumference: C = 2. Suppose x and y have the following measured values. Calculate A and B. State your final correctly rounded results with uncertainty. Show your work. atimas x=(7.00±0.05) cm y = (9.20±0.05) cm A = 2xy A = Area: A = B = B = 4x3 27
Transcribed Image Text:A. Error Propagation Exercises Calculate the 1. Suppose you measure the diameter of a circle to be (12.34 ± 0.05) cm. circumference and area of the circle. State your final correctly rounded results with uncertainty. Show your work. (Note: It is recommended that you first write the formulas for C and A in terms of the diameter instead of the radius.) Circumference: C = 2. Suppose x and y have the following measured values. Calculate A and B. State your final correctly rounded results with uncertainty. Show your work. atimas x=(7.00±0.05) cm y = (9.20±0.05) cm A = 2xy A = Area: A = B = B = 4x3 27
fbest=f(best best). The second step is to find the uncertainty in f. The rule for this depends on whether the calculation involves addition, subtraction, multiplication, or division. It turns out that the error propagation rules are the same for addition as they are for subtraction. They are also the same for multiplication as they are for division. For addition and subtraction, the absolute uncertainty in fis the sum of the absolute uncertainties in x and y. For multiplication and division, the fractional uncertainty in fis the sum of the fractional uncertainties in x and y. This is summarized symbolically below. Rule for Addition and Subtraction If either f=x+y or f=x-y, then fz6x+8y Rule for Multiplication and Division If either f = xy or Lab 2: Error Propagation &f=fre (You may have noticed that the rules are stated as approximations. These formulas are only approximations for statistical reasons that are beyond the scope of this class. The exact rules involve addition in quadrature, which is described in Appendix C, but you will not be required to use the exact rules in Physics 4AL.) It is important to keep in mind that the rule for multiplication and division only gives the fractional uncertainty. The final answer should be written in the form f=fee tof, thus it is necessary to find the absolute uncertainty by multiplying the fractional uncertainty by the best estimate, -) = Srce · fest = fbest &f=fest Ex + , then Ex 0 + = =+ Sy Ybest There are a couple of special cases that will come up often. Special Case #1: Multiplication by an exact number An exact number is one that has no uncertainty. You will frequently encounter exact numbers in formulas and definitions. For example, say you have a measured value for the radius of a circle and you wish to find the diameter. By definition, the diameter is exactly two times the radius. Cest 8f8x dy + Sest Xest Y'best (Note about the number : When the number a is used in a calculation, it is not strictly an exact number since z is irrational and you will only use a finite number of digits. We can treat it effectively as an exact number as long as we use several significant figures. It is best to use the a button on your calculator. If your calculator does not have this button, you should use 3.14159. Using only 3.14, or especially 22/7, can cause significant errors in your calculations.) 8f SC dx dx CXest Xest best Chest Xest If we want the absolute uncertainty in f, we have to multiply both sides by fe dx ( * )=C²₂ Xbest We will consider the general case, f = Cx, where C is an exact number and x is a measured number with uncertainty, x=xbest tox. As usual, the best estimate is best Cest Using the rule for multiplication, we find the fractional uncertainty in fis equal to the fractional uncertainty in x. 17 dx Xest Physics 4AL = Cox So, the fractional uncertainty in fis the same as the fractional uncertainty in x and does not depend on C, but the absolute uncertainty infdoes depend on C. It is important to remember this difference between the fractional uncertainty and the absolute uncertainty when multiplying by an exact number.
Transcribed Image Text:fbest=f(best best). The second step is to find the uncertainty in f. The rule for this depends on whether the calculation involves addition, subtraction, multiplication, or division. It turns out that the error propagation rules are the same for addition as they are for subtraction. They are also the same for multiplication as they are for division. For addition and subtraction, the absolute uncertainty in fis the sum of the absolute uncertainties in x and y. For multiplication and division, the fractional uncertainty in fis the sum of the fractional uncertainties in x and y. This is summarized symbolically below. Rule for Addition and Subtraction If either f=x+y or f=x-y, then fz6x+8y Rule for Multiplication and Division If either f = xy or Lab 2: Error Propagation &f=fre (You may have noticed that the rules are stated as approximations. These formulas are only approximations for statistical reasons that are beyond the scope of this class. The exact rules involve addition in quadrature, which is described in Appendix C, but you will not be required to use the exact rules in Physics 4AL.) It is important to keep in mind that the rule for multiplication and division only gives the fractional uncertainty. The final answer should be written in the form f=fee tof, thus it is necessary to find the absolute uncertainty by multiplying the fractional uncertainty by the best estimate, -) = Srce · fest = fbest &f=fest Ex + , then Ex 0 + = =+ Sy Ybest There are a couple of special cases that will come up often. Special Case #1: Multiplication by an exact number An exact number is one that has no uncertainty. You will frequently encounter exact numbers in formulas and definitions. For example, say you have a measured value for the radius of a circle and you wish to find the diameter. By definition, the diameter is exactly two times the radius. Cest 8f8x dy + Sest Xest Y'best (Note about the number : When the number a is used in a calculation, it is not strictly an exact number since z is irrational and you will only use a finite number of digits. We can treat it effectively as an exact number as long as we use several significant figures. It is best to use the a button on your calculator. If your calculator does not have this button, you should use 3.14159. Using only 3.14, or especially 22/7, can cause significant errors in your calculations.) 8f SC dx dx CXest Xest best Chest Xest If we want the absolute uncertainty in f, we have to multiply both sides by fe dx ( * )=C²₂ Xbest We will consider the general case, f = Cx, where C is an exact number and x is a measured number with uncertainty, x=xbest tox. As usual, the best estimate is best Cest Using the rule for multiplication, we find the fractional uncertainty in fis equal to the fractional uncertainty in x. 17 dx Xest Physics 4AL = Cox So, the fractional uncertainty in fis the same as the fractional uncertainty in x and does not depend on C, but the absolute uncertainty infdoes depend on C. It is important to remember this difference between the fractional uncertainty and the absolute uncertainty when multiplying by an exact number.
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