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50 Last/First Name (print): Section 3. In your calculator menu, you should see a setting that states DEG or RAD. What does this setting mean? When should you use one from the other? 4. In a right triangle, define the relationships of the following trigonometric functions in terms of the sides of the triangle. a) Sin 0- b) Cos 0- c) Tan 0- 5. Round the following to 3 significant figures: a) 25.132 b) 10.071 SAHd
does this setting Jsername ID (xxx9999): Last/First Name (print):) PHYS -Section Username ID (xxxxx9999): PRE-Lab Summary (Not Graded. Practice Use Only) Read the experiment before coming to lab. 1) Summarize the procedure for this experiment on the page below. 2) Include the purpose, procedure and calculations that you will need. 3) This summary should be in your own words in bullet format. You may use the back of this page as needed. ents and
U Last/First Name (print): Lab Orientation 1: Significant Figures: Applications to Measurements and Error Analysis Introduction "I often say that when you can measure what you are talking about, and express it in numbers, you know something about it, but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the state of science, whatever the matter may be." -Lord Kelvin (William Thomson) This statement is often quoted in support of the argument that science cannot be understood without mathematics. It is instead addressed to the issue of meaningful and accurate communication of information. While it is possible to conceptualize the various relationships. of science without attaching numbers to them, conveying these concepts in a meaningful way is quite difficult unless we first remove the uncertainty, or "fuzziness" associated with them. For example, mass is a concept one might reasonably well acquire without using numbers, but without a method for comparing masses, the ability to communicate such a concept would be limited. Phrases such as "a lot of mass," or "a very small mass" are subject to a wide range of interpretations. Purpose The purpose of orientation is to review mathematical skills that should have been learned before this course. A review is done for personal reflection and to see what kinds of mathematical tools will be used for the semester. This will be done with an application to measurements and graphing data. 3 لیا
Percent Error and Percent Difference Ofer doing an experiment, it is necessary to estimate how accurate the measurements were One way to do that is to calculate percent error. This can be done when the measured quantity has a known value called the standard value or the accepted value. Percent Error Experimental Value-Accepted Value Accepted Value x 100% Equation 1 The percent error is the absolute value of the difference between the accepted value and your bete, divided by the accepted value. When no accepted value is known or when comparing between two measurements, the percent difference is used. The percent difference is the absolute value of the difference between two experimental values divided by the average of the experimental values. Percent Difference: Value-Value₂l. Average Value x 100% Equation 2 In scientific work, most numbers are measured quantities and thus are not exact. All measured quantities are limited in significant figures (SF) by the precision of the instrument used to make the measurement. The measurement must be recorded in such a way as to show the degree of precision to which it was made- no more, no less. Calculations based on the measured quantities can have no more (or no less) precision than the measurements themselves. The answers to the calculations must be recorded to the proper number of significant figures. To do otherwise is misleading and improper. Determining Which Figures are Significant Non-zero integers are always significant. example: 23.4g and 234g both have 3 SF Captive zeroes, those bounded on both sides by non-zero integers, are always significant. example: 20.05 has 4 SF; 407 has 3 SF Leading zeros, those not bounded on the left by non-zero integers are never significant. Such zeros just set the decimal point; they always disappear if the number is converted to powers-of-10 notation. example: 0.04g has 1 SF; 0.00035 has 2 SF. They can be written as 4x10 and 3.5x10 respectively. Trailing zeros, those bounded only on the left by non-zero integers may or may not be significant. wapdam Nm To chaty wa
example: 45.0L has 3 SF; 450L has only 2 SF: 450 L has 3 SF. Note: To clarify whether a trailing zero is significant, it is preferable to use scientific notation to express the final answer. example: 450 L can be expressed as 4.50 x 10³ or 4.50102 whereas 450 L would be expressed as 4.5x10. Exact numbers are those not obtained by measurement but by definition of by counting numbers of objects. They are assumed to have an unlimited number of significant figures. Multiplication and Division Involving Significant Figures Calculations involving only multiplication and/or division of measured quantities shall have the same number of significant figures as the fewest possessed by any measured quantity in the calculation. example: 14.0 x 3-40, not 42, because one of the multipliers has only one SF. example: 14.0 x 3.0-42, because one of the multipliers has only two SF. example: 14.0/3-5, not 4.6, because the denominator has only one SF. Addition and Subtraction Involving Significant Figures Calculations where measured quantities are added or subtracted shall correspond to the position of the last significant figure in any of the measured quantities. That is, the final answer is only as precise as the decimal position of the least precise value. The number of significant figures can change during these calculations. example: 14.16 example: 46.6 +5.72 +3.2 52.32 17.36 17.4 is the correct answer 52.3 is the correct answer Combined Calculations In calculations involving addition/subtraction and multiplication division, significant figure guidelines must be applied after each calculation involving addition or subtraction. example: (3.2 x 4 x 0.035/7) + (12 x 0.5) - 0.06+6 6
