Physics · Gravitation and the universe · Grade 10

Law of universal gravitation

Do two lead spheres attract each other, and how can we “weigh” the Earth?

The situation

According to legend, Newton watched an apple fall to the ground and wondered whether the force pulling the apple to the Earth and the force holding the Moon in orbit around the Earth were one and the same. Then do two apples on a table attract each other too? In 1797–1798 Henry Cavendish set up a torsion balance inside a small building and watched it through a telescope from a hole in the wall: lead spheres of 158 kg turned the little 0.73 kg spheres by a barely noticeable amount. How can such a tiny twist be used to “weigh” the whole Earth?

Topic

Law of universal gravitation. Cavendish balance: two lead spheres attract each other

Key ideas

Gravitational constant G
Cavendish experiment
Field strength and potential

Formulas

F = G \frac{m_{1} m_{2}}{r^{2}}

The law of universal gravitation (Newton, 1687): any two bodies attract each other.

G = \frac{2 π^{2} L θ r^{2}}{M T^{2}}, θ = \frac{S}{2 D}

Cavendish balance: $L$ — rod length, $T$ — period of oscillation, $S$ — shift of the spot on the scale, $D$ — distance from the mirror to the scale.

g = \frac{G M}{{(R + h)}^{2}} = g_{0} \frac{R^{2}}{{(R + h)}^{2}}, M = \frac{g_{0} R^{2}}{G}

The field strength decreases with height; once $G$ is known — the mass of the Earth (“Weighing the Earth”).

φ = - \frac{G M}{r}, A = m (φ_{1} - φ_{2}) = G M m (\frac{1}{r_{2}} - \frac{1}{r_{1}})

The gravitational potential and the work done by gravity do not depend on the shape of the path.

m_{i} a = m_{g} g \Rightarrow a = g (m_{g} = m_{i})

The equivalence principle: inertial and gravitational mass are equal — all bodies fall with the same acceleration.

Theory

Law of universal gravitation

According to the law published by Newton in 1687, any two bodies attract each other with a force directly proportional to the product of the masses and inversely proportional to the square of the distance: $F = G m_{1} m_{2} / r^{2}$ . $G$ is very small: two 1 kg spheres 10 cm apart attract each other with a force of only $6.7 \cdot 10^{- 9}$ N. Near the Earth this force is the gravitational force (weight force) $F = m g$ ; weight is the force with which a body presses on its support or pulls its suspension (at rest $P = m g$ ; it changes in an accelerating lift — see the “Newton's laws” experiment). Nature has four fundamental interactions: gravitational, electromagnetic (the elastic and friction forces are of this nature too), strong and weak. Gravity is the weakest, but it acts at long range and is always attractive — it governs planets and galaxies.

Cavendish experiment

A light rod is hung from a thin wire, with small spheres at its ends. The large spheres attract them and twist the rod through an angle $θ$ , while the wire resists the twisting: $κ θ = F L$ . We find the stiffness of the wire from the period of free oscillation: $T = 2 π \sqrt{I / κ}$ , $I = m L^{2} / 2$ . Combining the two, $G = 2 π^{2} L θ r^{2} / (M T^{2})$ — the small mass $m$ cancels. Since the twist is only a few milliradians, an optical lever is used: if the mirror turns through $θ$ , the reflected beam turns through $2 θ$ , and on a scale at distance $D$ the spot moves by $S = 2 θ D$ . Moving the large spheres from “−” to “+” doubles the shift.

G and the mass of the Earth

Once $G$ is known, the mass of the Earth follows from $g = G M / R^{2}$ at the surface: $M = g R^{2} / G \approx 6 \cdot 10^{24}$ kg, and the mean density is about 5.5 g/cm³ — twice that of surface rocks, so the Earth's interior is heavier. $G$ is still the least precisely known fundamental constant: according to CODATA-2018, $G = 6.67430 (15) \cdot 10^{- 11}$ N·m²/kg². Model: the mass of the rod is neglected; the pull of the far large sphere and the change of $r$ during the twist are taken into account — they add a correction of about 0.3 % to the formula; there are no air currents or temperature changes.

g depends on height and latitude

With height, $g = g_{0} R^{2} / {(R + h)}^{2}$ : even at the height of the ISS (≈ 410 km), $g$ is about 88 % of its value at the surface — astronauts are weightless not because there is no gravity but because they are in free fall. With latitude, $g$ is 9.780 m/s² at the equator and 9.832 m/s² at the pole. There are two causes: the centrifugal inertial force due to the Earth's rotation (it reduces $g$ by $ω^{2} R \cos^{2} φ$ , ≈ 0.034 m/s² at the equator) and the Earth's flattening — the equatorial radius is about 21 km greater than the polar one. In “Where is g?” mode look at the two contributions separately.

The gravitational field: field strength and potential

Around a mass there is a gravitational field. Its strength $\vec{g} = \vec{F} / m$ is the force on 1 kg, directed towards the centre, $g = G M / r^{2}$ . The potential $φ = - G M / r$ is the potential energy of 1 kg (zero at infinity). The work done by gravity does not depend on the path: $A = m (φ_{1} - φ_{2})$ — the field is potential (conservative). Equipotential surfaces are perpendicular to the field lines. The depth of the “funnel” shows $φ$ : a probe launched straight up rises to the level of its total energy, and if $E \geq 0$ ( $v_{0} \geq 11.2$ km/s) it does not return. Orbits and cosmic speeds are in the “Newton's cannon: an artificial satellite” experiment.

Inertial and gravitational mass. Modern ideas

Inertial mass $m_{i}$ shows how much a body resists acceleration ( $F = m_{i} a$ ), and gravitational mass $m_{g}$ how strongly the field attracts it ( $F = m_{g} g$ ). In experiments they are equal: the Eötvös experiments (early 1900s, precision about $10^{- 8}$ – $10^{- 9}$ ) and the MICROSCOPE satellite (2022, precision $10^{- 15}$ ) found no difference. From this Einstein derived the equivalence principle: experiments inside an accelerating cabin cannot be distinguished from gravity. In the general theory of relativity gravity is the curvature of space-time; this is confirmed by the bending of light near the Sun, by the correction applied to GPS clocks, and by the gravitational waves detected by LIGO in 2015. In engineering, gravity is taken into account in satellite orbits, in GPS and in gravimetry (prospecting for underground resources).

Did you know?

Cavendish determined not $G$ but the mean density of the Earth: in his paper he wrote that it is 5.48 times that of water; an error in his calculation was found later, and his own measurements give 5.448. The present value is about 5.51.
On 2 August 1971 the Apollo 15 astronaut David Scott released a 1.32 kg geological hammer and a 30 g falcon feather together on the Moon — they landed on the surface at the same time.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The Solar System — Grade 7
Scales of the universe — Grade 9
Newton's cannon: an artificial satellite — Grade 10
Special relativity — Grade 10
Relativistic dynamics — Grade 10
Kepler's laws — Grade 10
Physics of stars — Year 1
The expanding universe — Year 1
Black holes — Year 4