Four particles of masses m, 2 m, 3 m and 4 m are kept in...

Four particles of masses $m, 2 m, 3 m$ and $4 m$ are kept in sequence at the corners of a square of side $a$. The magnitude of gravitational force acting on a particle of mass $m$ placed at the centre of the square will be (a) $\frac{24 m^{2} G}{a^{2}}$ (b) $\frac{6 m^{2} G}{a^{2}}$ (c) $\frac{4 \sqrt{2} G m^{2}}{a^{2}}$ (d) Zero

Four particles of masses $m, 2 m, 3 m$ and $4 m$ are kept in sequence at the corners of a square of side $a$. The magnitude of gravitational force acting on a particle of mass $m$ placed at the centre of the square will be
(a) $\frac{24 m^{2} G}{a^{2}}$
(b) $\frac{6 m^{2} G}{a^{2}}$
(c) $\frac{4 \sqrt{2} G m^{2}}{a^{2}}$
(d) Zero

Key Concepts

Newton's Law of Universal Gravitation

This fundamental law describes the attraction between two masses. It states that every two masses exert an attractive force on each other that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This concept is crucial to understand how gravitational forces are calculated between bodies.

Vector Nature of Gravitational Forces

Gravitational forces are vector quantities, which means they have both magnitude and direction. When analyzing multiple forces acting on a single object, it is important to consider the direction of each force and perform vector addition to determine the net force acting on the object.

Principle of Superposition

The superposition principle states that when several forces act on an object, the net force is the vector sum of all individual forces. This allows one to calculate the combined effect of multiple gravitational attractions by summing their vector contributions from each mass acting on the object.

Geometric Symmetry in Force Analysis

Understanding the geometry of the configuration, such as the distances involved and the angles between force vectors, is key to resolving forces into components and simplifying their vector addition. In many arrangements, symmetry can be exploited to simplify the calculation of net forces, although uneven mass distributions require careful component-wise analysis.

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