Measurement. 2. The magnitude of the quantity. Even with the same instrument, every order of magnitude jump will generate one more significant digit.

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1 Measurement Significant digits form the basis of any physical science, especially when performing experiments. Physics is an exact science. This does not mean that the true value 1 of all quantities is known; it just means that measurements and calculations are always reported to the proper number of significant digits - no more or no less. When it comes to measured values there are a number of factors that influence the total number of significant digits. 1. The instrument used. If you measure the same quantity with two different instruments, the one with a smaller graduation will produce more significant digits. 2. The magnitude of the quantity. Even with the same instrument, every order of magnitude jump will generate one more significant digit. 3. Whether or not you include an estimate in your measurement. Most analog instruments allow you to estimate beyond their graduation; if you do you will get an extra significant digit. The first two are straightforward but you may not have any experience with the third. In general, if an instrument allows you to estimate beyond the graduation and you can reasonably do so, then you should - always. This will take some practice but always ask yourself whether or not it makes sense to estimate the quantity you are measuring beyond the graduation of the instrument you are using. When it comes to significant digits in calculated values there are basic rules. 1. When you add or subtract values, the final answer can have no more decimal places than the value with the least number of decimal places used in the calculation. 2. When multiplying and dividing, the final answer can have no more significant digits than the value with the least number of significant digits used in the calculation. Use scientific notation for vary large or very small values to help you keep track of the significant digits in both. Remember to keep an extra significant digit or two in any intermediate calculations only rounding for your final answer; this will assure no round-off error along thw way. If you have a calculator, just let it keep intermediate values to whatever internal accuracy it generates. This is a simple experiment, but that does not mean it will be easy. You can make reasonably accurate (close to what you should be getting) measurements, but if you record them to the incorrect number of significant digits your measurements are only half correct and will be graded accordingly. If significant digits were not important there would be no need for different instruments! In the first procedure you will measure the length l and width w of a sheet of paper; you will then calculate the surface area A of the sheet. Remember that 1 A mythical beast, by the way. 1

2 A = lw (1) In the second procedure, you will measure the height h, diameter d, and mass m of a cylinder; you will then calculate the density ρ of the cylinder. Remember that where V is the volume and that for a cylinder ρ = m V Thus V = 1 4 πd2 h ρ = 4m πd 2 h (2) Apparatus Rule, Sheet of paper, Vernier caliper, Micrometer, Triple-beam balance, Cylinder. Procedure 1. Take out a sheet of letter-sized paper. Measure the length of the page three times; these measurements should be done by different people at different locations on the paper. To avoid any bias, do not share the measurements until all have been made. Do not forget the estimated digit. Record these values on the Data Sheet and calculate the average length. Repeat for the width. 2. Calculate the surface area of your sheet using the average values of length and width; note that you will do this for two different units. 3. Measure the height, diameter, and mass of the cylinder similarly. Use the vernier caliper for the height, the micrometer for the diameter, and the triple-beam balance for the mass. Do not forget the estimated digit on your micrometer and triple-beam measurements. 4. Calculate the density of the cylinder and use this to identify the material of which it is made using the list provided. 2

3 Data Sheet l (cm) w (cm) A (cm 2 ) A (m 2 ) h (cm) d (cm) m (g) ρ (g/cm 3 ) Material Densities (g/cm 3 ): Maple 0.77 Acrylic 1.17 Teflon 2.20 Steel 7.85 Polypropylene 0.90 Polyurethane 1.23 Aluminum 2.71 Brass 8.52 Polystyrene 1.03 Phenolic 1.32 Zinc 7.14 Copper 8.96 Nylon 1.15 PVC 1.37 Tin 7.27 Lead

4 Analysis 1. You calculated the surface area of the sheet of paper using two different units. Is the number of significant digits in these values the same? Do you think this will always be the case; i.e., does the number of significant digits change when you change SI units? Why or why not? 2. Give an example of a situation where an instrument with a smaller graduation could give less significant digits than an instrument with a larger graduation. 4

5 Pre-Lab: Measurement Name Section Answer the questions at the bottom of this sheet, below the line (only) - continue on the back if you need more room. Any calculations should be shown in full. 1. Read the lab thoroughly; check the lab manual for any additional information. 2. What is the least count of a measuring instrument? 3. What is the index on a vernier caliper? 4. You have l = 12.65cm and w = 8.30cm; what is A? 5. You have h = 5.210cm, d = cm, and m = 32.05g; what is ρ? 6. What is the reading on the following micrometer? 5