JEE Physics 2026 Question with Solution

Given: Four particles of masses $M, 2M, 3M$ and $4M$ are placed at the corners of a square. The net gravitational force at the center is $F_1$ in the first arrangement and $F_2$ after interchanging $3M$ and $4M$.

Find: The value of $\alpha$ if

$$\frac{F_1}{F_2} = \frac{\alpha}{\sqrt{5}}$$

Each corner of the square is at the same distance from the center, so the gravitational force due to each mass at the center is proportional to that mass.

Let the side of the square be $a$. Distance from center to each corner is

$$r = \frac{a}{\sqrt{2}}$$

Resolve the forces along two perpendicular axes.

Case I — Original configuration:

Horizontal component is proportional to

$$(3M - M)$$

Vertical component is proportional to

$$(4M - 2M)$$

Therefore,

$$F_1 \propto \sqrt{(2M)^2 + (2M)^2} = 2M\sqrt{2}$$

Case II — After interchanging $3M$ and $4M$:

Horizontal component is proportional to

$$(4M - M)$$

Vertical component is proportional to

$$(3M - 2M)$$

Therefore,

$$F_2 \propto \sqrt{(3M)^2 + (M)^2} = M\sqrt{10}$$

Now,

$$\frac{F_1}{F_2} = \frac{2M\sqrt{2}}{M\sqrt{10}} = \frac{2}{\sqrt{5}}$$

Comparing with

$$\frac{F_1}{F_2} = \frac{\alpha}{\sqrt{5}}$$

we get

$$\alpha = 2$$

Therefore, the correct option is D and the value of $\alpha$ is $2$.

Given: All four masses are at equal distance from the center of the square.

Find: Compare the magnitudes of the resultant gravitational forces in the two arrangements.

Because all corner distances from the center are equal, the common factor in gravitational force,

$$\frac{Gm}{r^2}$$

is the same for every corner contribution. Hence only the mass values matter in forming the resultant.

For the first arrangement, the imbalance along one axis is $2M$ and along the perpendicular axis is also $2M$. So the resultant is proportional to

$$\sqrt{(2M)^2 + (2M)^2}$$

For the second arrangement, the imbalances become $3M$ and $M$. So the resultant is proportional to

$$\sqrt{(3M)^2 + (M)^2}$$

Thus,

$$\frac{F_1}{F_2} = \frac{\sqrt{8M^2}}{\sqrt{10M^2}} = \sqrt{\frac{8}{10}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}$$

Therefore,

$$\alpha = 2$$

So the correct option is D.