Acceleration Due to Gravity You are probably familiar with the motion of a pendulum, swinging back and forth about some equilibrium position. A simple pendulum consists of a mass m suspended by a string with negligible mass. The physics involved here is a bit complex and you will explore this motion in some detail later on. However, one of the things you will discover is that the period T - the time for one complete oscillation - is dependent only on the length L of the pendulum (at relatively small amplitudes). In the figure above T would be the time from amplitude A, through equilibrium position B and over to amplitude C, then back to A. This relationship is given by T = 2π and you will use it to calculate a value for the acceleration due to gravity g since L and T are measurable. L g g = 4π2 L T 2 (1) Actually you will make several measurements of T, which will yield a range of times. Therefore there will be a range of values for your calculated values of g. One way to quantify the spread of these values about the mean is with the standard deviation σ. You expect that your measurements for T will fall both above and below the mean value hence your calculated values of g should do the same 1. With your calculated values of g, the equation for σ gives 1 i.e., you expect the values to be randomly scattered about the mean. 1
σ = (gi ḡ) 2 n (2) Thus you expect about 68% of your calculated values of g to fall in the range ḡ ± σ (95% within ḡ ± 2σ and 99% within ḡ ± 3σ). Apparatus Vertical support, Pendulum clamp, String, Hooked mass, Stopwatch, Meter stick. Procedure 1. The table clamp and vertical rod should be clamped to the side of the table. The pendulum clamp attaches to the vertical rod. 2. Cut a piece of sting about 120cm long and attach it to the middle and farthest clamps on the pendulum clamp. These are designed so that they hold the string in place when it is threaded between them and the pendulum clamp as shown below. 3. Loosen the clamps, thread the string, then re-tighten. What you should now have is a loop of string hanging below the pendulum clamp. Hook the mass at the bottom of the loop. This will ensure a more vertical plane for the motion of your pendulum. Adjust one of the strings as necessary so that the length of the pendulum (bottom of clamp to center of the hooked mass) is about 60cm. Measure this distance with the meter stick and record it on the Data Sheet. 4. Pull back the mass (perpendicular to the pendulum clamp) about 10-15cm and release it. Let the mass swing a few times to stabilize and then time 10 complete oscillations. Remember that 1 oscillation is from amplitude on on side, over, and then back. Record this time and divide by 10 to get the period. 5. Repeat Step 4 for a total of 10 trials. Once you release the mass you should be able to time 3 10-oscillation sets before the entire process will need to be repeated. 2
6. Calculate a value of g for each value of T, keeping one extra significant digit in the values (giving you 4 total). Calculate the mean of these values ḡ. 7. Calculate the standard deviation in your values of g - a Worksheet has been provided to assist you in your calculations 2. Start by transferring your values of g and ḡ then use these for the remainder of the calculations. 2 Most calculators have this ability; if you know how, check your answer; if, not, learn how! 3
Data Sheet L (cm) Time (s) T (s) g (cm/s 2 ) ḡ (cm/s 2 ) 4
Standard Deviation Worksheet ḡ (cm/s 2 ) i g i (cm/s 2 ) (g i ḡ) (cm/s 2 ) (g i ḡ) 2 (cm 2 /s 4 ) 1 2 3 4 5 6 7 8 9 10 (gi ḡ) 2 (cm 2 /s 4 ) σ 2 (gi ḡ) 2 = n (gi ḡ) 2 σ = n (cm 2 /s 4 ) (cm/s 2 ) 5
Analysis 1. The standard deviation applies to your entire data set; what would be nice is a value that applies to the mean. A value that does is known as the standard deviation of the mean, or simply the standard error or uncertainty, and is given by σ n In general, this is a bit beyond what we will do in this lab - but by coincidence here the uncertainty in the mean to 99% confidence is roughly the same as your calculated σ. Round your σ using the rule for percentages, then round your ḡ to the same number of decimal places. e.g., σ = 2.66(cm/s 2 ) and ḡ = 971.6(cm/s 2 ) yields (972 ± 3)(cm/s 2 ). Display the results below. 2. Interpret your answer in Question 1. 6
Pre-Lab: Acceleration Due to Gravity 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 a simple pendulum? 3. What is the period of a pendulum? 4. For a simple pendulum of length 62.7cm you time 10 oscillations as 15.37s. What is g? 5. What is the standard deviation in the values 83.2, 86.4, 79.5, 80.7, and 82.5? 7