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Explanation of the Cavendish Experiment to Measure Gravitational Constant by Ron Kurtus - Succeed in Understanding Physics. **Key words:** physical science, Isaac Newton, Henry Cavendish, Universal Gravitation Equation, torsion balance, torque, oscillation period, moment of inertia, School for Champions. Copyright © Restrictions

# Cavendish Experiment to Measure Gravitational Constant

by Ron Kurtus (5 August 2010)

A major element in the *Universal Gravitation Equation*, **F = GMm/R ^{2}**, is the

*Universal Gravitational Constant*,

**G**. The constant was not determined until many years after Isaac Newton formulated his equation, as a result of what is called the Cavendish experiment.

This experiment used a torsion balance device to attract lead balls together, measuring the torque on a wire and equating it to the gravitational force between the balls. Then by a complex derivation, the value of **G** was determined.

Questions you may have include:

- What is the background of the Cavendish experiment?
- What is the experimental setup?
- What is the derivation of
**G**?

This lesson will answer those questions. Useful tool: Units Conversion

## Background of experiment

After Isaac Newton formulated the *Universal Gravitation Equation* in 1687, there really wasn't much interest in **G**. Most scientists simply considered it a proportionality constant. They were more interested in gravity than in gravitation.

In 1798, Henry Cavendish performed an experiment to determine the density of the Earth, which would be useful in astronomical measurements. He used a torsion balance invented by geologist John Mitchell to accurately measure the force of attraction between two masses. From this measurement, he determined the mass of the Earth and then its density. In Cavendish's published paper on the experiment, he gave the value for the density and mass of the Earth but never mentioned the value for **G**.

It wasn't until 1873 that other scientists repeated the experiment and documented the value for **G**. The value for **G** implied from Cavendish's experiment was very accurate and within 1% of present-day measurements.

Because his experiment ultimately determined the value for **G**, Cavendish has been often incorrectly given credit for determining the gravitational constant.

## Cavendish experiment setup

The Cavendish experiment uses a torsion balance to measure the weak gravitational force between lead balls. A torsion balance consists of a bar suspended from its middle by a thin wire or fiber. Twisting the fiber requires a torque that is a function of the fiber width and material.

When the gravitational force pulls the balls on the bar toward the stationary balls, the bar turns until the torque of the fiber balances the gravitational force. The magnitude of the force can then be calculated from the angle through which the bar turned. This angle is accurately determined by a mirror placed on the fiber.

However, the inertia of the balls causes them to go slightly beyond the balance point and thus create a harmonic oscillation around that point. This can also be measured by the light reflected from the mirror. The rate of oscillation is used to determine the spring constant of the wire, which is necessary in the final calculation of **G**.

It is truly a clever experiment.

Cavendish experiment uses torque to measure gravitation

The most recent measurement used a device called an atomic interferometer to measure **G**.

## Derivation of G

The derivation of the equation for **G** from the experiment is fairly complex. Variables measured in the experiment are:

**M**is the mass of the larger ball in kg**m**is the mass of the smaller ball in kg**R**is initial separation between the balls in meters**L**is the length of the balance bar in meters**θ**(small Greek letter omega) is the angle from the rest position to the equilibrium point measured in radians**T**is the oscillation period in seconds

The Universal Gravitation Equation for the balls is:

F = GMm/R^{2}

where

**F**is the force of attraction between the balls in newtons (N)**G**is the Universal Gravitational Constant in in N-m^{2}/kg^{2}or m^{3}/kg-s^{2}

Solving for **G**:

G = FR^{2}/Mm

In order to get a value for **G**, the force must be determined.

### Force related to torque

The force **F** is related to the torque on the fiber. The equation for torque is the applied force times the moment arm. Since there are two moment arms of **L/2**, the torque is:

τ = FL

where **τ** (small Greek letter tau) is the torque in N-m. Thus:

F = τ/L

### Torque related to torsion coefficient

However, the torque is also related to the torsion coefficient of the fiber or wire:

τ = κθ

where **κ** (small Greek letter kappa) is the torsion coefficient in newton-meters/radian. Thus:

F = κθ/L

### Torsion coefficient related to oscillation period

The unknown factor is the torsion coefficient, which is calculated by measuring the resonant oscillation period of the wire.

When the balance bar is initially released and the moving balls approach the larger balls, the inertia of the smaller causes them to overshoot the equilibrium angle. This results in the torsion balance oscillating back-and-forth at its natural resonant oscillation period:

T = 2π√(I/κ)

where

**T**is the oscillation period in seconds**π**(small Greek letter pi) is 3.14...**I**is the moment of inertia of the smaller balls in kg-m^{2}

Note: the mass of the bar is considered negligible and not a factor in the inertia.

### Oscillation period related to moment of inertia

The moment of inertia of the smaller balls is:

I = mL^{2}/2

Substitute inertia in the torque equation:

T = 2π√(mL^{2}/2κ)

Solve for **κ**:

T^{2}= 4π^{2}(mL^{2}/2κ)

2κT^{2}= 4π^{2}mL^{2}

κ = 2π^{2}mL^{2}/T^{2}

Substitute **κ** in the equation for **F**:

F = κθ/L

Thus:

F = 2π^{2}mL^{2}θ/LT^{2}

^{}Find G

Substitute for **F** in the equation for **G**:

G = FR^{2}/Mm

G = 2π^{2}mL^{2}θR^{2}/LT^{2}Mm

Simplify the equation:

G = 2π^{2}LθR^{2}/T^{2}M

The calculated value of **G** from this experiment is:

G= 6.674*10^{−11 }m^{3}/kg-s^{2}

Since a newton is equivalent to kg-m/s^{2}, **G** also is defined as:

G= 6.674*10^{−11}N-m^{2}/kg^{2}

## Summary

Henry Cavendish performed an experiment to find the density of the Earth. Other scientists used his experimental setup to determine the value of **G**. The setup consisted of a torsion balance to attract lead balls together, measuring the torque on a wire and then equating it to the gravitational force between the balls. Then by a complex derivation, the value of **G** was determined.

Seek to find out the reasons for things

## Resources and references

### Websites

**Cavendish Experiment** - Harvard University Natural Science Lecture Demonstrations

**The Cavendish Experiment** - Good illustrations of experiment from Leyden Science

**Cavendish experiment** - Wikipedia

### Books

**Top-rated books on Gravitation**

## Questions and comments

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