Gravitational Constant Brett Waite and Daniel Klinge

The universal gravitational constant was discovered by Henry Cavendish during the years of 1797-1798. His goal was to determine the density of the earth. The apparatus he used was a torsion balance which is good for measuring very weak forces. The balance was made of a six-foot wooden rod suspended from a wire, with a 2-inch diameter 1.61-pound lead sphere attached to each end. Two 12-inch, 348-pound lead balls were located near the smaller balls, about 9 inches away, and held in place with a separate suspension system. Cavendish used this to measure the gravitational force between the two masses to calculate the density of the Earth, which later led to a value for the gravitational constant. This was the first experiment to measure the force of gravity between masses in the laboratory. In order to measure the unknown force, the spring constant of the torsion fiber used in the balance must be known. Cavendish accomplished this by a method of measuring the resonant vibration period of the balance. If the free balance is twisted and released, it will oscillate slowly back and forth as a harmonic oscillator, at a frequency that depends on the moment of inertia of the beam and the elasticity of the fiber. Since the inertia of the beam can be found from its mass, the spring constant can be calculated. The attraction of the masses caused the arm of the balance to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the masses. By measuring the angle of the rod, and knowing the twisting force of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the Earth to be calculated, using Newton’s law of gravitation. Cavendish’s only objective was to measure the density of the Earth. He did not report a value for G in any of his work, but his experiment allowed it to be determined, in 1873. In the late 1800’s scientists began to recognize G as a fundamental constant of God’s creation. Using Cavendish’s results, a value for G was calculated to be

, which differs by less than 1% from the currently accepted value of .

The following is a method of how the gravitational constant can be determined using Cavendish’s results. From Hook’s law, the torque on the torsion wire is proportional to the deflection of the balance. The torque is where is the torsion coefficient. The torque can also be written as a product of the attractive forces and the

distance to the wire. Since both sets of spheres experience force F at distance from the

axis of the balance, the torque is LF. Equating the two formulas for torque gives the equation

Using Newton’s law of universal gravitation to express the attractive force, F, between the large and small balls

Substituting F into the previous equation gives

The torsion coefficient, , of the wire was determined by Cavendish by measuring the natural resonate oscillation period, T , of the balance

If the mass of the torsion beam is ignored, the moment of inertia of the balance is just due to the small balls

Where r is equal to

Substituting this value into the oscillation equation

Solving for

Plugging into the equation gives

Solving this for G gives the equation for the universal gravitational constant

Apparatus The piece of equipment we used to measure the gravitational constant is called a torsion balance.

The torsion balance has a support base (pictured above in black) with two 1.5 kg lead balls resting on a large swivel support. Inside the blow up box above on the bottom right is one of the two smaller 0.038 kg lead balls. Inside the long black stem is a torsion ribbon made of a mixture of beryllium and copper. The role of this ribbon provides the force by twisting the zero adjust knob which in turn adjusts the torsion ribbon head at the top of the long black stem, which puts the two small lead balls in equilibrium so that neither one is touching the case in which they are contained; thus giving us a way to measure the attractive force between the large and small lead balls. Refer to drawing below: The little round mirror in between the two lead balls about one inch up from the rod connecting the two balls (i.e. pendulum bob arm) is responsible for reflecting a laser beam onto the wall about six and a half meters away which provided a method for us to measure the period of the two smaller lead balls slightly swaying back and forth.

First, we setup the apparatus with the laser reflecting off the mirror onto the wall. We allowed the kinetic energy of the pendulum bob arm to go to zero (macroscopically) so that the laser beam on the wall was stationary. Then we used the zero adjust knob to turn the ribbon which put the pendulum bob in equilibrium. After equilibrium was reached, we rotated the two larger balls 180 degrees. This caused a change between the attractive force between the small and large masses such that the pendulum bob arm came to a new equilibrium. While the apparatus was setting its new equilibrium we measured the period of each oscillation by the laser moving back and forth between points on the wall. We measure the distance between each point in centimeters with the middle portion of a meter stick so as to decrease experimental error. We also measure the distance from the wall to the mirror on the torsion balance; knowing these value allow us to calculate the gravitational constant using the equations below. Results To calculate the gravitational constant we used several measurements as well as known constants. The equation we used was:

in which is the distance from its first equilibrium point to its second equilibrium point. The first equilibrium point is when the two larger lead balls are in its starting position and the laser is stationary on the wall. The second equilibrium position is when the two large masses have been rotated 180 degrees which causes the mirror to turn and the laser to come to rest at a new location on the wall. The constant, b, is the distance from the center of the small ball to the center of the large ball, when the large balls are up against the casing. The distance from the mirror to the wall is L. In the denominator of the fraction, little r is the radius of the small balls. Little d is the distance from the center of the small balls to the mirror. Capital T is the period of the mirror’s oscillations while the torsion balance is reaching equilibrium. In the denominator, m1 is the mass of the large ball. The values we used in this equation are as follows:

.

Using our numbers the value we calculated for the gravitational constant is:

.

The following equation accounts for the fact that the small sphere is attracted not only to the large sphere adjacent to it; it is attracted to the more distant large sphere, though with a much smaller force. is the corrected value that accounts for the systematic error. So, is equal to

. where is equal to the following:

.

Calculating using the equation above,

gives us a value of 0.07496.

Then for , the corrected value is . We calculated the percent error using the equation:

error.

Discussion

Getting the torsion balance to come to an equilibrium point was the most challenging part of the experiment. Turning the zero adjust knob just right so the pendulum bob arm is not touching a face plate took a lot of trial and error; in the end teaching us patience and perseverance.

One interesting point to bring to light is the fact that our periods of oscillation were very close together. We stopped timing the oscillations after four trials and they all differed by one second, which gave us hope we were doing the experiment correctly. In addition, we were relieved that everything remained intact at the completion of the experiment, thus doing our part to keep the physics department under its budget.

We were very surprised that our value of big G was within one order of magnitude of the actual value of the known gravitational constant. With a number as small as big G (notice the play on words), it makes it difficult to get within the same order of magnitude. As mentioned above, our Percent Error was 593%. Possible causes include the table not being steady, vibrations from people walking in the building above and next to it and we did not use the copper wire to ground it and dissipate the charge. We would like to thank Henry Cavendish and his ingenuity in calculating the gravitational constant with a one percent error.

Reference Page

1. W. R. Aykroyd. Three Philosophers. Westport: Greenwood Press, 1970. Print. 2. J. G. Crowther. Scientists of the industrial revolution. London: Cresset, 1962. Print.