Problem 11 - Master Resource Book in JEE Main Physics

Question

The resultant of two forces, each \(\mathbf{P}\), acting at an angle \(\theta\) is (a) \(2 P \sin \frac{\theta}{2}\) (b) \(2 P \cos \frac{\theta}{2}\) (c) \(2 P \cos \theta\) (d) \(P \sqrt{2}\)

Step-by-Step Solution

Answer

The resultant force is \( 2P \cos \left(\frac{\theta}{2}\right)\). Option (b).

Step 1: Identify the Problem

We have two forces, each with a magnitude of \( P \), acting at an angle \( \theta \) to each other. We need to find their resultant force.

Step 2: Apply the Formula for Resultant of Two Forces

The formula for the resultant \( R \) of two forces \( P_1 \) and \( P_2 \) acting at an angle \( \theta \) to each other is given by: \[ R = \sqrt{P_1^2 + P_2^2 + 2P_1P_2 \cos \theta} \] In this case, both forces have the same magnitude \( P \).

Step 3: Substitute Values into the Formula

Since both forces have the same magnitude \( P \), substitute \( P_1 = P \) and \( P_2 = P \) into the formula: \[ R = \sqrt{P^2 + P^2 + 2P \cdot P \cdot \cos \theta} \]Simplify this to \[ R = \sqrt{2P^2 + 2P^2 \cos \theta} \]

Step 4: Factor Out the Common Terms

Factor out \( 2P^2 \) from under the square root: \[ R = \sqrt{2P^2(1 + \cos \theta)} \] This simplifies further to \[ R = P\sqrt{2(1 + \cos \theta)} \]

Step 5: Use the Trigonometric Identity

Recall the trigonometric identity: \[ 1 + \cos \theta = 2 \cos^2 \left(\frac{\theta}{2}\right) \]Substitute this identity into the expression: \[ R = P \sqrt{2 \cdot 2 \cos^2 \left(\frac{\theta}{2}\right)} \]This simplifies to \[ R = P \cdot 2 \cos \left(\frac{\theta}{2}\right) \]

Step 6: Conclude with the Correct Answer

The resultant of the two forces each \( P \) is \[ 2P \cos \left(\frac{\theta}{2}\right) \] which matches option (b).

Key Concepts

Vector Addition

Vector addition is a fundamental process in physics and engineering used to find the resultant of two or more vectors. This is done by combining vector magnitudes and directions to form a new vector.

• When vectors act at an angle to each other, they can be added using graphical methods like the head-to-tail method or using analytical methods like vector components.
• For analytical vector addition, the law of cosines is often used when dealing with forces or velocities that are not perpendicular.

To find the resultant vector using vector addition, consider each vector's magnitude and direction. If they are represented by coordinates, add their respective components, otherwise apply vector formulas.
In the case of equal magnitude forces acting at an angle, as in our exercise, you use the cosine of the angle to handle their interaction.

Trigonometry Identities

Trigonometric identities are crucial in simplifying expressions and solving equations involving trigonometric functions in physics problems. One of the most useful identities in vector addition is the identity for cosines and sum of angles.

• The identity: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \) is essential for dealing with angles that occur between vectors.
• In simplification processes, the identity \( 1 + \cos \theta = 2 \cos^2(\frac{\theta}{2}) \) is often used, as demonstrated in the provided solution.

This specific identity can transform trigonometric expressions into equivalent forms that are easier to work with. Understanding and using these identities allows completing complex calculations and simplifications more seamlessly.

Physics Problem Solving

Physics problem-solving often involves applying mathematical techniques and physical laws to find unknowns from given information. Modern physics combines both qualitative understanding and quantitative computation.

• Define the problem: Identify the quantities given and what you need to find. Determine the relationship between these quantities using known principles or formulas.
• Apply relevant formulas: In force problems, like the one solved, start by writing the equation for the resultant force as a function of the component forces and their angles.
• Simplify and solve: Use arithmetic and algebraic manipulations to isolate the unknowns, applying identities and vector rules as necessary.

Breaking down a problem into smaller steps makes it manageable and reduces the chance of mistakes. Practice with structured examples builds both theoretical knowledge and practical skills.