Tips for Rounding Off Numbers A number is rounded off to the desired number of significant figures by dropping one or more digits to the right. The following guidelines should be observed when rounding off When the first digit dropped is less than 5, the last digit remains unchanged. When the first digit dropped is more than or equal to five, the last digit retained is increased by I. 17.9-18 243-240 examples: Plotting of Data Most physical relationships in nature can be graphically analyzed using one of three different types of plotted curves. Fortunately, all three of these can be graphed as a linear relationship with only minor mathematical manipulation. They are: Linear Relationships-such as y-mx + b Power Relationships-such as y=ax" Exponential Relationships-such as y-be Before you start graphing your data, it is important to properly scale and label your graph paper. Plot the independent variable along the x-axis and the dependent variable along the y- axis. For example, our relationship y-mx +b has the form such that y is our dependent variable and x is our independent variable. Is the distinction between independent and dependent variables just a matter of how our equation is expressed? Could we have rearranged the equation where x was dependent on y? Yes, but the distinction between dependent and independent variables is actually a consequence of what you can actually measure in the laboratory. For the relationship y=mx+b, we were evidently able to vary x and measure y. Therefore, y is dependent on x. Once you have decided which variable belongs to which axis, you must choose a scale. There are two important steps in scaling your graph, first, use a scale that allows your plotted curve to fill as much of the graph paper as possible. Secondly, use a scale that is convenient and easy to read such as one that increases as a factor of either 1, 2, or 5 of your units of measure. Example: Your data ranges from 1.103 meters to 2.047 meters. The graph paper is divided into twenty, evenly spaced grids. If you subtract your two extreme data points, your values encompass a range of 944 meters. This value evenly distributed among all twenty divisions of your graph paper would give .0472 meters per grid, a difficult number with which to work. Make the graphing simpler by rounding up the .0472 to the next multiple of 1, 2, or 5. In this case you would choose .05. Label your graph
axis starting with 1.10 meters and continue in increments of 05. Your net so would be 1.15, 1.20, and so on. Once you have plotted all your points, be sure to label all axes with the appropriate units, give your graph an appropriate title, and include i any additional scaling information as required. y and x are variables plotted on your graph . m is a constant value that represents the slope of your graph b is the y value when x-0, or the y intercept. The slope of your graph is the ratio of the change in the y values to the change in the x values. The slope of your graph is determined from the line of best fit" that you draw on your graph paper. The line of best fit is a straight line you think best depicts the average of your values. 20 27-** 18 10 14 12 6 2 7 8 9 5 6 10 3 4 Time (5) Figure 1. "Line of Best Fit The power relationship is difficult to graph and difficult to work with in the form y-ax" However, if you take the Log of both sides of the equation you obtain; Log(y) - Log(ax") rearranging gives Log(y)-m Log(x) + Log(a). In this form, if you plot Log(y) versus Log(x), the slope of your graph will be m. This typ plot is commonly referred to as a Log-Log plot. If m were always an integer it would be 7 In the linear relationship y-mx + b. Voty (ms) 10 (XY). 2
almost as easy to plot y versus x". The slope would then become the value of a. This metho is certainly satisfactory as long as you are sure that m should be an integer. The third of our relationships, the exponential relationships, can be handled by taking the natural log of the equation y-be In(y)-In(be) rearranging gives In(y)-In(e)-In(b) In(y)-mx+ln(b) In this form, a plot of Inty) versus x would give a linear graph with slope m and y intercept of In(b). A graph of this type is a semi-log graph. of practice. The only way to truly understand and appreciate the usefulness of graphical Becoming proficient at graphing your data and properly interpreting the results is a matter analysis is to work your way through some labs. The most important point of any graphing exercise is to be meat. When graphing, clearly label your axis and name your graph. When labeling your axis, use the appropriate units as well as magnitudes. The "line best fit is really just an approximation of what you think the average values represent. The slope that is determined from your line of best fit does have physical significance and must be calculated with the units. Other Useful Math Tools for this course Other techniques that you TA will cover include the following: . FOIL • System of Equations . Using Exponentials . Trigonometry (SINE, COSINE, and TANGENT functions and their inverses) Unit Circle Difference between Degrees and Radians If you do not remember how to use these mathematical tools, please review these before coming to class.