Problem 21 - 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

Option (C): \(\frac{P}{OA} + \frac{Q}{OB} - \frac{R}{OC} = 0\).

Step 1: Understanding the Problem

We have forces \(P\), \(Q\), and a resultant \(R\) that act at a point \(O\). A transversal cuts their lines of action at points \(A\), \(B\), and \(C\), respectively. We need to determine a relation involving their magnitudes and distances \(OA\), \(OB\), and \(OC\).

Step 2: Conceptual Analysis Using the Principle of Transverse Lines

According to the principle of moments (Varignon's theorem) applied to transverse lines, the algebraic sum of moments of any system of coplanar forces about any point is equal to the moment of their resultant about that point. If force lines are transversed by a line, the sum of the components of forces perpendicular to the transversal can be considered.

Step 3: Setting Up the Equation

Since forces \(P\), \(Q\), and \(R\) have components perpendicular to the transversed line at points \(A\), \(B\), and \(C\), we can write\[ \frac{P}{OA} + \frac{Q}{OB} = \frac{R}{OC} \]. Here, the left side sums the contributions of \(P\) and \(Q\), while the right side relates the resultant \(R\) with distance \(OC\).

Step 4: Manipulating the Equation

The equation derived from the principle is \(\frac{P}{OA} + \frac{Q}{OB} = \frac{R}{OC} \). Rearranging this equation gives \(\frac{P}{OA} + \frac{Q}{OB} - \frac{R}{OC} = 0\). This matches option (C).

Step 5: Conclusion

The derived equation \(\frac{P}{OA} + \frac{Q}{OB} - \frac{R}{OC} = 0\) confirms that the correct answer is option (C).

Key Concepts

Principle of Moments

The Principle of Moments, also known as Varignon's theorem, is a fundamental concept in physics that facilitates our understanding when dealing with coplanar forces and their effects.

This principle states that for any system of forces acting on a body, the algebraic sum of the moments about any point is equal to the moment of the resultant force about that same point.

A moment, in simple terms, relates to the rotational effect produced by a force about a pivot point or axis. This is calculated as the product of the force and the perpendicular distance from the point to the line of action of the force.

• Moment = Force × Distance

This is immensely useful because, in problems where multiple forces act on a body, instead of analyzing each force separately to understand its rotational effect, you can find the resultant force and consider the moment of this resultant force.

This principle is utilized frequently in mechanical physics to solve problems involving equilibrium and can also help simplify complex systems of forces by focusing on the resultant effects.

Coplanar Forces

Coplanar forces refer to a set of forces that lie within the same plane and affect an object accordingly.

Such forces are common in physics problems where you are analyzing forces acting on two-dimensional objects or within a specific plane of three-dimensional space.

Coplanar forces can be concurrent, parallel, or non-parallel:

• Concurrent forces: Forces that meet at a single point.
• Parallel forces: Forces that run parallel to each other.
• Non-parallel forces: Forces that neither meet at a point nor are parallel.

The analysis often revolves around resolving these forces into their components, calculating their resultant, and understanding the effect they collectively have — such as rotation or translation. Problems involving coplanar forces can typically be addressed using vector addition and the principles of equilibrium.

When forces are coplanar, their resultant can be simplified as they interfere on the same plane, making the analysis manageable and accessible.

Resultant Force

The resultant force is the single force that represents the combined effect of all the individual forces acting on a body. Think of it as the sum total of all forces, capturing their collective impact in both magnitude and direction.

To find the resultant force, you often need to perform vector addition, taking into account all the forces' magnitudes and directions.

The methods to find a resultant force usually involve:

• Graphical methods: Using scale diagrams to add vectors visually.
• Analytical methods: Applying mathematical equations, such as components method or using trigonometry.

A resultant can cause a body to translate (move in a straight line) and/or rotate, depending on how it acts on the object. It's crucial in physics as it helps to predict the actual motion and behavior of objects under the action of multiple forces.

Once determined, the resultant force simplifies the problem-solving process by reducing the complexity of multiple forces into a single, more manageable entity.