The Cavendish Experiment

Background

Isaac Newton's (1642 - 1727) theory of gravitation explained the motion of terrestrial objects and celestial bodies by positing a mutual attraction between all pairs of massive objects proportional to the product of the two masses and inversely proportional to the square of the distance between them. In modern notation, the law of universal gravitation is expressed

        (1)

where M and m are the masses of the two objects, r the distance separating them, and G is the universal constant of gravitation. Newton was not particularly concerned to evaluate the constant of proportionality, G, for two reasons. First, a consistent unit of mass was not in widespread use at the time. Second, he judged that since the gravitational attraction was so weak between any pair of objects whose mass he could sensibly measure, being so overwhelmed by the attraction each feels toward the center of the earth, any measurement of G was impractical.

Notwithstanding Newton's pessimism, towards the latter half of the 18th century several scientists attempted to weigh the earth by observing the gravitational force on a test mass from a nearby large mountain. These efforts were hampered, however, by very imperfect knowledge of the composition and average density of the rock composing the mountain. Spurred by his interest in the structure and composition of the interior of the earth, Henry Cavendish in a 1783 letter to his friend Rev. John Michell discussed the possibility of devising an experiment to "weigh the earth." Borrowing an idea from the French scientist Coulomb who had investigated the electrical force between small charged metal spheres, Michell suggested using a torsion balance to detect the tiny gravitational attraction between metal spheres and set about constructing an appropriate apparatus. He died in 1793, however, before conducting experiments with the apparatus.

The apparatus eventually made its way to Cavendish's home/laboratory, where he rebuilt most of it. His balance was constructed from a 6-foot wooden rod suspended by a metal fiber, with 2-inch-diameter lead spheres mounted on each end of the rod. These were attracted to 350-pound lead spheres brought close to the enclosure housing the rod, roughly as illustrated in the figure below. He began his experiments to "weigh the world" in 1797 at the age of 67, and published his result in 1798 that the average density of the earth is 5.48 times that of water. His work was done with such care that this value was not improved upon for over a century. The modern value for the mean density of the earth is 5.52 times the density of water. Cavendish's extraordinary attention to detail and to the quantification of the errors in this experiment has lead C. W. F. Everitt to describe this experiment as the first modern physics experiment. In this experiment you will use a torsional balance similar to Cavendish's to "weigh the earth" by determining a value for G.

Theory

The Cavendish torsional balance is illustrated in the figure at the right. Two small metal balls of mass m are attached to opposite ends of a light, rigid, horizontal rod which is suspended from a torsion fiber. When the "dumbbell" formed by the rod and masses is twisted away from its equilibrium position (angle), the fiber generates a restoring twist (torque) proportional to the angle of twist,

In the absence of damping, the dumbbell executes an oscillatory motion whose period is given by

where I is the rotational inertia of the dumbbell,

In this expression, r is the radius of the small masses m, and d is the distance from the center of the rod to the center of one of the masses, and we have neglected the mass of the thin rod. Knowledge of m, d, and r, and a careful measurement of the period of oscillation T allows one to calibrate the torsion fiber, obtaining its spring constant k. From k and a measurement of the twist caused by the large masses M you can deduce the gravitational force between the masses, and hence G.

Gravitational Torque

When the large metal spheres are positioned as shown in the figure, the gravitational attraction between the large and small spheres produces a torque (twist) that rotates the dumbbell clockwise. Only the component of the force on each mass that is perpendicular to the horizontal bar produces a torque about the center of the rod. The magnitude of the torque is given by

where the factor of 2 comes from the fact that the torque is equal on the two masses m. This torque displaces the equilibrium angle of the dumbbell by an amount given by

Hence, if one can measure the angle q very carefully, one can deduce the gravitational force that produces the torque and finally G.

Light Lever

Cavendish mounted a finely ruled scale near the end of the dumbbell, which he could read to one-hundredth of an inch resolution using a telescope. The telescope allowed him to remain outside the experimental chamber, thus eliminating air currents and his gravitational influence on the oscillator.

We will take advantage of a light lever to magnify the dumbbell's tiny rotation into an easily observed displacement on a far screen. The light lever is produced by bouncing a HeNe laser beam off a mirror mounted to the dumbbell. When light bounces off a mirror, the angle the incoming beam makes with the normal (perpendicular) of the mirror is equal to the angle the outgoing beam makes with the normal. If the mirror rotates through a small angle a, the outgoing beam rotates through twice the angle a, since both the incoming and outgoing angles change by the same amount a. By measuring the motion of the laser spot on a far screen, and knowing the distance between the mirror and the screen, you can determine the angle a, from which you can infer the rotation of the dumbbell, q.

Damping

In the absence of damping, the motion of the dumbbell is a sinusoidal oscillation with the period given by

        (2)

        (3)

where f_o is the initial phase of the motion, q_o is the equilibrium position, and A is the amplitude of the motion. By measuring q(t) and fitting your data to this equation, you can determine both T and q_o, from which you can determine G. For guidance in performing the fit, see the Analysis page.

Parameters

According to Pasco, the parameters of the apparatus include the following.

Safety

Danger! The laser pointer that forms the light lever for this experiment is a class III laser capable of damaging retinas. Do not look directly into the beam. Please ensure that nobody looks into the beam. Note that it is safe to look at the diffuse spot the beam produces as it reflects from an object, such as a meter stick.

Procedure

1. See the instructions next to the apparatus for proper alignment of the oscillator. You should be sure that when the freely suspended dumbbell remains at rest, it is equidistant from the front and back faces of the enclosure. This alignment should be done with the large lead balls removed.
2. Carefully weigh the large lead balls. Place one of the styrofoam trays on the electronic scale and tare it. Then gently place the lead ball into the tray. If the lead balls are dropped, they will become misshapen, which will severely compromise the accuracy of the experiment.
3. Lock the dumbbell using the screws on the bottom of the enclosure, then gently place the large lead balls in the armature. Rotate the armature until the balls just touch the sides of the enclosure. Gently lower the support screws until the dumbbell rotates freely. If it rotates so much that the dumbbell bounces off the sides of the enclosure, use the support screws to settle the motion.
4. Once the dumbbell oscillates freely, begin recording the position of the reflected laser spot as a function of time. Record at least two full periods.
5. Gently rotate the armature until the large balls once again touch the sides of the enclosure and record the position of the reflected laser spot as a function of time for at least another two periods.
6. From the distance between the mirror and the screen on which you measured the laser spot position, you can deduce the angle of rotation of the dumbbell. As described above, fit a damped sinusoid to each set of data and extract the two centers of rotation. From their difference and the data given in the parameters table above, you can deduce the value of G.

Updated 10/22/99 by Peter N. Saeta .