Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length a. When released…

Year	2026	2025	2024	Total
Questions	5	9	8	22

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

By Dimensional Analysis or Scaling of Kepler's Third Law:

T \propto m^x G^y a^z

Core Logic

Let's perform a dimensional matching to express the collision time $T$ $T$ in terms of physical scaling variables $m$ $m$, $G$ $G$, and $a$ $a$:

[T] = [M]^x [M⁻¹L³T⁻²]^y [L]^z

Equating dimensions on both sides:

Mass ( $M$ $M$): $x - y = 0 x = y$ $x - y = 0 \implies x = y$
Length ( $L$ $L$): $3y + z = 0 z = -3y$ $3y + z = 0 \implies z = -3y$
Time ( $T$ $T$): $-2y = 1 y = -1/2$ $-2y = 1 \implies y = -1/2$

Solving these equations:

x = -1/2, y = -1/2, z = 3/2

Step 1: Scaling Formula of Time

Therefore, the scaling relationship for time $T$ $T$ is:

T \propto m-1/2 G-1/2 a3/2 T \propto \sqrt((a³)/(m))

Let's write the ratio for two cases:

(T₂)/(T₁) = \sqrt(((a₂)/(a₁))³ \cdot ((m₁)/(m₂)))

Step 2: Calculating Final Time

Given values:

$a₁ = a$ $a_1 = a$, $a₂ = 2a$ $a_2 = 2a$
$m₁ = m$ $m_1 = m$, $m₂ = 2m$ $m_2 = 2m$
$T₁ = 4~s$ $T_1 = 4\mathrm{~s}$

(T₂)/(4) = \sqrt(((2a)/(a))³ \cdot ((m)/(2m))) = \sqrt(2³ \cdot (1)/(2)) = \sqrt(4) = 2

T₂ = 4 \times 2 = 8~seconds

Pattern Recognition

Kepler's Third Law / free-fall collapse scaling: whenever a orbit or a direct gravitational collapse scale is involved, the time scales as $T \propto \sqrt((R³)/(GM))$ $T \propto \sqrt{\frac{R^3}{GM}}$. Thus doubling $R$ $R$ multiplies time by $\sqrt(8)$ $\sqrt{8}$ and doubling $M$ $M$ divides time by $\sqrt(2)$ $\sqrt{2}$. The combination results in a clean doubling of time: $\sqrt(8)/\sqrt(2) = 2$ $\sqrt{8}/\sqrt{2} = 2$.

Chapter Mix

Class 11 Physics: Gravitation Class 11 Physics: Units and Measurements: Dimensional Analysis

More Gravitation Previous-Year Questions — Page 5

Q46 jee_main_2024_31_jan_evening Escape Velocity

The mass of the moon is

1/144

times the mass of a planet and its diameter

1/16

times the diameter of a planet. If the escape velocity on the planet is v, the escape velocity on the moon will be:

A. $(v)/(3)$ $\frac{v}{3}$
B. $(v)/(4)$ $\frac{v}{4}$
C. $(v)/(12)$ $\frac{v}{12}$
D. $(v)/(6)$ $\frac{v}{6}$

Solution

Related Formula

vescape = \sqrt((2GM)/(R))

Core Logic

For the planet: $v = \sqrt((2GMₚ)/(Rₚ))$ $v = \sqrt{\frac{2GM_p}{R_p}}$ For the moon: $Mm = (Mₚ)/(144)$ $M_m = \frac{M_p}{144}$ and $Rm = (Rₚ)/(16)$ $R_m = \frac{R_p}{16}$.

Step 1: Setup the Ratio

vm = \sqrt((2G Mm)/(Rm))

vm = \sqrt((2G ((Mₚ)/(144)))/(((Rₚ)/(16))))

vm = \sqrt((2G Mₚ)/(Rₚ) \times (16)/(144))

Step 2: Simplification

vm = \sqrt((2G Mₚ)/(Rₚ)) \times \sqrt((1)/(9))

vm = v \times (1)/(3) = (v)/(3)

Pattern Recognition

Escape velocity scales as $\sqrt(M/R)$ $\sqrt{M/R}$. If $M$ $M$ scales by $x$ $x$ and $R$ $R$ scales by $y$ $y$, velocity scales by $\sqrt(x/y)$ $\sqrt{x/y}$. Here, $\sqrt((1/144)/(1/16)) = \sqrt(16/144) = \sqrt(1/9) = 1/3$ .

Chapter Mix

Class 11 Physics: Gravitation

Q jee_main_2024_31_jan_morning Superposition Principle

Four identical particles of mass

m

are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is

( 2\sqrt(2) + 132) Gm²L²

, the length of the sides of the square is

A. $L2$ $\frac{\mathrm{L}}{2}$
B. $4 L$ $4 L$
C. $3L$ $3L$
D. $2 L$ $2 L$

Solution

Related Formula

F = (G m₁ m₂)/(r²)

Core Logic

Superposition Principle diagram for Q37 - JEE Main 2024 Morning

Let the side length of the square be $a$ $a$. Considering one corner mass, it experiences forces from the adjacent two masses (distance $a$ $a$) and the diagonally opposite mass (distance $\sqrt(2)a$ $\sqrt{2}a$).

The forces from the two adjacent masses are at $90^°$ $90^\circ$ to each other:

F = (Gm²)/(a²)

The resultant of these two is $\sqrt(2)F = \sqrt(2) (Gm²)/(a²)$ $\sqrt{2}F = \sqrt{2} \frac{Gm^2}{a^2}$, directed along the diagonal.

Step 2: Total Force Equation

The force from the diagonal mass is:

F' = Gm²(\sqrt(2)a)² = (Gm²)/(2a²)

Total resultant force $Fₙₑₜ = \sqrt(2)F + F'$ $F_{\text{net}} = \sqrt{2}F + F'$:

Fₙₑₜ = \sqrt(2) (Gm²)/(a²) + (Gm²)/(2a²) = (Gm²)/(a²) ( \sqrt(2) + (1)/(2) )

Fₙₑₜ = (Gm²)/(a²) ( 2\sqrt(2) + 12 )

Equating this to the given force value:

( 2\sqrt(2) + 132)(Gm²)/(L²) = (Gm²)/(a²) ( 2\sqrt(2) + 12 )

(1)/(32 L²) = (1)/(2 a²)

$a² = 16 L²$ $a^2 = 16 L^2$ $a = 4L$ $a = 4L$

Chapter Mix

Class 11 Physics: Gravitation

Solution & Explanation

Related Formula

Core Logic

Step 1: Scaling Formula of Time

Step 2: Calculating Final Time

Pattern Recognition

Chapter Mix

Reference Study Guides

More Gravitation Previous-Year Questions — Page 5

Solution

Related Formula

Core Logic

Step 1: Setup the Ratio

Step 2: Simplification

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 2: Total Force Equation

Chapter Mix

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Solution & Explanation

Related Formula

Core Logic

Step 1: Scaling Formula of Time

Step 2: Calculating Final Time

Pattern Recognition

Chapter Mix

Reference Study Guides

More Gravitation Previous-Year Questions — Page 5

Solution

Related Formula

Core Logic

Step 1: Setup the Ratio

Step 2: Simplification

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 2: Total Force Equation

Chapter Mix

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