Problem 20 - Master Resource Book in JEE Main Physics

Question

If, the resultant of two forces \((A+B)\) and \((A-B)\) is \(\sqrt{A^{2}+B^{2}}\), then the angle between these forces is (a) \(\cos ^{-1}\left[-\frac{\left(A^{2}-B^{2}\right)}{A^{2}+B^{2}}\right]\) (b) \(\cos ^{-1}\left[-\frac{\left(A^{2}+B^{2}\right)}{\left(A^{2}-B^{2}\right)}\right]\) (c) \(\cos ^{-1}\left[-\frac{A^{2}+B^{2}}{2\left(A^{2}-B^{2}\right)}\right]\) (d) \(\cos ^{-1}\left[-\frac{2\left(A^{2}+B^{2}\right)}{A^{2}-B^{2}}\right]\)

Step-by-Step Solution

Answer

The angle between the forces is 180°, corresponding to \( \cos^{-1}(-1) \). The correct option is (a).

Step 1: Understanding the Problem

We are given two forces, \( A + B \) and \( A - B \), with a resultant \( \sqrt{A^2 + B^2} \). We need to find the angle between \( A \) and \( B \).

Step 2: Resultant Force Equation

The equation for the resultant of two vectors \( \vec{R} = \vec{C} + \vec{D} \) is given by \( R^2 = C^2 + D^2 + 2CD \cos \theta \), where \( \theta \) is the angle between them.

Step 3: Applying the Formula for Resultant

We are given \( C = A + B \) and \( D = A - B \). Therefore, the equation becomes: \[ (\sqrt{A^2 + B^2})^2 = (A + B)^2 + (A - B)^2 + 2(A + B)(A - B) \cos \theta \] This simplifies to:\[ A^2 + B^2 = (A^2 + 2AB + B^2) + (A^2 - 2AB + B^2) + 2((A^2 - B^2) \cos \theta) \]

Step 4: Simplifying the Equation

On simplification:\[ A^2 + B^2 = 2A^2 + 2B^2 + 2(A^2 - B^2) \cos \theta \]Subtract \( 2A^2 + 2B^2 \) from both sides:\[ 0 = 2(A^2 - B^2)(1 + \cos \theta) \]

Step 5: Solving for \( \cos \theta \)

For the equation \( 2(A^2 - B^2)(1 + \cos \theta) = 0 \) to hold true, either \( A^2 - B^2 = 0 \) or \( 1 + \cos \theta = 0 \). Since \( A^2 - B^2 eq 0 \) (assumed implicit),\[ \cos \theta = -1 \]Thus, the angle \( \theta = \cos^{-1}(-1) = 180^\circ \).

Step 6: Final Answer Identification

The correct option based on \( \cos \theta = -1\) is option (a):\[ \cos ^{-1}\left[-\frac{A^2 - B^2}{A^2 + B^2}\right] \] which matches with our condition at the solution \( \cos \theta = -1 \).

Key Concepts

Vector Addition

Vector addition is a fundamental concept in physics and mathematics, allowing us to determine a resultant vector when combining two vectors. In simple terms, when you add two vectors, you are stacking them head to tail and finding the vector that runs from the tail of the first to the head of the second. This process helps determine a single vector that has the same effect as combining the two original vectors.

Here's what you need to remember about vector addition:

• Vectors have both a magnitude and a direction. You can't simply add them like regular numbers.
• To find the resultant of two vectors, place them so that one vector's head meets the other's tail.
• The resultant vector, or sum, is the vector connecting the tail of the first vector to the head of the second.
• Mathematically, this involves adding the corresponding components of the vectors.

In physics problems, understanding vector addition is key to solving for quantities like resultant forces, which often occur in systems with multiple interacting forces.

Angle Between Vectors

The angle between vectors is a crucial component in determining how they interact in terms of both magnitude and direction. When two vectors form an angle, their relative orientation affects the result of their combination, whether through addition, subtraction, or other operations.

Here are a few important points to understand about angles between vectors:

• The angle between vectors is typically measured in degrees or radians.
• Knowing the angle helps in using trigonometric functions to solve vector problems, like determining resultant forces.
• In the context of the given problem, finding the angle where the cosine value is needed is vital for determining the resultant of vector addition.
• The relationship is shown in the equation for resultant forces, where the cosine of the angle directly influences the magnitude of the resulting vector.

Understanding angles between vectors helps us accurately predict the resultant force when vectors are combined.

Cosine of Angle

In many physics problems involving vectors, the cosine of the angle between them plays a critical role. It affects how much two forces or vectors either work together or oppose each other, impacting the resultant of their combination.

Consider the following about the cosine of an angle between vectors:

• Mathematically, the cosine function ranges from -1 to 1.
• It's used in the expression for the dot product of two vectors: \( ext{Dot product} = A \cdot B = |A| |B| \cos \theta\).
• When the cosine of the angle is -1, as found in the problem solution, the vectors are 180 degrees apart, meaning they are pointing in exactly opposite directions.
• In the case of forces, a negative angle (cosine of the angle = -1) leads to results indicating opposing forces, resulting in the simplification of equations like in this exercise.

Understanding how the cosine of an angle affects vector interactions is essential for accurate physics problem solving.

Physics Problem Solving

The art of physics problem solving often requires combining multiple concepts and applying them to find a solution. It involves not only understanding theories but also practical applications and mathematical manipulations.

Key techniques in physics problem solving include:

• Breaking down complex problems into smaller, manageable parts.
• Using mathematical equations and known formulas to relate different quantities.
• Understanding the relationship between different physical quantities, such as force, vectors, and angles.
• Frequently checking units and dimensions to ensure calculations are consistent.

By using these techniques, one can systematically approach and solve even the most challenging physics exercises. The process often involves logical thinking, identifying known quantities, and calculating unknowns by using appropriate formulas, as illustrated in the given exercise with vector components and angles.