• Yazar: Russell McCormmach
• Google books
• Copying the relevent parts:
• page:280
• From a comparison of the mutual attraction of the lead balls to the attraction between the balls and the earth, Cavendish calculated the weight and mean density of the Earth.
• His value, 5.48 times the density of water, is within 1% of the accepted value today, 5.52.
• From the small variation in the computed density from one experimental run to another, Cavendish estimated that the true density does not differ from the mean by as much as 1/14th part of the whole.
• He thought that the narrowrange of values for the density showed that the average density of the Earth was "determined hereby, to great exactness."
• After his death, it was noticed that in averaging over the runs of his experiment, he had made an arithmetic error; the corrected average, 5.45, is within 1.3% of today's value, 5.52 (Fig. 6.8).
• 6.11.2 Theory of the Experiment
• Except to persons who are knowledgeable in physics, the reasoning behind the experiment on the density of the earth is far from obvious.
• The Experiment, like its forerunner, the Royal Society's experiment on the deflection of a plumb bob by a mountain, does not look like a weighing of anything, and Cavendish's brief description of the theory of the experiment is not transparent to a modern reader.
• To go through his calculations here would take up a good deal ou space and woul likely lose the reader in details, and I will instead give the broad outline of his reasoning in the footnote below. (269)
• (begin footnote 269) Cavendish deduces the density of the Earth in two steps. The first step assumes the laws of pendular motion.
• The second step assumes the inverse square law of gravitation.
• Step 1. Cavendish here draws on two laws: the period of vibration of a pendulum is proportional to the square root of the length of the pendulum, and it is inversely proportional to the square root of the restoringforce on the pendulum.
• With the aid of an analogy between the horizontal torsion pendulum and an imagined vertical simple pendulum beating seconds, the length of which is known, Cavendish expresses the force required to move the small balls at the ends of the torsion arm, with its observed period of vibration, through any observed angle of deflection of the arm in termi of the weight of a ball.
• Step 2. Cavendish here invokes Newton's law of gravitation twice, once to express theattarcation between a small ball and the nearby larger ball, or "weight", and once to express the attraction between the small ball and the Earth.
• The latter attraction is written so as to include the to-be-determined average density of the Earth.
• Forming a ratio of the two attractions, he expresses the attraction of the "weight" on the ball in terms of attraction of the Earth on the same ball.
• Finally, he combines Step 1 and 2.
• The force of the twisted wire from Step 1 is equal to the forceof attraction between the small balls and the "weights" from Step 2.
• By dividing one force by the other, Cavendish arrives at the desired result: the density of the Earth, expressed in termi of the density of water, is equal to a numerical factor times the square of the period of vibration of the torsion arm divided by the deflection of the arm.
• By this reasoning, Cavendish brings the world into his laboratory. [end footnote 269]
• We can achieve clarity by formulating the teory in modern notation and setting out a few equations.
• The experiment is a comparison of two forces, one the vertical gravitational pull of the Earth on a mass \(m\), which is its weight, \(w\), and the other a horizontal pull on mass \(m\) by a force \(f\).
• The average density of the Earth \(D\) enters the experiment through the vertical pull of the Earth, \(w\).
• By Newton's law of gravitation, which is expressed mathematically in the earlier discussion of Michell's work on double stars, the pull of the Earth on \(m\) is \(w = GmM/r^2\), where again, \(M\) is the mass of the Earth, \(r\) is the radius of the Earth, and \(G\) is the gravitational constant.
• Because the object of the experiment is to determine the density of the Earth, not its mass–the experiment is not, strickly speaking, directed at "weighing the world"– we rewrite the equation, replacing the mass of the Earth, \(M\), with the product of the volume of the Earth and its average density:

\begin{align*} w = \frac{Gm(4/3 \times \pi r^3 D)}{r^2} = G m (4/3 \; \pi r D) \end{align*}

$

• In the experiment, the mass \(m\) is a small metal ball, and the horizontal force, which is also a gravitational force, acts between this mass ant the mass of a large metal ball, the "weight", of mass \(\mu\)
• The horizontal force between a ball and a "weight", therefore, is \(f = G m \mu / d^2\), where \(d\) is the distance between the centers of the ball and "weight" after the arm is deflected; this force is balanced by the restoring force of the twisted wire.
• Because there are two pairs of balls and "weights", one at each end of the arm, the total deflecting force is \(2 f\).
• By combining the equations for the two forces– dividing the first by the second– the gravitational constant, \(G\), and \(m\) are eliminated, yielding an equation for the average density of the Earth:

\begin{align*} D = \frac{3\mu}{4\pi r d^2} \; \frac{w}{2f} \end{align*}

• [ If you are going to eliminate why are you writing those decorative terms? Since they are eliminated they must be decorative terms. ]