Physics 244

Measurement Uncertainties Lab Introduction In Part I of this exercise you will measure the mass and dimensions of a solid sphere of steel. From your measurements, you will calculate the density of the sphere. Assume that the scales are accurate to 0.1%. You will have to take multiple measurements of the diameter of the sphere in order to calculate the uncertainty in diameter and the corresponding uncertainty in volume. For your Lab Report This is not meant to be a normal lab report. In writing up Part I, give a one paragraph summary of your results, and include in a sample calculation of the density of steel and its uncertainty. About 75% of your grade will be determined by your performance on the problems in Part II. In your answers, be sure to show your work. Materials Part I: steel sphere, scale, meter stick, vernier calipers, Excel Reference Measurement Uncertainties handout. Giancoli, Physics 6th Edition: Chapter 1, sections 5 Theory Part I: The density (symbolized by the Greek letter, ρ or “rho” ) of an object is defined as the mass divided by the volume:

Vm

=ρ

The volume of a sphere is dependent on its radius, r, or its diameter, d:

33

634 drV ππ

==

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Procedure Part I: Measurement of a steel sphere's density. 1. Determine the value of and the uncertainty in diameter, d, of the steel sphere by measuring it at least 10 times with a meter stick. The meter stick is used in order to have a larger uncertainty. 2. Calculate the average diameter and standard deviation. To use Excel for the calculation, enter the diameter values in cells A2 to A11. In cell A12 enter "=average(A2:A11)" and in cell A13 enter "=stdev(A2:A11)". 3. Calculate the volume and the uncertainty in volume using the average diameter and uncertainty in the diameter. Convert the results to kg/m3 . 4. Calculate the density of steel and its uncertainty. Within how many standard deviations does

your result agree with the accepted density of steel? ⎟⎟⎠

⎞⎜⎜⎝

⎛× 3

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mkg1080.7

Figure 1: Vernier Calipers

5. The vernier caliper can be read to an accuracy of 201 mm. To measure the sphere with the

vernier caliper, close the jaws of the caliper lightly on the sphere as shown in Figure 1. Do your best to make sure that you are measuring the widest cross-section of the sphere. 6. Notice that there are two blades to the caliper: one is fixed, in Figure 1 it is the left-hand blade of the caliper and it is connected to the fixed measuring scale. The other blade slides and has 10 tick marks on its base; it is on the right in Figure 2. There are also two different units: on the upper edge are inches, on the lower edge centimeters. Use the lower scale.

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Figure 2 7. To read centimeters and millimeters, use the leftmost tick mark on the very bottom short scale as a pointer. This will point to a length on the scale just above it. For a rough approximation, you might read off the length to the closest mm. However with more careful examination you can approximate exactly where the pointer tick is pointing to, to a tenth of a mm (or to two or three tenths of a mm).

8. Try to estimate the answer to 101 of a millimeter. This is clearly impossible for most humans,

but start with this estimate, and in Step 8 you will use the vernier scale to get an accurate reading. As an example, for Figure 2, a rough estimate would be 1.89cm (a tiny bit smaller than 1.9cm). 9. To use the vernier scale, look at the other tick marks on the very short bottom scale next to the point tick. Find the tick mark on that tiny scale that is perfectly aligned with a tick mark on the scale just above it. There may be more than one, so just estimate which is closest to perfect. 10. Count the number of ticks from the perfectly aligned tick mark to the left most tick mark (the pointer tick). This gives you the tenths of millimeters that you must add to the closest millimeter mark that lies to the left of the leftmost tick. In Figure 2, the final reading is 1.88cm. Part II: Problems

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1. A certain device has made 7 measurements of some quantity, x: 6.03, 5.65, 8.01, 6.88, 7.35, 6.51 and 7.27. Compute the average value of x and xσ . Compare the results using (N-1) to using N in the denominator. 2. Find the percentage uncertainty for the values with given uncertainties below.

Value Uncertainty 13.3 m m4.0± 9.4 m/s m/s5.0± 20.28 s s55.0±

3. Find the uncertainty for the values with given percentage uncertainties below.

Value Uncertainty 33.21 m 10± % 14.0 m/s ± 0.8 %

8.59 s ± 4% 4. Length A is measured to be 4.5cm± 0.3 cm and length B is 4.0cm ± 0.7cm. What is the difference between the two lengths, A - B, and the uncertainty of the difference? Calculate the percent uncertainty of A, B and their difference. Is the percent uncertainty of the difference much different than that of either A or B? If the uncertainty in length A is reduced from 0.3cm to 0.1cm, a factor of 3 improvement, what is the new uncertainty in the difference of B - A? Can you explain the unexpected results? 5. Assume y = , and use Equation (8) in the “Measurement Uncertainties Reference” to calculate

2xyσ for x = 10 1. Now this as rewritten as ± xxy ×= , and use Equation (5) to find yσ .

Explain why the results are different and which method is correct.

6. Assuming x, t, and a are related by the equation 2

2txa = , find the value for % aσ , if x =

20m 2.0m and t = 7.00s 0.30s. ± ± 7. An experimental measurement has a result 7.95 ± 0.01, and a predicted value of 8.00. How many standard deviations is the experimental measurement from the predicted value? What is the percent uncertainty for the measurement? What is the percent difference between measured and predicted value? 8. If the uncertainty of a single length measurement is ± 0.4 cm, how many measurements, N, of the same length are necessary so that the average of the measurements has an uncertainty of

0.04cm? ± 9. If , what is the value of oo 425 ±=θ σ for sin (θ )?

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