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AST 101 LAB: CALCULATION OF THE EARTH’S MASS AST 101 LAB: CALCULATION OF THE EARTH’S MASS INSTRUCTIONS AND WORKSHEET Names__________________________________________________________________ __________________________________________________________________ ______________________________________________________ Section _____ Objective This lab teaches you how you can find the mass of the Earth with reasonable accuracy through its effect on a pendulum. It also teaches you a bit about numerical precision. Overview How can we find the mass of the Earth? A giant scale? But what would it rest on? A much better way (pun intended) is to measure the effect of the Earth’s mass on other objects. The Earth’s gravity pulls everything near it, including the Sun (there is a small wobble of the Sun caused by the Earth), the Moon, yourself, and other objects such as pendulums. We are going to use a pendulum for this experiment because this experiment has three goals: 1) measure the Earth’s mass as accurately as possible, 2) show how scientists achieve increased precision, and 3) use materials commonly available. The period of a pendulum’s swing is the time it takes the pendulum to swing from a certain point all the way back to that same point, moving in the same direction it started out from. To measure this set your timer to 0, start timing when you release the pendulum from a given point, and stop timing when the pendulum returns to the same point. To make the pendulum you will get about 3 feet of thread or lightweight string, support it somewhere where it can hang freely (without hitting anything or being affected by air currents from windows, radiators, etc), and attach some kind of weight at the bottom of the string. You might use a plumbing washer, plumb bob, marble, or similar “roundish” object – the rounder the better. Carefully measure the distance from the balance point (normally the center) of the round object to the top of your support. You can find the balance point by finding the point under which you can balance the entire object (when turned on its side). For example, in the not-quite- round object at the right, the balance point is about at the position shown. If your yardstick can measure in centimeters, enter the value here to 3 place precision (e.g. 96.2 cm) L = string length in centimeters: _______________________ Otherwise if your yardstick only measures in inches, put the measurement here (to 3 place precision, e.g. 35.3) and then convert it to centimeters: string length in inches ________________ × 2.54 = _________________cm Question 1: What is the length of your string, in centimeters? _________________ Question 2: Do you think your measurement of L was as accurate as it could have been? List 3 things that could have improved the accuracy: _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ Now let’s time our pendulum. Hold it at the bottom, move it out a bit to the side, and time one swing. Note you want to keep your swings small – the formula I’ll be using below is only correct for small swings. Try to estimate to at least one digit (e.g. 2 seconds) if you are using a watch, or three digits if you are using a stopwatch (e.g. 2.12 seconds) T = period = ___________________________ secs Question 3: Do you think your measurement of T was as accurate as it could have been? List 3 things that could have improved the accuracy: _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ The next part is a bit mathematical – if you are interested in how I combine three formulas to get this result, see Appendix A. The result of combining the three formulas is: G r T Lme 2 2 24π = MAIN FORMULA Thus now if you know the period T (which you’ve measured), the length L (which you’ve measured), the distance to the center of the Earth (the Earth’s radius) r, and the universal gravitational constant G, you can find me, the mass of the Earth. You should now get your first estimate for the mass of the Earth. You know L and T for the above equation, but don’t know r and G. Question 4: Look up the diameter of the Earth in Universe and divide it by 2 to get the radius of the Earth (hint: each chapter describing a planet begins with a table which gives the main characteristics of the planet) Diameter in kilometers _______________________________________ Radius r in kilometers = diameter/2 = ____________________________ You will also need Newton’s gravitational constant G = 6.6742×10-11 m3kg-1s-2 (see http://en.wikipedia.org/wiki/Gravitational_constant). This is a more accurate value than appears in your text. Lastly, let’s check if all the units work out – you should end up with kg: kg m kmcm kgm kmcm skgm km s cmconst G r T Lme 3 2 13 2 213 2 2 2 2 2 ))((4 ⋅ = ⋅ === −−− π As you see, we don’t. There is an important lesson here – all variables in a formula should be measured in consistent units – if you use meters in the definition of G, then all your other values should also be in meters. Mentally replace the cm and the km in the above formula with m – you’ll see that then me will be measured in kg. Question 5: Rewriting our value of r gives: r = ____________________________ meters (multiply prior value by 1000) Question 6: And rewriting L gives: L = ____________________________ meters (divide prior value by 100) If you are calculating this using a TI-83 or similar calculator, o to raise a number to a power, you use the ^ key (right above the ÷key). If you’ve never done this, try it now: enter 4 ^ 2 and press Enter – the answer displayed should be 16. o to enter a number in scientific notation with a negative exponent, e.g. 6×10-2 , you use the shifted EE key (on comma key, just above the number 7). Try it now: enter 6, press the orange 2nd key to shift to orange keys, press the EE key, then the negative key (-) at the bottom of the keypad (next to the ENTER key), then press 2 and Enter. The calculator should show your result as .06. http://en.wikipedia.org/wiki/Gravitational_constant If you are using Excel, you can also use the ^ key to denote a power, and E for an exponent. For example, you would enter 6.6742×10-11 into Excel as 6.6742E-11. Question 7: Calculate the mass of the Earth me using the corrected units = __________________kg Improved Estimate You noted several possible sources of inaccuracy in taking your measurements. Let’s try to do better. First take L, the string length. Might it have changed (because of stretching out or loosening) since you first fastened it? Measure it again without looking at your previous length. Write your answer in meters (i.e. divide cm by 100): 2nd length measurement = __________________________________ m (If you measure in inches, convert to meters by multiplying by .0254). Now ask someone else to measure it, and write the length down (if no one else is available, remeasure this yourself): 3rd length measurement = ___________________________________ m Now take the average of all your measurements. Important: Say you took your measurements to 3 significant digits. Then your average is still only good to 3 significant digits (even though your calculator might give you 14 places – the calculator does not know that you are dealing with measured, approximate numbers, and this leads to errors). I’ll explain later how to correctly get additional significant digits and a more accurate answer. Question 8: What is the average length of the three pendulum measurements________ m Question 9: This tests your understanding of significant digits: If Mars takes 686.98 days to orbit the Sun, how long (in days) does it take Mars to go 1/3 of the way around the Sun? Be careful to use the right number of significant digits. _______________________________________________________________________ Now let’s go back to our lab and get a better timing for the swing. The first thing we can do is use a stopwatch, in case you’ve been using a watch (if you don’t have a stopwatch and are doing this on your own, let go of the string exactly when the seconds on your watch = 0). The 2nd thing we can do is to time a series of swings, rather than a single swing. Again, it is useful here to have one person work the timer, and the other work the pendulum. So what you’ll want to do is let go of the pendulum, count 10 complete swings, and then stop. The procedure is then to start the stopwatch with the first swing, measure the time it takes to do 10 swings, and then divide by 10 to get the average time per swing. By measuring 10 swings instead of 1, you are measuring for a much longer time. So if your watch measures in seconds, instead of say 2 seconds, you might measure 21 seconds, which then becomes an average of 2.1. So you’ve added one place of precision by taking 10 measurements. Question 10: Record your average here for 10 swings of the pendulum. Hint: don’t forget to divide the total time by 10. Use the