Derivation of G and The Earth's Mass
- For the definitions of terms, see the drawing below and the table at the end of this section.
The following is not the method Cavendish used, but shows how modern physicists would use his results. From Hooke's law, the torque on the torsion wire is proportional to the deflection angle of the balance. The torque is where is the torsion coefficient of the wire. However, the torque can also be written as a product of the attractive forces between the balls and the distance to the suspension wire. Since there are two pairs of balls, each experiencing force F at a distance L / 2 from the axis of the balance, the torque is LF. Equating the two formulas for torque gives the following:
For F, Newton's law of universal gravitation is used to express the attractive force between the large and small balls:
Substituting F into the first equation above gives
To find the torsion coefficient of the wire, Cavendish measured the natural resonant oscillation period T of the torsion balance:
Assuming the mass of the torsion beam itself is negligible, the moment of inertia of the balance is just due to the small balls:
- ,
and so:
Solving this for , substituting into (1), and rearranging for G, the result is:
Once G has been found, the attraction of an object at the Earth's surface to the Earth itself can be used to calculate the Earth's mass and density:
| Definition of terms | ||
| SYMBOL | UNITS | DEFINITION |
| Deflection of torsion balance beam from its rest position | ||
| Gravitational force between masses M and m | ||
| Gravitational constant | ||
| Mass of small lead ball | ||
| Mass of large lead ball | ||
| Distance between centers of large and small balls when balance is deflected | ||
| Length of torsion balance beam between centers of small balls | ||
| Torsion coefficient of suspending wire | ||
| Moment of inertia of torsion balance beam | ||
| Period of oscillation of torsion balance | ||
| Acceleration of gravity at the surface of the Earth | ||
| Mass of the Earth | ||
| Radius of the Earth | ||
| Density of the Earth | ||
Read more about this topic: Cavendish Experiment
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