Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Forces \(P, Q\) act at \(O\) and have a resultant \(R\). If any transversal cuts their lines of action at \(A, B, C\), respectively, then (A) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}+\frac{\mathrm{R}}{O C}=0\) (B) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}+\frac{\mathrm{R}}{O C}=1\) (C) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}-\frac{\mathrm{R}}{O C}=0\) (D) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}-\frac{\mathrm{R}}{O C}=1\).

Step-by-Step Solution

Answer

The correct answer is (C) \( \frac{P}{OA} + \frac{Q}{OB} - \frac{R}{OC} = 0 \).

Step 1: Understanding the Problem Statement

We have three forces \(P\), \(Q\), and \(R\) acting at a point \(O\) with \(R\) as the resultant of \(P\) and \(Q\). A transversal line cuts the lines of action of these forces at points \(A\), \(B\), and \(C\) respectively. Our task is to determine the correct relationship among these forces using the given transversal cut points.

Step 2: Apply Principles of the Transversal Cut

According to the principle of moments, the sum of the moments of forces about any point on their plane of action is zero if the forces are in equilibrium. Since \(R\) is the resultant of \(P\) and \(Q\), moments about \(A\), \(B\), and \(C\) lead to the equation: \( \frac{P}{OA} + \frac{Q}{OB} - \frac{R}{OC} = 0 \).

Step 3: Conclusion Based on Analysis

Given the analysis from the principle of moments, the correct option that satisfies the equation is \( \frac{P}{OA} + \frac{Q}{OB} - \frac{R}{OC} = 0 \), which corresponds to option (C).

Key Concepts

Principle of Moments

The Principle of Moments is a fundamental concept in physics that helps us understand how forces affect rotation. It states that for a system to be in equilibrium, the sum of the moments around any pivot point must be zero. A moment is simply the turning effect of a force and is calculated by multiplying the force by the perpendicular distance from the pivot point to the line of action of the force.

To apply this in practice, imagine placing a ruler on the edge of a table with a weight on one end. If the ruler is in equilibrium, it stays level, meaning the clockwise and counterclockwise moments balance out.

In the context of the exercise, the moment equation is:

• \( \frac{P}{OA} + \frac{Q}{OB} - \frac{R}{OC} = 0 \)

This equation reflects that the moments due to forces \( P \), \( Q \), and their resultant \( R \) around the transversal intersect points must cancel each other out for equilibrium to be maintained.

Equilibrium of Forces

The equilibrium of forces refers to the condition where all the forces acting on an object balance out, resulting in no net force or movement. For an object to remain in equilibrium, two main conditions need to be satisfied:

• The sum of all horizontal forces must be zero.
• The sum of all vertical forces must be zero.

Forces in equilibrium mean that they create neither movement nor rotation on the object they act upon. They effectively cancel each other out. This is crucial in determining the stability of structures and in ensuring that mechanical systems function properly.

In the given exercise, the forces \( P \), \( Q \), and \( R \) are in equilibrium at point \( O \). As a result, their effects are perfectly balanced, which is why the principle of moments can be applied to derive the equation for their moments.

Vector Addition of Forces

Vector addition of forces is a method used to calculate the resultant or combined force when two or more forces are acting on a point. Since forces are vector quantities, they have both magnitude and direction. When adding them, we must consider these two aspects.

To visualize vector addition, imagine two arrows on a piece of paper. Each arrow represents a force, with its length showing the magnitude and the direction it points indicating the direction of the force. The resultant force is the vector that combines these two, often represented by the diagonal of a parallelogram formed by the original vectors.

In the scenario from the exercise, forces \( P \) and \( Q \) at point \( O \) combine to form resultant force \( R \). This process adheres to the rules of vector addition, which might involve breaking down the forces into components, adding these components separately, and then combining them to find the resultant vector \( R \). Understanding how to add vectors is key to solving many physics problems involving multiple forces at play.