Supposing Newton's law of gravitation for gravitation forces $\mathbf{F}_1$ and $\mathbf{F}_2$ between two masses $m_1$ and $m_2$ at positions $\mathbf{r}_1$ and $\mathbf{r}_2$ read $\mathbf{F}_1 = -\mathbf{F}_2 = -\dfrac{\mathbf{r}_{12}}{r_{12}^{3}}\, G M_0^{2} \left(\dfrac{m_1 m_2}{M_0^{2}}\right)^{n}$ where $M_0$ is a constant of dimension of mass, $\mathbf{r}_{12} = \mathbf{r}_1 - \mathbf{r}_2$ and $n$ is a number. In such a case, (Note: more than one of the given options may be correct.) (a) the acceleration due to gravity on earth will be different for different objects. (b) none of the three laws of Kepler will be valid. (c) only the third law will become invalid. (d) for $n$ negative, an object lighter than water will sink in water.

Supposing Newton's law of gravitation for gravitation forces mathbfF₁ and mathbfF₂ between two masses m₁ and m

NCERT Exemplar · Q14

\mathbf{F}_1

(a) the acceleration due to gravity on earth will be different for different objects.

(b) none of the three laws of Kepler will be valid.

(d) for

n

negative, an object lighter than water will sink in water.

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With the force scaling as $(m_1m_2)^n$ instead of $m_1m_2$ , the acceleration of a body near Earth depends on its own mass unless $n=1$ — so different objects fall differently, making option (A) correct. Kepler's first and second laws still hold (the force stays central and still falls off as $1/r^2$ in distance), but the third law breaks because the orbiting body's own mass no longer cancels out — so only the third law is invalidated, making (B) false and (C) correct. For $n$ negative, the weight-vs-buoyancy comparison flips the usual density inequality, so an object lighter than water can sink — (D) is correct.

Setting up

The modified force law is

\mathbf{F}_1=-\mathbf{F}_2=-\frac{\mathbf{r}_{12}}{r_{12}^3}\,GM_0^2\left(\frac{m_1m_2}{M_0^2}\right)^n

In the usual case $n=1$ , this reduces to Newton's law, and the force is directly proportional to $m_1m_2$ . For $n\neq1$ , the force instead scales as $(m_1m_2)^n$ — a genuinely different dependence on mass.

(A) Acceleration due to gravity on Earth

For an object of mass $m$ on Earth's surface (mass $M_E$ , radius $R_E$ ), the force magnitude is

F=\frac{GM_0^2}{R_E^2}\left(\frac{M_Em}{M_0^2}\right)^n

so the acceleration is

a=\frac{F}{m}=\frac{GM_0^{2(1-n)}}{R_E^2}M_E^n\,m^{n-1}

Unless $n=1$ , this depends on the object's own mass $m$ — different objects would fall with different accelerations. (A) is correct.

(B) and (C) Kepler's laws

Kepler's first law (elliptical orbits) and second law (equal areas in equal times) both follow from the force being central and varying purely with distance as $1/r^2$ — neither property has changed here, only the mass-dependence has. So the first and second laws still hold.

Kepler's third law, however, comes from equating the force to the centripetal requirement for a planet of mass $m$ orbiting a star of mass $M$ :

$m\frac{4\pi^2}{T^2}r = \frac{GM_0^2}{r^2}\left(\frac{Mm}{M_0^2}\right)^n$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2026Set ANNUAL1 mark
$Q. Question number 33 (continued) is a case-study based question. Read the case carefully and answer the question that follows: Recently, on 7th September 2025, a lunar eclipse was observed on the day of full moon. Some people in a village avoided eating or cooking during the eclipse, believing it could bring ill effects. A young boy, who had studied about eclipses in science, knew that solar and lunar eclipses are natural celestial phenomena caused by the motion of the Earth, Moon, and Sun and have no harmful effects on living beings. When he explained this to his mother, she ignored his reasoning and followed the traditional belief. On his way to a friend's house, he noticed people performing prayers and rituals to ward off the so-called "ill effects" of the eclipse. The boy felt disheartened at the superstitions and decided to study more about the scientific reason behind such phenomena.$
(ii) Name the natural phenomenon caused by the gravitational force between the Earth and the Moon. OR Alternative case study (Or), continued: Ajay was always fascinated by the flights of airplanes and rockets. He often wondered how rockets could travel into space and not return, while everything he threw upward would always fall back due to Earth's gravitational pull. Curious about this, he discussed the matter with his elder brother, who explained that rockets are launched based on a scientific principle. When the rocket is projected with increasing speed, a point comes when its speed becomes high enough to overcome Earth's gravitational pull — this speed is called the escape velocity. At this speed, the rocket can move out of Earth's gravitational field and never return on its own. Understanding this concept inspired Ajay to dream of becoming an astronaut in the future.
(ii) State the principle of launching of satellite.
›Reveal solutionSolution

