Briefly explain the Cavendish’ s experiment for the determination of the universal constant G.

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.