Cavendish7

Henry Cavendish (1731-1810)

English eccentric; from a "noble family" (Dukes of Devonshire). Educated at Cambridge University but never took his degree. At age 40, he inherited a fortune of over 1 million pounds; spent almost none of it.

He spent 60 years doing scientific research in physics and chemistry. Experiments in electricity were half a century ahead of anyone else, but remained unknown until Maxwell published his papers (posthumously). Measured electric current by estimating how much pain it caused. Discovered hydrogen and showed it to be a component of water (which was therefore not an element!) Discovered argon as a component of air (which wouldn't react with added oxygen when an electrical spark was passed through the gas).

Cavendish determined the universal constant of gravitation G in 1798, using an apparatus with two small lead spheres (mass m, on an arm of length 2r) attracted by two large lead spheres (mass M, whose centre is a distance a from the centre of the small spheres). The angular deflection (theta) gives the gravitational force if the torsional constant (k) of the supporting fibre is known: torque = F.r = k (theta), where F = G m M/a^2 It is possible to determine the constant k from the period T of torsional oscillation: T= 2 (pi) / omega, where omega = (k/I)^1/2 and I = 2 mr^2 is the moment of inertia of the torsional beam

The value of G obtained by Cavendish was about 1% different from modern values, measured using similar methods but increasingly refined apparatus. From G, one can calculate the mass of the earth: M g = G M Me / re^2 gives Me = re^2(g/G), so the Cavendish experiment is sometimes known as "weighing the earth".

Is G absolutely constant with time? Some theories of cosmology link value G to the age of the universe, in which case G would be decreasing by about 1 part in 10^10 per year. To measure such a small effect, the Cavendish-type experiment is far too inaccurate. But small changes in G can be inferred from changes in radius of the planetary orbits, or the orbit of the moon about the earth. Many corrections have to be applied (e.g. tidal effect, for the moon); so far, the experimental evidence suggests that G changes by no more than 0.5 in 10^10 /year.

---

Gravitational and Inertial Mass

Is the mass m which occurs in Newton's 2nd law really the same as that which occurs in his equation of universal gravitation ?

In other words, is the inertial mass mi precisely equal to the gravitational mass mg in the equations: F = mi a and F = G mg Me / r^2 ?

If mg and mi differ by a factor which is always constant, the question merely concerns the definition of units. But if (mg/mi) varies between different materials, for example, the free fall of objects would occur at different rates: a = F/mi = (mg/mi) G Me/r^2

Easier to observe is the motion of a pendulum. For a simple pendulum and small angle of swing (theta): Restoring force = - mg g (theta) = - mg g (x/L) = - k x where k =mg g/L This is SHM with angular frequency (omega) = root(k/mi),
period T = 2(pi)/(omega) = 2(pi) (L/g)^1/2 (mi/mg)^1/2

Galileo constructed pendulums of equal length, but with bobs of lead and cork, and reported equal periods over several hundred swings.

Newton did a more careful experiment using identical pendulums with hollow bobs (equal air resistance) and containing different substances of equal weight (so that friction at the pivot would be the same) and observed equal periods T to within 1 in 1000.

Subsequent experiments, some involving torsional pendulums, have established that mi = mg to within 1 part in 10^12. Equality of these two masses (called the Principle of Equivalence) is central to Einstein's theory of gravitation: the General Theory of Relativity.