Notes : Determination of Gravitational Constant (G) – Cavendish Experiment

Determination of Gravitational Constant (G) – Cavendish Experiment

The value of the universal gravitational constant (G) entering Newton's law of universal gravitation was first determined experimentally by Henry Cavendish in 1798. The experiment demonstrated that gravitational forces between ordinary laboratory objects can be measured directly.

Cavendish Experiment Principle

The experiment is based on the principle that the tiny gravitational attraction between two pairs of masses produces a measurable torque on a suspended rod. This gravitational torque twists the suspension wire until it is balanced by the restoring torque of the wire. By measuring the angle of twist, the value of the gravitational constant G can be determined.

Experimental Arrangement

The apparatus consists of a light horizontal rod AB carrying two small lead spheres, each of mass m, at its ends. The rod is suspended from its centre by a fine torsion wire attached to a rigid support, allowing it to rotate about a vertical axis.

Two large lead spheres, each of mass M, are placed near the small spheres but on opposite sides. The centres of the four spheres lie approximately in the same horizontal plane. The distance between the centres of a large sphere and its neighbouring small sphere is d.

Working of the Experiment

Initially, the equilibrium position of the rod is noted.
The large spheres are brought close to the small spheres.
Each large sphere attracts the neighbouring small sphere with a gravitational force F.
These equal and opposite forces form a couple, producing a gravitational torque.
The suspended rod rotates, twisting the torsion wire.
The twisted wire develops a restoring torque opposite to the gravitational torque.
At equilibrium, the restoring torque becomes equal to the gravitational torque.
The angle of twist θ is measured experimentally.
To improve accuracy, the positions of the large spheres are reversed and the corresponding deflection is observed.

Gravitational Force Between the Spheres

If:

M = mass of each large sphere,
m = mass of each small sphere,
d = distance between the centres of a large sphere and its neighbouring small sphere,

then the gravitational force between them is given by

$$F=\frac{GMm}{d^2}$$

Gravitational Torque

If L is the distance between the two small spheres, then the gravitational torque is

$$\tau_g=F\times L$$

Substituting the value of F,

$$\tau_g=\frac{GMm}{d^2}\times L$$

If the distance of each small sphere from the axis of suspension is l, then L = 2l, and

$$\tau_g=\frac{GMm}{d^2}\times 2l$$

Restoring Torque

If τ is the restoring couple per unit radian twist of the suspension wire and θ is the angle of twist, then restoring torque is

$$\tau_r=\tau\theta$$

Condition for Equilibrium

At equilibrium,

$$\text{Gravitational Torque}=\text{Restoring Torque}$$

Therefore,

$$\frac{GMm}{d^2}\times L=\tau\theta$$

or,

$$\frac{GMm}{d^2}\times 2l=\tau\theta$$

Expression for Gravitational Constant

Rearranging the above equation,

$$G=\frac{\tau\theta d^2}{MmL}$$

or,

$$G=\frac{\tau\theta d^2}{Mm(2l)}$$

Thus, by measuring the angle of twist θ and knowing the values of τ, M, m, d, and L, the value of the gravitational constant can be calculated.

Value of Gravitational Constant

The currently accepted value of the universal gravitational constant is

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

Importance of Cavendish's Experiment

It provided the first experimental determination of G.
It verified Newton's law of universal gravitation.
It enabled scientists to determine the mass and average density of the Earth.
It demonstrated that extremely small gravitational forces between ordinary objects can be measured accurately.

Frequently Asked Questions (FAQs)

1. Who determined the value of the gravitational constant for the first time?

Answer: Henry Cavendish determined the value of the gravitational constant experimentally in 1798.

2. Which instrument is used in Cavendish's experiment?

Answer: A torsion balance is used to measure the tiny gravitational force between the spheres.

3. Why are lead spheres used in the experiment?

Answer: Lead has a high density, resulting in larger gravitational forces that are easier to detect.

4. What is the restoring torque in the experiment?

Answer: The restoring torque is the torque developed in the twisted suspension wire and is given by:

$$\tau_r=\tau\theta$$

5. What is the formula for gravitational force between the spheres?

$$F=\frac{GMm}{d^2}$$

6. What is the SI unit of the gravitational constant?

Answer: The SI unit of G is N m² kg^-2.

7. What is the accepted value of G?