Q.Three equal masses of $m$ kg each are fixed at the vertices of an equilateral triangle $ABC$.

Concept & Intuition

The key idea is that gravity is a fundamental, long-range force that acts directly between masses. Unlike electric or magnetic forces, gravity is not affected by the presence of a dielectric or any other medium — it does not get “screened” or “reduced” by water, air, or vacuum. This is because the gravitational interaction is mediated by spacetime curvature, not by particles that interact with the medium. So, moving the setup from vacuum to water changes nothing about the gravitational force.

Step-by-step reasoning

1. Recall Newton’s law of universal gravitation In a vacuum, the gravitational force between two point masses $m_1$ and $m_2$ separated by distance $r$ is given by

where $G$ is the universal gravitational constant.

2. Consider the effect of a medium For forces like electrostatic force, the presence of a medium introduces a dielectric constant $\varepsilon_r$ that reduces the force:

However, gravity has no such “permittivity” or “permeability” — there is no gravitational analogue of a dielectric constant. The constant $G$ is universal and does not change with the medium.

3. Apply to the water medium Water is a material medium, but it does not alter the gravitational interaction. The masses remain the same, the distance remains the same, and $G$ remains the same. Therefore: $F_w = G \frac{m_1 m_2}{r^2} = F_v$ …

The gravitational force between the two particles is $F=\dfrac{GMm}{r^2}$. This same force accelerates the heavier particle (mass $M=4\ \text{kg}$), so its acceleration is

which depends only on the lighter mass $m$ and the separation $r$ (the heavier mass cancels). …

Concept and Intuition

The key idea is that gravitational force is a universal interaction that does not depend on the medium between the masses. Unlike electrostatic or magnetic forces, which can be screened or altered by a dielectric or magnetic material, gravity passes through any substance unchanged. The density of the liquid is irrelevant here — it only affects buoyancy, not gravity. Many students mistakenly think the liquid reduces the force, but that’s a confusion with electrostatic forces in a dielectric medium.

Step-by-step reasoning

1. Recall Newton’s law of universal gravitation The gravitational force between two point masses $M_1$ and $M_2$ separated by a distance $d$ in vacuum (or air) is given by

where $G$ is the universal gravitational constant.

2. Identify what the law depends on

   Notice that $F$ depends only on the masses and the distance — not on any property of the medium (like density, refractive index, or permittivity). This is because gravity is a fundamental force that couples directly to mass-energy, and no intervening material can “block” or “alter” it.

3. Consider the effect of the liquid …

The key idea here is the difference between theorizing a constant and measuring it. Newton’s law of universal gravitation ($F = G \frac{m_1 m_2}{r^2}$) was published in 1687, but Newton had no way to measure $G$ — he could calculate the product $GM_\text{Earth}$ from the Moon’s orbit, but not $G$ alone. The constant $G$ remained unknown until an experiment could detect the tiny gravitational force between everyday objects.

1. Why not Newton?

   Newton formulated the law, but he lacked the technology to measure the extremely weak attraction between laboratory masses. He famously estimated that the gravitational force between two 1 kg spheres 1 m apart is about $6.7 \times 10^{-11}$ N — far too small for 17th-century instruments. So he never determined $G$ numerically.

2. Cavendish’s breakthrough (1798)

   Henry Cavendish used a torsion balance — a horizontal rod with small lead spheres suspended by a thin wire. When large lead spheres were brought near, the tiny gravitational twist of the wire was measured. From the twist angle and the known properties of the wire, he calculated the force and thus $G$. His result was within 1% of today’s accepted value ($6.674 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2$).

3. Why not Einstein or Galileo? …

Concept: superposition (the 'cavity' trick). Remaining solid = full sphere MINUS the small sphere that was removed. Forces subtract likewise. Both the full sphere and the removed sphere are spherically symmetric, so each attracts the external mass m as if its mass were at its own centre.

Geometry from the figure: the cavity has diameter R (radius R/2) and is internally tangent to the sphere on the side facing m, so its centre lies at distance R/2 from the sphere's centre, ON the line towards m. The mass m is at 3R from the sphere's centre.

Step 1 - full sphere:

F = G M m / (3R)^2 = G M m / (9 R^2). So G M m / R^2 = 9F.

Step 2 - removed portion: …