Weighing the Earth: Measuring G

G is so small that two people standing a metre apart attract each other with well under a millionth of a newton. Henry Cavendish measured it in 1798 with little more than a stick, a wire and four lead balls.

The torsion balance#

Two small balls of mass m are fixed to the ends of a light rod of length l, which hangs horizontally from a very fine quartz fibre. Two large balls of mass M are placed beside the small ones, on opposite sides, all four centres on one horizontal circle. Each small ball feels a sideways pull

where r is the centre-to-centre distance (a uniform sphere acts like a point at its centre). The two pulls are perpendicular to the rod and turn it the same way, giving a torque $2F\cdot\tfrac l2 = Fl$. As the rod turns, the fibre twists and pushes back with a torque proportional to the twist, kθ. The rod settles where the two balance:

Making a tiny angle visible#

θ is far too small to read off a protractor. So a small mirror is fixed to the fibre and a light beam is bounced off it onto a distant scale. A mirror turned through θ turns the reflected beam through 2θ.

To double the effect again, the big balls are moved to the mirror-image position on the other side. The rod swings from +θ to −θ, a total of 2θ, so the beam sweeps through 4θ. If the spot moves a distance d on a scale a distance D away, $4\theta = d/D$, and

Every quantity on the right can be measured. (k itself comes from timing the rod’s slow twisting oscillations.)

Weighing the Earth#

Once G is known, the Earth’s mass follows from the acceleration of a falling apple: $g = GM/R^2$, so