# The Cavendish Experiment

To calculate the force of gravity between two objects you need to have the masses of the two objects, the distance between the two objects, and the gravitational constant.
This means to be able to prove the law of gravitation you need to be able to calculate the gravitational constant (G).

Remember that the strength of attraction between two small masses will be extremely small. Despite the weakness of the attraction, Henry Cavendish was able to perform an experiment to measure the force between two small objects which allowed him to measure the gravitational constant (G) .
For his experiment in 1798, Cavendish hung a dumbell from a fine string. He then placed two large lead weights below the dumbell, and was able to see a small twisting in the string. From this small twist in the string he was able to measure the force between the objects. After measuring the force, masses, and distance, the gravitational constant could be calculated. Below is a modern version of the Cavendish experiment.

## The Torsion Balance

A dumbbell consisting of two small lead spheres fastened to a thin rod is hung from a fine fiber. The dumbbell will twist until the torque from the fiber is equal to any external torque applied to the dumbbell. A small mirror is attached to the dumbbell to help determine the angle of the twist.

We shine a laser beam at the mirror, reflecting onto a screen. The location of the laser spot on the screen allows us to determine the twist of the dumbbell. By placing the screen at a reasonable distance away, we can detect even a small twist.

Two big lead spheres are placed near the small spheres on the dumbbell. The gravitational force between the large spheres and the nearby small spheres twists the dumbbell. The laser spot consequently shifts to one side. When the dumbbell reaches equilibrium, the torque from the fiber balances the torque from the gravitational forces on the dumbbell.

## A Simulation of the Experiment

First, we allow the balance to come to equilibrium with a clockwise torque as seen from a topview. Next, we move the big spheres to the opposite side to give an equal torque in the counterclockwise direction. The dumbbell then moves and after oscillating settles onto a new equilibrium. Click on the picture to see an animation.

## The Actual Balance

The actual balance is enclosed in a metal case with glass windows to protect the torsion balance from air currents while allowing the laser beam to bounce from the mirror. Click on the picture to see an animation.

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## The Various Parts of the Experiment

The various components of the experiment are: the torsion balance, the laser, and a ruler, which acts as the screen. In the last part of the movie, one can see the laser spot at about 34 cm. There is a counter below the ruler, which will measure the elapsed time in seconds in our actual run. Click on the picture to see an animation. .

## Start of Measurements

To start our measurement, we allow the dumbbell to settle at equilibrium 1, with the big spheres twisting the dumbbell clockwise. We then move the big spheres to the opposite positions, so that they will twist the dumbbell with an equal torque counterclockwise. Click on the picture to see an animation.

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## Time-lapse Measurement of Motion

After the big spheres are moved, the dumbbell leaves equilibrium 1, oscillates, and settles onto the new equilibrium 2. The movie shows the position of the laser spot on the ruler. In the lower part of the picture, we plot the spot's position against the elapsed time in seconds, which can be read from the counter. Click on the picture to see an animation.

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## The Observed Value of G

We determined G from the measurements of:

• the masses of the spheres: m and M,
• the separation of the spheres: r,
• the length of the moment arm of the torsion balance: d,
• the stiffness of the suspension fiber: D,
• the distance from the mirror to the screen: L,
• the separation of the two equilibrium positions: S,
• and a correction factor to account for the attractions between each small sphere and its more distant big sphere: beta.