Cavendish experiment

The Cavendish experiment or the torsion balance implicitly obtained in 1798 the first measurement of the universal gravitational constant G and, with this data, from Isaac Newton's law of universal gravitation and from the orbital characteristics of the bodies of the solar system, the first determination of the mass of the planets and of the Sun. It should be noted that Henry Cavendish did not calculate this directly constant (since he did not need it for his measurements; this was done much later, taking advantage of his experiences), since his objective was to determine the density of the Earth, or, more specifically, "weigh the Earth", which he managed to achieve with exceptional precision for his time. However, since the product of the universal constant and the mass of the Earth had been known since Newton's time, Henry Cavendish was able to give the first estimate of the value of G.

The gravitational constant does not appear in Cavendish's paper, and there is no indication that he envisioned this calculation as an experimental purpose. One of the earliest references to G appeared in 1873, 75 years after Cavendish's work.

History

An early version of the experiment was proposed by John Michell, who went so far as to build a torsion balance to measure the force of attraction between two masses. However, he died in 1793 unable to complete his experiment and the instrument he had built was inherited by Francis John Hyde Wollaston, who, in turn, gave it to Henry Cavendish.

Cavendish took an interest in Michell's idea and rebuilt the apparatus, performing several very careful experiments in order to determine the average density of the Earth. Reports of him appeared published in 1798 in the Royal Society journal Philosophical Transactions .

The Experiment

Drawing from the vertical section of the Cavendish torsion scale, including the enclosure on which it was located. Large spheres were suspended from a frame, so that they could be directed from the outside to small spheres through a pulley system. Figure 1 of Cavendish's writing.

Detail showing the arm of the torsion scale (m), large sphere (W attached), small area (x), and isolated enclosure (ABCDE).

The instrument built by Cavendish consisted of a torsion balance with a horizontal wooden arm six feet (1.8288 m) long, from whose ends hung two small lead spheres (x) of identical mass (0, 73kg). This rod hung suspended from a long wire. Near the test spheres, Henry Cavendish arranged two large lead spheres (W) of 158 kg each, whose gravitational action was to attract the masses of the balance, producing a small rotation on them. To prevent disturbances caused by drafts, Cavendish placed his balance in a windproof room and measured the smallest torsion of the balance using a small telescope.

The two large lead spheres were placed on alternate sides of the horizontal wooden arm of the balance. The mutual attraction on the small balls caused the arm to rotate, in turn twisting the supporting wire of 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 large and small lead spheres. By measuring the angle of twist of the rod, and knowing the twisting force (torque) of the wire for a given angle, Cavendish was able to determine the force of attraction between the two pairs of masses. Since the gravitational force exerted by the Earth on each small ball could be measured directly by weighing, the ratio of the two forces allowed the density of the Earth to be calculated, using Newton's law of universal gravitation.

Cavendish deduced that the density of the Earth was 5.448 ± 0.033 times that of water (due to a simple arithmetic error, detected in 1821 by Francis Baily, the erroneous value of 5.48 ± 0.038 appears in Cavendish's writing).

To determine the modulus of torsion of the wire (i.e., the torque exerted by the wire for a given angle of twist), Cavendish timed the period of oscillation of the balance rod as it rotated slowly clockwise and counterclockwise. counterclockwise against twisting the wire. The period was about 20 minutes. The modulus of torsion could be calculated from this data, knowing the mass and the dimensions of the balance. In reality, the rod was not at rest; Cavendish had to measure the angle of deflection of the rod while he was swinging.

The equipment Cavendish designed was extraordinarily sensitive for its time. The torque involved in turning the balance was very small, on the order of 1.74 × 10^-7 N, about 1/50,000,000 of the weight of the small balls, or about the weight of a large grain of sand.

To prevent drafts and temperature changes from interfering with the measurements, Cavendish placed the entire apparatus inside a 2-foot (0.6 m) thick, 10-foot (3 m) wooden box. tall, and 10 feet (3 m) wide, all in a locked shed on his farm. Through two holes in the walls of the hut, Cavendish used telescopes to observe the movement of the horizontal bar of the torsion balance. The movement of the rod was only 0.16 in (4.1 mm). Cavendish was able to measure this small deflection to the nearest one-hundredth of an inch using vernier scales on the ends of the rod. Accuracy The achievement of Cavendish was not surpassed until the experiments of Charles Vernon Boys in 1895. In time, Michell's torsion balance became the dominant technique for measuring the gravitational constant (G) and most contemporary measurements they continue to use variations of it.

Mathematical formulation

Diagram of the torsion scale used in the " Cavendish experiment", performed by Henry Cavendish himself in 1798:
The force of gravity was measured between the M and m masses, calculating the density of the Earth.
Tags: (Mmass of stationary lead balls, (mmass of mobile lead balls, (F) gravitational force between each pair of balls, (θ θ {displaystyle theta }) equilibrium angle regarding neutral position, (κ κ {displaystyle kappa }) wire torsion module, (r) distance between the centers of the balls when they are in balance, (L) distance between the wire and each of the small balls (length of the rod: 2L).

