Question

If \( 3p \) and \( 4p \) are resultant of a force \( 5p \), then the angle between \( 3p \) and \( 5p \) is:

• A. \( \sin^{-1} \left( \frac{3}{5} \right) \)
• B. \( \sin^{-1} \left( \frac{4}{5} \right) \)
• C. \( 90^\circ \)
• D. None of these

Hint:

In a triangle of forces, use the law of cosines to find the angle between two forces when the magnitudes of the forces and the resultant are given.

Answer 1

The Correct Option is B

Step 1: Understanding the vector addition.
We are given that the magnitudes of two forces are \( 3p \) and \( 4p \), and the resultant force is \( 5p \). According to the triangle of forces, we use the law of cosines to find the angle between the two forces. The law of cosines states: \[ R^2 = A^2 + B^2 - 2AB \cos \theta, \] where \( R \) is the resultant force, \( A \) and \( B \) are the magnitudes of the two forces, and \( \theta \) is the angle between them. Substituting the known values: \[ (5p)^2 = (3p)^2 + (4p)^2 - 2(3p)(4p) \cos \theta. \]
Step 2: Simplifying the equation.
Simplifying the equation: \[ 25p^2 = 9p^2 + 16p^2 - 24p^2 \cos \theta, \] \[ 25p^2 = 25p^2 - 24p^2 \cos \theta. \] Canceling out \( 25p^2 \) from both sides: \[ 0 = -24p^2 \cos \theta, \] \[ \cos \theta = 0. \]
Step 3: Conclusion.
Since \( \cos \theta = 0 \), the angle \( \theta \) is \( 90^\circ \), and the correct answer is (c) \( 90^\circ \).