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PHYSICS 3900F/G the CAVENDISH EXPERIMENT 1. Introduction 2

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PHYSICS 3900F/G the CAVENDISH EXPERIMENT 1. Introduction 2 PHYSICS 3900F/G THE CAVENDISH EXPERIMENT 1. Introduction The gravitational constant G is involved in the calculation of the gravitational force between two bodies of masses m1 and m2 according to Newton’s law of universal gravitation, Gm m 1 2 , (1) F = 2 r where r is the distance between the bodies. Despite the importance of G in, for example, orbital mechanics and the evolution of stars and galaxies, it is the least well-determined of any of the fundamental physical constants, with a relative uncertainty of 0.012% (compare this with the value of Planck’s constant, which is known to an accuracy of 2.3 x 10–6). The value of G was first measured by Henry Cavendish in 1798 by measuring the force between relatively small forces in the laboratory using a torsion balance. So well constructed was Cavendish’s apparatus, that the value of G derived from his results differs by only 1% from the currently accepted value. In this experiment, you will use a scaled-down version of the Cavendish apparatus to repeat the experiment for yourself. 2. Apparatus The Cavendish apparatus consists of a light horizontal rod suspended by a thin metal ribbon. Attached to each end of the rod is a lead ball of mass m, as shown in Fig. 1(a). If the rod is twisted from its equilibrium position, a torque due to the ribbon acts to return the rod to equilibrium. In order to minimize disturbances due to air currents, this torsional pendulum is enclosed in a box. Two larger lead balls, each of mass M are positioned outside of the box at the same height is the smaller masses. When these are brought close to the small masses (the minimum distance, denoted “Position A,” is determined by the box enclosing the pendulum), the gravitational force between the large and small masses exerts a torque on the pendulum, shifting the equilibrium angle, as shown in Fig. 1(b). If the large masses are then moved to the opposite sides of the pendulum box (Position B), the equilibrium angle will shift. Phyiscs 3900F/G The Cavendish Experiment 1/1 (a) (b) Fig. 1 (a) Side view of the Cavendish experiment, in which two lead balls of mass m and radius r are attached to the ends of a light rod suspended by a metal ribbon of torsional spring constant !. The centres of the masses are a distance d from the rotational axis of the torsional pendulum. (b) Top view, showing two lead balls of mass M positioned a perpendicular distance b from the equilibrium position of the torsional pendulum. The balls are shown in Position A, with Position B indicated as a dashed outline. Because the gravitational force between the lead balls is small, the changes in angular position are also small. To measure the angular position of the pendulum, a laser beam is reflected from a mirror attached to the ribbon onto a scale mounted on a distant wall. By measuring the position of the laser spot on the wall (and applying some trigonometry ...), the angular motion of the pendulum can be calculated. Table I. Experimental One more quantity is needed to determine the parameters in Fig. 1. gravitational force: the torsional spring constant ! of the ribbon. This is can be accurately determined by m 38.3 ± 0.2 g measuring the period of the angular oscillations of r 9.53 mm the pendulum and calculating its rotational moment d 50.0 mm of inertia. M 1500 ± 10 g b 46.5 mm Note: The Cavendish experiment is extremely sensitive to vibrations, and can take as long as 3 h to settle down after being jostled. Be careful not to disturb the set-up unnecessarily. Phyiscs 3900F/G The Cavendish Experiment 2/2 3. Pre-Laboratory Preparation Your understanding of the experimental procedure your efficiency in analyzing your data will be enhanced if you complete the following exercises before beginning the laboratory. (a) Using the data provided in Table I, calculate the moment of inertia of the torsional pendulum. You may assume that the horizontal rod has negligible mass, and that the vertical ribbon is thin enough that it does not contribute to the moment of inertia). (b) Derive an expression for the period of the torsional pendulum in terms of the torsional spring constant of the ribbon and the moment of inertia of the suspended bob. (c) Derive an expression for the torque exerted on the pendulum by the two large masses (i.e., due to gravitational attraction) when the large masses are a perpendicular distance b from the horizontal rod. You may assume that the deviation of the pendulum from its equilibrium position is negligible. (d) Derive an expression for the change in the equilibrium position of the pendulum when the large masses are moved from Position A to Position B. Include these calculations in your lab report. 4. Experimental Procedure The most obvious way to use the Cavendish apparatus is to measure the initial pendulum angle with the large masses in Position A, and the final position long after the large masses are moved to position B. This method has the disadvantage that it will take a long time for the new equilibrium to be reached. Instead, we will measure the angular position of the pendulum as a function of time both before and after the position of the masses is changed. The equilibrium positions and period of motion can then be determined by numerical fits to the data. (a) Begin by aligning the diode laser provided so that its beam reflects from the mirror on the pendulum onto the wall. Attach a distance scale to the wall, making sure that it covers the entire range of motion of the reflected spot. Note: Be sure not to touch the Cavendish apparatus itself during this process. Warning: Laser is dangerous to eyes. Do not look directly into the beam, and ensure that the laser is never pointed in a direction where it can be a hazard to others! Phyiscs 3900F/G The Cavendish Experiment 3/3 (b) Record the position of the reflected laser spot at regular intervals (of perhaps 15 s or 20 s) for several minutes. The laser spot should either be at a constant position, or — more likely — be oscillating about an equilibrium position. If the latter, be sure to measure the position through several cycles so the equilibrium position can be accurately determined. Note that the period of motion should several minutes. It would be wise to plot the data as you obtain it. If the resulting curve does not “look” sinusoidal, it is likely that the pendulum is oscillating in a combination of modes (e.g., swinging as well as twisting) and/or striking the limits of its motion (rather than coming to a gradual stop and turning around). (c) Carefully swivel the large masses to Position B. They should be touching the enclosing box, but be careful not to disturb the system. (d) Again record the position of the reflected laser spot at regular intervals through several cycles of motion. Determine the equilibrium positions of the laser beam for Positions A and B, and hence the change in equilibrium angle of the torsional pendulum, from your plots of the position of the laser spot. Determine the period of oscillation of the pendulum, and hence its torsional spring constant, from your plots of position vs. time. Use your results to determine the value of the gravitational constant. Keywords: Newton’s law of gravity gravitational constant Cavendish apparatus torsional pendulum moment of inertia Phyiscs 3900F/G The Cavendish Experiment 4/4 .
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