The objective of the experiment is to measure the twist in the torsion balance produced by the force of gravity exerted between the external spheres and the masses arranged at the ends of the balance, which allows deducing the value of all the forces involved:

The recovery force in the balance (Δ Δ {displaystyle tau }), can be written according to the angle rotated on the balance position, θ θ {displaystyle theta }and the wire torsion module k:

Δ Δ =− − kθ θ {displaystyle tau =-ktheta {frac {}{}{}}}

The angle θ θ {displaystyle theta } can be measured by a mirror located in the torsion fiber. Yeah. M represents the mass of the outer spheres and m the mass of the spheres in the balance of torsion, the force of torsion can be matched with the force of attraction exercised by the spheres through the formula:

Δ Δ =2GMmr2L{displaystyle tau =2{frac {GMm}{r^{2}}}L}

where G is the universal gravitational constant, L is the distance between the torsion thread and the spheres m (ie the torque arm); and finally r is the distance between the centers of the spheres M and m . Therefore, combining the previous equations, it turns out that:

G=kθ θ r22MmL(1){displaystyle G={frac {ktheta r^{2}{2MmL}}qquad qquad qquad (1),}

We have that k can be measured from the oscillation period of the torsion balance, T. Therefore, if:

T=2π π I/k{displaystyle T=2pi {sqrt {I/k}}}}

and assuming that the mass of the elements of the torsion balance is negligible, the moment of inertia of the balance is that of the two small balls:

I=mL2+mL2=2mL2{displaystyle I=mL^{2}+mL^{2}=2mL^{2},}

then:

T=2π π 2mL2k{displaystyle T=2pi {sqrt {frac {2mL^{2}{k}}}}{,}

Solving this equation for k{displaystyle k}Turns out:

k=4π π 2(2mL2)T2{displaystyle k={frac {4pi ^{2}(2mL^{2}{2}}{T^{2}}}}{,}

Substituting this expression in (1), and solving for G, the result is:

G=4π π 2r2Lθ θ MT2{displaystyle G={frac {4pi ^{2}r^{2}Ltheta }{MT^{2}}}{2}}}}}

Once determined Gthe force of attraction on an object on the surface of the Earth (mg{displaystyle mg}), can be used to calculate the mass (MT{displaystyle M_{T}}) and Earth density (ρ ρ T{displaystyle rho _{T}}) known terrestrial radio (RT{displaystyle R_{T}}(c):

mg=GmMTRT2{displaystyle mg={frac {GmM_{T}}{R_{T}^{2}}}}{,}

Mass of the Earth:

MT=gRT2G{displaystyle M_{T}={frac {gR_{T}^{2}{G}}}}{,}

Density of the Earth:

ρ ρ T=MT(4/3)π π RT3=3g4π π RTG{displaystyle rho _{T}={frac {M_{T}}}{(4/3)pi R_{T^{3}}}}}{frac {3g}{4pi R_{T}G}}}}{,}

Common mistake

It is common to find books that erroneously state that Cavendish's purpose was to determine the gravitational constant, G, and this error has been pointed out by various authors. In reality, Cavendish's only purpose was to determine the density of the Earth. He called this "weighing the world." Cavendish's method used to calculate the density of the Earth was to measure the force on a small sphere due to a larger sphere of known mass and compare this to the force on the small sphere due to the Earth. In this way, the Earth could be described as N times more massive than the large sphere without the need to obtain a numerical value for G.

In Cavendish's time, G did not have the importance among scientists that it has since. This value was simply a constant of proportionality in Newton's law of universal gravitation. Instead, the purpose of measuring the force of gravity was to determine the Earth's density. This amount was required in 18th century century astronomy, since, once known, the densities of the Moon, the Sun and the rest of the planets could be found from it.

A further complication was that by the mid-19th century, physicists did not use a specific unit for force. This fact unnecessarily tied G to the mass of the Earth, instead of recognizing G as a universal constant. However, although Cavendish did not report a value for G , the results of his experiment allowed it to be determined. In the late 19th century scientists began to recognize G as a fundamental physical constant, calculating it from the Cavendish results. Therefore:

G=gRTierra2MTierra=3g4π π RTierraρ ρ Tierra{displaystyle G=g{mathrm {Earm} }^{2}{M_{mathrm {Earm}}}}}}}{frac {3g}{4pi R_{mathrm {Earm} }rho _{mathrm {Earm}},}

After converting to International System units the value obtained by Cavendish for the density of the Earth (5.45 g/cm³), as well as the rest of the data collected, the value G = 6.674×10^-11 N m²/kg², so the error is within 1% of the currently accepted value; very approximately 0.9889%.