Four particles, each of mass $M$ and equidistant from each other, move along a circle of radius $R$ under the action of their mutual gravitational attraction. Calculate the speed of each particle.

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the acti

IV. Exercises · Q8

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Step 1. Place the four equal masses $M$ at the corners of a square inscribed in the circle of radius $R$ (i.e. at $90^\circ$ intervals). Consider one particle; the other three sit at: one directly opposite (distance $2R$ , the diameter), and two "adjacent" corners (each at distance $R\sqrt2$ , the side of the square).

Step 2. The pull from the opposite mass has magnitude $F_{opp}=\dfrac{GM^2}{(2R)^2}=\dfrac{GM^2}{4R^2}$ , directed straight toward the centre.

Step 3. Each adjacent mass pulls with magnitude $F_{adj}=\dfrac{GM^2}{(R\sqrt2)^2}=\dfrac{GM^2}{2R^2}$ , but at $45^\circ$ to the centre-ward direction; adding the two adjacent contributions as vectors, only their components along the centre-ward direction survive (the perpendicular components cancel by symmetry), giving a combined magnitude of $2\times F_{adj}\cos45^\circ=2\cdot\dfrac{GM^2}{2R^2}\cdot\dfrac{1}{\sqrt2}=\dfrac{GM^2}{\sqrt2\,R^2}$ . …

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

Showing the 12 most recent of 32 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$Q. Who formulated the law of universal gravitation?$
(a) Albert Einstein
(b) Galileo Galilei
(c) Isaac Newton
(d) Johannes Kepler
›Reveal solutionSolution

Isaac Newton formulated the law of universal gravitation, stating that every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

While Johannes Kepler discovered the empirical laws of planetary motion (Kepler's three laws), it was Isaac Newton who generalised these observations into a single universal law of gravitation:

F = G m1 m2 / r^2

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. If the distance between two objects is doubled, how does the gravitational force between them change?$
(a) It becomes half
(b) It becomes one-fourth
(c) It remains the same
(d) It becomes twice
›Reveal solutionSolution

F is proportional to 1/r^2 (inverse-square law); doubling r reduces F to 1/4 of its original value.

Newton's law of gravitation: F = G m1 m2 / r^2

If the distance is doubled, r' = 2r:

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. The gravitational force between two objects is F. If masses of both the objects are halved without altering the distance between them, the gravitational force would become$
(a) F/4
(b) F/2
(c) F
(d) 2F
›Reveal solutionSolution

F is proportional to m1 x m2; halving each mass multiplies the force by (1/2)(1/2) = 1/4.

Newton's law of gravitation: F = G m1 m2 / r^2

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. If distance between two bodies is doubled, what is the effect on gravitational force between two bodies:$
(a) become half
(b) become one fourth
(c) become double
(d) become four times
›Reveal solutionSolution

Since $F=Gm_1m_2/r^2$ , doubling $r$ divides $F$ by $2^2=4$ .

Newton's law of gravitation gives the force between two point masses as:

$F = \frac{Gm_1m_2}{r^2}$

If the separation is doubled ( $r\to2r$ ), the new force is: …

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CBSE 2026Set ANNUAL1 mark
$Q. What is the mass of an object, whose weight is 98 N on the earth? OR Define Universal Gravitational Constant (G).$
›Reveal solutionSolution

$m=W/g=98/9.8=10\text{ kg}$ .

Weight is the gravitational force on an object, $W=mg$ , where $g$ is the acceleration due to gravity ( $\approx9.8\text{ m/s}^2$ on Earth).

…

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CBSE 2026Set ANNUAL1 mark
$Q. Write the answer in one word/one sentence: Write the C.G.S. value of universal gravitational constant G.$
›Reveal solutionSolution
G ≈ 6.674 × 10⁻⁸ dyne·cm²/g² in the C.G.S. system.