The gravitational pull between the Earth and the Moon causes ocean tides on Earth.

(This is the primary part of the case-study question; the alternative — the principle of satellite launching — is not required since the primary is fully answerable.)

The Moon exerts a gravitational force of attraction on every part of the Earth, including its oceans. Because the strength of this gravitational pull is slightly different at different points on Earth (stronger on the side facing the Moon, weaker on the far side), it creates a net stretching (differential) effect on the Earth's oceans, causing the water level to bulge outward on both the side of the Earth facing the Moon and the opposite side.

…

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Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2025Set ANNUAL1 mark
$Q. Gravitational force is the weakest force in nature but still it is considered the most important force. Why?$
›Reveal solutionSolution
Gravity is the weakest force between two individual particles, but because it is always attractive and has unlimited range, it never cancels out as matter accumulates — making it the dominant force at the scale of planets, stars, and galaxies.

Of the four fundamental forces (gravitational, electromagnetic, strong nuclear, weak nuclear), gravity is indeed the weakest — for example, the gravitational attraction between two protons is about $10^{36}$ times weaker than the electric repulsion between them. Yet gravity governs the motion of planets, the structure of stars and galaxies, tides, and the overall large-scale structure of the universe. This happens because of two special properties gravity has that the other forces (at macroscopic scale) do not:

It is always attractive, never repulsive. Electric charges can be positive or negative, so in any large, bulk body the positive and negative charges very nearly cancel each other's electric field — matter is essentially electrically neutral overall. But mass has only one "sign", so gravitational effects of every particle in a body always add up in the same direction and never cancel.

It has an infinite range. Unlike the strong and weak nuclear forces (which act only over the size of a nucleus), gravity's influence extends without limit, falling off only as $1/r^2$ . …
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Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2022Set ANNUAL1 mark
$Q. The gravitational force between two bodies is 1 N if the distance between them is doubled, what will be the force between them ?$
›Reveal solutionSolution

Since gravitational force varies as the inverse square of the distance, doubling the separation reduces the force to $1/4$ of its original value — from 1 N to 0.25 N.

By Newton's law of gravitation, $F = \dfrac{Gm_1m_2}{r^2}$ , so for fixed masses $m_1, m_2$ : $F \propto \dfrac{1}{r^2}$ .

Let the original force be $F_1 = 1\text{ N}$ at distance $r$ . When the distance is doubled to $2r$ :

$F_2 = \dfrac{Gm_1m_2}{(2r)^2} = \dfrac{Gm_1m_2}{4r^2} = \dfrac{F_1}{4}$

…

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Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2021Set ANNUAL1 mark
$Q. The gravitational force between two bodies is 1 N if the distance between them is doubled, what will be the force between them ?$
›Reveal solutionSolution

Since $F \propto \dfrac{1}{r^2}$ , doubling the separation reduces the gravitational force to one-fourth: $F' = 1/4$ N $= 0.25$ N.

By Newton's Universal Law of Gravitation, the force between two point masses $m_1, m_2$ separated by distance $r$ is:

$F = \dfrac{G m_1 m_2}{r^2}$

Here $F = 1\ \text{N}$ at separation $r$ . Now the distance is doubled to $r' = 2r$ : …

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Concept understanding — Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation

Setting up

(A) Acceleration due to gravity on Earth

(B) and (C) Kepler's laws

Setting up

(A) Acceleration due to gravity on Earth

(B) and (C) Kepler's laws

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