Measurement of the Gravitational Constant (G)

The gravitational constant (G) is a fundamental value in physics. Its magnitude is determined by measuring the force of gravitational attraction between two bodies of known masses, m₁ and m₂, separated by a distance 'L'. This measurement is performed using a specialized and sensitive instrument. The method was pioneered using an apparatus known as the Cavendish balance.

The gravitational constant is the proportionality constant G in Newton's Law of Universal Gravitation, which relates the force of gravitational attraction between two masses and the distance separating them.

The magnitude of the force of attraction (F) between a big sphere (Mass M) and a neighbouring small sphere (Mass m) separated by distance r is:

The torque (τ) generated by the force of attraction is:

\[\tau = F \cdot L = G \frac{mM}{r^2} L\]

The Cavendish balance setup has the following key components and characteristics:

• It uses a light rigid rod supported at the center.
• The support is a fine vertical metallic fibre about 100 cm long.
• Two small lead spheres (s₁, s₂) of equal mass m (diameter about 5 cm) are mounted at the ends of the rod.
• A small mirror (M) is fastened to the metallic fibre to measure the angle of twist.
• Two large lead spheres (L₁, L₂) of equal mass M (diameter about 20 cm) are brought close to the small spheres.

The Cavendish balance is an instrument used to measure the minute gravitational attraction between masses to determine G.

1. Attraction: Two large lead spheres (L₁ and L₂) are positioned close to the small spheres (s₁ and s₂) on opposite sides.

2. Force: The big spheres attract the nearby small spheres by equal and opposite gravitational forces (\[\vec F\]).

3. Torque Generation: This pair of equal and opposite forces creates a torque (τ = F · L) on the rod, but exerts no net force on the bar.

4. Twisting of Wire: Due to this torque, the rod turns, and the central suspension wire gets twisted.

5. Equilibrium: The twisting continues until the restoring torque (τ_restoring = Kθ) due to the elastic property of the wire becomes equal and opposite to the gravitational torque (τ).

• The gravitational force between the spherical balls is treated as if their masses are concentrated at their centres.

6. Equilibrium Equation: At equilibrium, the balance condition is \[G\frac{mM}{r^{2}}L=K\theta\].

Aim:

To determine the value of the Gravitational Constant (G).

Requirements (The Apparatus):

Cavendish balance, consisting of a rigid rod, fine metallic fibre, small spheres (s₁, s₂), large spheres (L₁, L₂), and a mirror (M) with a light beam/scale arrangement.

Proof/Procedure (Calculation of G):

1. Determine K (Restoring Torque per Unit Angle):

• Apply a known torque (τ₁).
• Measure the corresponding angle of twist (α).
• Calculate K using the formula: K = τ₁ / α.

2. Achieve Equilibrium: Allow the large spheres to attract the small spheres until the gravitational torque equals the restoring torque, resulting in the equilibrium condition:

3. Measure Variables: Measure the equilibrium angle of twist (θ) using the mirror and light scale. The values of m, M, L, and r are known from the setup.

4. Calculate G: Rearrange the equilibrium equation to solve for G:

Result (Measured Value):

The gravitational constant measured using this method is found to be:

• Determines G: Provides a direct method for measuring the magnitude of the fundamental gravitational constant G.
• Confirms Law: Demonstrates the universality of Newton's Law of Gravitation by measuring the force between small, terrestrial objects.