Problem 9 - Master Resource Book in JEE Main Physics

Question

If three particles each of mass \(M\) are placed at the three corners of an equilateral triangle of side \(a\), the forces exerted by this system on another particle of mass \(M\) placed (i) at the mid point of a side and (ii) at the centre of the triangle are respectively (a) 0,0 (b) \(\frac{4 G M^{2}}{3 a^{2}}, 0\) (c) \(0, \frac{4 G M^{2}}{3 a^{2}}\) (d) \(\frac{3 G M^{2}}{a^{2}}, \frac{G M^{2}}{a^{2}}\)

Step-by-Step Solution

Answer

The forces exerted in both cases are 0,0 (option a).

Step 1: Understanding the Problem

We have an equilateral triangle with three masses, each of mass \(M\), at each corner. We need to find the gravitational forces on a fourth mass, also of mass \(M\), placed at two different positions: (i) at the midpoint of a side, and (ii) at the center of the triangle.

Step 2: Force Calculation at Midpoint of a Side

Consider the mass \(M\) placed at the midpoint of side of the triangle. Two of the triangle's corners will exert opposite forces along the same line on this mass due to symmetry. These forces will cancel out each other because they are equal in magnitude and opposite in direction. As a result, the net force exerted by these two on the midpoint mass is zero.

Step 3: Force Calculation at Center of the Triangle

The center of the equilateral triangle is equidistant from all three masses. Each mass exerts a force on the central mass \(M\) along a line joining the center and the respective mass. Due to symmetry, these forces cancel each other out in terms of both direction and magnitude, resulting in a net force of zero.

Key Concepts

Equilateral Triangle

An equilateral triangle is a geometric shape where all three sides have the same length, and all three angles are equal, each measuring 60 degrees. This shape is known for its symmetry, which plays a crucial role in problems involving forces and balance. In scenarios like the one in the provided problem, analyzing the forces exerted by masses located at the vertices of an equilateral triangle often involves using symmetry to simplify calculations. In our specific case, the triangle consists of three particles, each with mass \(M\), placed at the corners. Understanding the properties of an equilateral triangle helps in predicting the symmetry of forces when another mass \(M\) is placed in relation to these corners.

Newton's Law of Gravitation

Newton's Law of Gravitation states that every point mass attracts every other point mass by a force acting along the line intersecting both points. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, it can be expressed as:

• \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \)

where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the two bodies, and \(r\) is the distance between the centers of the two masses.
In the exercise, the law helps us understand how forces work between the three masses at the vertices of the triangle and a fourth mass placed either at the midpoint of one side or at the center of the triangle. Using the law, the force due to each mass can be calculated, and with an understanding of symmetry, we can determine that these forces ultimately cancel each other in certain positions, resulting in no net force.

Symmetry in Forces

Symmetry is a powerful tool for simplifying problems involving forces, especially in configurations like an equilateral triangle. The symmetry of an equilateral triangle means any particle placed symmetrically in relation to the triangle's vertices will experience net forces in specific patterns. At the midpoint of one of the sides, two forces cancel out because they are equal in magnitude and opposite in direction due to the triangle's symmetry. When considering the center of the equilateral triangle, the symmetry is even more profound. Here, each force originates from one of the triangle's corners, and because of their equal distances and equal angles from all vertices, the forces cancel each other out in all directions.
This concept of symmetry significantly reduces the complexity of calculations as it allows us to predict the resultant forces without computing each interaction's minutiae. Thus, the understanding of symmetry in forces is crucial for applying Newton's Law efficiently in scenarios with symmetrical geometrical figures like equilateral triangles.