PC1141 Physics I Compound Pendulum 1 Purpose Determination of the acceleration due to gravity 2 Equipment Kater pendulum Photogate timer Vernier caliper Meter stick 3 Theory One of the most important physical constants is the acceleration due to gravity g. It denes the unit of force for mechanics and consequently underlies all mechanical measurements. The value of the acceleration due to gravity can be measured directly to a reasonable accuracy (about 1%), although a well-conducted experiment on a free-fall apparatus or on a good airtrack will usually not be quite accurate. To improve the accuracy, other methods must be used. In 1817, Kater, following a suggestion of Bessel, developed a reversible pendulum that made possible the accurate measurement of g. A reversible pendulum is a pendulum that can be swung from either of two pivot points. When the mass distribution of the pendulum is adjusted so that the periods are the same from either pivot, the period is the same as a simple pendulum having a length equal to the distance between the pivots. While this may seem to be a simple property, it has great power in determining the value of g to high accuracy. A generalized compound pendulum is shown in the Figure 1. The pivot of the pendulum is is at O and the center of mass at G. If this pendulum is moved from rest, the restoring torque is Mgh 1 sin θ. Upon release of the pendulum, the motion will be described by d 2 θ I O dt = Mgh 2 1 sin θ where I O is the moment of inertia of the pendulum about the pivoting axis through O. If θ is small, the period is T 1 = 2π IO Mgh 1 Page 1 of 5
Compound Pendulum Page 2 of 5 Figure 1: Generalized pendulum. The parallel axis theorem states that I O = I G + Mh 2 1 where I G is the moment of inertia about a parallel axis through the center of mass, so that IG T 1 = 2π + h 1 Mgh 1 g Inverting the pendulum by using S as pivot and following a similar argument, one nds the period T 2 = 2π IS Mgh 2 where I S = I O + Mh 2 2 is the moment of inertia about the pivoting axis through S. Therefore, IG T 2 = 2π + h 2 Mgh 2 g By comparing equations (1) and (3), it is clear that if h 1 and h 2 are chosen so that then h 1 h 2 = I G M T 1 = T 2 = 2π h1 + h 2 This period is the same as that of a simple pendulum of length h 1 + h 2 = OS which is the distance between the two pivots. g (1)
Compound Pendulum Page 3 of 5 The mechanical design of the pendulum is not important to the derivation of the theory; it is only necessary that the pendulum can be swung from the two pivot points. The actual form of Kater's pendulum consists of a long bar with two knife edges placed near the ends. The pivots are knife edges so that the distance between them could be measured with accuracy. Attached to the bar are two bobs (one is lighter than the other) that can be adjusted on the bar to nd the point of equal period. The end of the pendulum holding the large bob is called the `A' end and the pendulum should swing freely in a vertical plane with `A' up or `A' down. Another end of the bar is called `B' end. There is no single bob position for the equal period requirement to be met. Rather, one bob can be positioned to make the periods equal for a range of positions of the other bob. The resulting period of the pendulum will be the same, and will only depend on the distance between the knife edges and the value of g. The distance between the knife edges is L = 100.03 cm. Once the position of the bobs for equal periods is determined, the pendulum can be moved from place to place to survey the variation of the acceleration due to gravity with location. The greatest variation of g is due to the latitude of the location. This latitude variation is due to two causes. The rst is the dierence in the radius of the earth with latitude the earth being fatter at the equator. Since the gravitational force is inversely proportional to the square of the distance between the centers of mass, gravity is smaller at the equator. The second cause is the centrifugal force of the earth's rotation which is a maximum at the equator and zero at the poles. Both eects are in the same direction, so g increases with latitude, being smaller at the equator. At a latitude Φ, the value of g is given by g = 879.0490 cm/s 2 ( 1 + 0.0052884 sin 2 Φ 0.0000059 sin 2 2Φ ) The latitude of Singapore is 1.2971 degrees. Figure 2: Schematic diagram of the Kater pendulum.
Compound Pendulum Page 4 of 5 4 Experimental Procedure 1. Position the large bob at a distance 5 cm from one end of the bar. This end of the bar will be called `A' end. The distance is to be measured from the `A' end to the outer edge of the large bob. 2. Position the small bob at an initial distance 60 cm from the `B' end. The distance is to be measured from the `B' end to the outer edge of the small bob. 3. Tighten the screws to fasten the bobs in place. Gently place the bar on the platform with `A' up position. 4. Set the photogate timer to PENDULUM mode. Adjust the height of the photogate so the pendulum interrupts the photogate beam as it swings. 5. Set the pendulum swinging so that its amplitude is a few centimeters. Let the pendulum swing for about half a minute to settle down. 6. Press the RESET button on the photogate timer. Note the rst time displayed. This is the period for one-complete oscillation. Record the period of oscillation as T in the Data Table 1. 7. Without removing the pendulum from the platform, repeat the measurement of the period of oscillation when the small bob is moved in about 10 cm steps towards the `B' end. Do it for SIX times and record the values in the Data Table 1. Note: When the small bob is near to the knife edge, take the bob o and replace it outside of the knife edge. 8. Reverse the pendulum and repeat the measurements of the period of oscillation in the `A' down position with the same small bob positions as in the `A' up measurements. Record the values in the Data Table 1. 9. Position the small bob at a distance 10 cm from the `B' end and place the bar on the platform with `A' up position. 10. Use a photogate timer, measure the period of oscillation THREE times and record the values in the Data Table 2. Note: Remember to wait for about half a minute for the oscillation to settle down before taking any measurement. 11. Repeat steps 910 for EIGHT dierent positions of the small bob until it is about 15 cm from the `B' end. 12. Reverse the pendulum and repeat the measurements of oscillation in the `A' down position with the same small bob positions as in the `A' up measurements. Record the values in the Data Table 2. 13. Repeat steps 912 with the large bob at distances 15 cm from the `A' end. Record the values in the Data Table 3.
Compound Pendulum Page 5 of 5 5 Data Analysis D1. Enter your data in the Data Table 1 into the Excel spreadsheet. Plot a graph of the period versus the position of the small bob for `A' up and `A' down respectively in the spreadsheet. Notice that the intersection of these two curves should be within 515 cm. D2. Enter your data in the Data Table 2 into the Excel spreadsheet. Calculate the average period of the pendulum for each position of the small bob with `A' up and `A' down respectively in the spreadsheet. D3. Perform a linear least squares t to the data for `A' up and `A' down respectively, with the period of the pendulum T as the y-axis and the position of the small bob measured from the `B' end x as the x-axis. Determine the slope and intercept with the corresponding uncertainties of the least squares t to the data. D4. Based on your results from the least squares t in D3, determine the experimental value of acceleration due to gravity g with the corresponding uncertainty. Use percentage discrepancy to compare your experimental value of g with the theoretical value. Hint: The percentage discrepancy is dened as Percentage discrepancy = Experimental value Theoretical value Theoretical value 100% D5. Enter your data in the Data Table 3 into the Excel spreadsheet. Perform a similar analysis as in D2D4. D6. Use percentage dierence to compare your experimental values of g obtained. Hint: The percentage dierence is dened as Percentage dierence = g 1 g 2 (g 1 + g 2 )/2 100%