The Gravitational Constant

What G is

Newton's universal law states that two point masses $m_1$ and $m_2$ separated by a distance $r$ attract each other with a force whose magnitude is

$$F = G\,\dfrac{m_1 m_2}{r^2}.$$

The proportionality factor $G$ is the universal gravitational constant. It converts the geometric quantity $m_1 m_2 / r^2$ into an actual force in newtons. The word "universal" is doing real work here: $G$ has the same value for any pair of bodies, anywhere in the cosmos, regardless of their composition, the medium between them, or where the measurement is made. A pair of lead spheres in a London laboratory and two galaxies in deep space are governed by the same $G$.

This universality is what makes Newton's law a law rather than a fitted formula. Once $G$ is measured a single time, the gravitational force between any two masses at any separation is fully determined. Numerically $G$ is extremely small, which is the formal statement of a familiar fact: gravity is by far the weakest of the fundamental interactions, and the gravitational pull between everyday objects is utterly negligible compared with their pull toward the Earth.

Value and units of G

NCERT §7.4 states that since Cavendish's experiment the measurement of $G$ has been refined, and the currently accepted value is

$$G = 6.67\times10^{-11}~\text{N m}^2~\text{kg}^{-2}.$$

The chapter summary quotes the slightly more precise figure $G = 6.672\times10^{-11}~\text{N m}^2~\text{kg}^{-2}$; for NEET, $6.67\times10^{-11}~\text{N m}^2~\text{kg}^{-2}$ is the value to use in calculations. The units follow directly from the force law: rearranging $F = G m_1 m_2 / r^2$ gives $G = F r^2 /(m_1 m_2)$, so the unit is $\text{N}\cdot\text{m}^2 / \text{kg}^2$, written $\text{N m}^2~\text{kg}^{-2}$.

Because $G$ is so small, the gravitational force between laboratory masses is minute — which is precisely why measuring $G$ accurately is one of the hardest tasks in precision physics, and why it remains the least well-determined of the fundamental constants.

The dimensions of G

The dimensions of $G$ are best derived, not memorised — the derivation is itself a favourite NEET question. Start from the force law and solve for $G$:

$$G = \dfrac{F r^2}{m_1 m_2}.$$

Now substitute the dimensions of each quantity. Force is $[MLT^{-2}]$, $r^2$ is $[L^2]$, and the product of two masses is $[M^2]$:

$$[G] = \dfrac{[MLT^{-2}]\,[L^2]}{[M^2]} = [M^{-1}L^{3}T^{-2}].$$

So $[G] = [M^{-1}L^{3}T^{-2}]$. This exact result was tested verbatim in NEET 2022 (Match List-I with List-II), where candidates had to pair $G$ with $[M^{-1}L^3T^{-2}]$ while distinguishing it from gravitational potential $[L^2T^{-2}]$, gravitational intensity $[LT^{-2}]$, and gravitational potential energy $[ML^2T^{-2}]$.

The Cavendish experiment

The value of $G$ cannot be deduced from theory; it must be measured. The first laboratory determination was made by the English scientist Henry Cavendish in 1798 using a torsion balance. The idea is to convert the almost imperceptible attraction between lead spheres into a measurable angle of twist.

The mechanism, following NCERT, is as follows. Each large sphere attracts its neighbouring small ball with a force whose magnitude — treating the spheres as point masses at their centres, separated by $d$ — is

$$F = G\,\dfrac{Mm}{d^2}.$$

The two forces are equal and opposite at the two ends, so there is no net force on the bar, only a torque equal to $F$ times the bar length $L$. This torque twists the suspension wire until the wire's elastic restoring torque, which is proportional to the twist angle and equal to $\tau\theta$ (where $\tau$ is the restoring couple per unit twist, measured independently), balances it. At equilibrium:

$$\tau\theta = G\,\dfrac{Mm}{d^2}\,L.$$

Every quantity except $G$ is measurable: $\tau$ by applying a known torque and noting the twist; $\theta$ by observing the deflection; $M$, $m$, $d$ and $L$ directly. Rearranging,

$$G = \dfrac{\tau\theta\,d^2}{M m L}.$$

Observation of the twist angle $\theta$ thus yields $G$. The genius of the design is that the long, very thin wire makes the restoring torque tiny, so even the feeble gravitational torque produces a readable deflection. Reversing the big spheres to the opposite side flips the torque and roughly doubles the swing, improving precision.

G versus g — the central distinction

The single most reliable confusion NEET exploits in this chapter is the difference between the universal constant $G$ and the local acceleration $g$. They share a name and a root, but almost nothing else. The relation that links them, derived for a mass on the Earth's surface, is $g = GM_E / R_E^2$ — but knowing this link is exactly what tempts students to treat them as interchangeable. They are not.

The cleanest way to keep them apart: $G$ is the universal conversion factor in the force law and never changes; $g$ is a property of a particular place near a particular body, computed from $G$, the body's mass and radius. Change the planet, climb a mountain, descend a mine — $g$ changes; $G$ does not.

Weighing the Earth

The surface relation $g = GM_E / R_E^2$ is more than a definition — it is the practical pay-off of measuring $G$. Rearranged, it reads

$$M_E = \dfrac{g\,R_E^{\,2}}{G}.$$

The acceleration $g$ is readily measured by timing a falling body or a pendulum, and the Earth's radius $R_E$ has been known since antiquity. The one missing ingredient was $G$ — and Cavendish's experiment supplied it. With $G$ in hand, the mass of the Earth follows immediately. NCERT records that this is the origin of the popular remark that "Cavendish weighed the earth." The same logic, applied to a planet's satellite via Kepler's third law, lets us "weigh" the Sun and other planets too.

Worked examples