In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant 𝐺. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass 𝑚, attached to the ends of a lightweight rod, forming a dumbbell. This rod is suspended vertically by a thin fiber. Two larger lead spheres, each of mass 𝑀, are placed near the smaller spheres, ensuring all four spheres lie in a horizontal plane. The small spheres are attracted to the larger spheres by the gravitational force, given by:F = G Mm/r²
where 𝑟 is the distance between the centers of a large sphere and its neighboring small sphere.
Torque and Equilibrium:
The forces on the two small spheres form a couple that exerts a torque on the dumbbell. This torque causes the rod to rotate and twist the suspension fiber until the restoring torque of the fiber balances the gravitational torque. The angle of deflection (𝜃) is measured using a lamp and scale arrangement, which detects the deflection of a light beam.
Deflecting Torque: tau_deflecting = F . L = G Mm/r² . L,
where 𝐿 is the length of the rod.
Restoring Torque:
tau_restoring = k𝜃,
where 𝑘 is the torsion constant of the fiber (restoring torque per unit angle of twist).
In rotational equilibrium, the two torques are equal and opposite:
G Mm/r² . L = k𝜃,
Rearranging, the value of 𝐺 is:
𝐺 = k𝜃r² / MmL
Determination of 𝐺
By measuring all the quantities on the right-hand side during the experiment, the value of 𝐺 can be calculated. Cavendish’s experiment laid the foundation for precise determination of
𝐺, and modern methods have refined its measurement. The currently accepted value is:
G = 6.67 × 10⁻¹¹ Nm² kg⁻².

In 1687, Newton published the universal law of gravitation in his book Principia. The law states: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the two particles.

Consider two bodies with masses m ₁ and m₂ separated by a distance r.
According to the law:
F ∝ m ₁ m₂ and F ∝ 1/r²
F ∝ m ₁m₂ /r²
or F = 𝐺 m ₁ m₂ / r²
where 𝐺 is the universal gravitational constant.
Definition of 𝐺 :
If m ₁ = m₂ = 1 and r = 1, then F = G. The universal gravitational constant is defined as the force of attraction between two bodies, each of unit mass, placed 1 unit distance apart.
Units of 𝐺 :
SI Unit: 𝐺 = 6.67 x 10⁻¹¹ Nm² kg⁻²
Cgs Unit, 𝐺 = 6.67 x 10⁸ dyn cm²g⁻².
Dimensions of 𝐺 :
From the formula 𝐺 = F r² / m ₁ m₂ , the dimensions of 𝐺 are:
[G] = MLT⁻² x L² / M x M = [M⁻¹ L³T⁻²]
Properties of 𝐺 :
The value of 𝐺 is constant and does not depend on the nature, size, or composition of the interacting bodies, making it a universal constant.