In SI units: G = 6.674 × 10⁻¹¹ N·m²/kg²

Convert each part to C.G.S.:

1 N = 10⁵ dyne

1 m² = 10⁴ cm²

1 kg² (in the denominator) = (1000 g)² = 10⁶ g²

So: G = 6.674 × 10⁻¹¹ × (10⁵ dyne) × (10⁴ cm²) / (10⁶ g²) …
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CBSE 2026Set ANNUAL1 markMCQ
$Q. The dimensional formula of the universal gravitational constant is$
(a) [M^-1 L^3 T^2]
(b) [M^-1 L^3 T^-1]
(c) [M^-1 L^3 T^-2]
(d) [ML^2 T^-3]
›Reveal solutionSolution
G has dimensions [M^-1 L^3 T^-2]. Answer (C).

Newton's law of gravitation: F = G m1 m2 / r^2, so G = F r^2 /(m1 m2).

Dimensions:

F -> [MLT^-2]

r^2 -> [L^2]

m1 m2 -> [M^2] …
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CBSE 2026Set ANNUAL1 markMCQ
$Q. Two artificial satellites A and B are revolving around a planet in circular orbits of radii 9R and R respectively. If speed of A is 2v, then the speed of B will be$
(a) 2v
(b) 4v
(c) 6v
(d) 9v
›Reveal solutionSolution

v proportional to 1/sqrt(r) gives v_B = 6v. Answer (C).

For a circular orbit, orbital speed v = sqrt(GM/r), so v is proportional to 1/sqrt(r).

Taking the ratio for the two satellites:

…

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Dimension of gravitational constant is (A) M^-1L^3T^-2 (B) ML^3T^-2 (C) M^3L^3T^3 (D) LT^-2$
›Reveal solutionSolution

The gravitational constant G has dimensional formula M⁻¹L³T⁻².

From Newton's law of gravitation:

$F = \dfrac{Gm_1m_2}{r^2} \Rightarrow G = \dfrac{Fr^2}{m_1m_2}$

Substituting dimensions: $[F] = MLT^{-2}$ , $[r^2] = L^2$ , $[m_1m_2] = M^2$ :

…

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CBSE 2025Set ANNUAL1 mark
$Q. Two 20 kg masses are separated by a distance of 2 metre. Gravitational force between them is .............. .$
›Reveal solutionSolution

Plugging the two 20 kg masses and 2 m separation into Newton's law of gravitation gives a force of about $6.67 \times 10^{-9}$ N — vanishingly small, which is why gravity between everyday objects is never noticeable.

Newton's law of gravitation:

$F = Gm_1m_2/r^2$

Here: $G = 6.674 \times 10^{-11}$ N·m²/kg², $m_1 = m_2 = 20$ kg, $r = 2$ m

$F = \left(6.674 \times 10^{-11}\right) \times \left(20 \times 20\right) / (2)^2$

…

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CBSE 2025Set ANN1 mark
$Q. Acceleration due to gravity is independent of ______ . (mass of earth / mass of body)$
›Reveal solutionSolution

g depends on the Earth's mass and radius, not on the mass of the falling body, so it is independent of the mass of the body.

For a body of mass m at the Earth's surface, the gravitational force is F = GMm/R squared, where M is the Earth's mass and R its radius. The acceleration this force produces is

a = F/m = GMm/(R squared . m) = GM/R squared = g.

…

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CBSE 2024Set ANNUAL1 markMCQ
$Q. The radius and mass of the earth are R and M respectively. The ratio of acceleration due to gravity g and G will be (A) R/M (B) R^2/M (C) M/R^2 (D) R^2M$
›Reveal solutionSolution

$g/G=M/R^2$ .

Acceleration due to gravity at the earth's surface: $g=\dfrac{GM}{R^2}$ , where $M$ is earth's mass and $R$ its radius. Dividing both sid …

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Q.Four particles, each of mass $M$ and equidistant from each other, move along a circle of radius $R$ under the action of their mutual gravitational attraction. Calculate the speed of each particle.

Concept understanding — Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation

Q.Four particles, each of mass MMM and equidistant from each other, move along a circle of radius RRR under the action of their mutual gravitational attraction. Calculate the speed of each particle.

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Q.Four particles, each of mass $M$ and equidistant from each other, move along a circle of radius $R$ under the action of their mutual gravitational attraction. Calculate the speed of each particle.

Unlock everything free for 14 days