Question

Two forces P and Q acting at \(\theta\) the resultant force R is given by

• A. \(R = \sqrt{P^2 + Q^2 + 2PQ\sin2\theta}\)
• B. \(R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta}\)
• C. \(R = \sqrt{P^2 + Q^2 - 2PQ\sin2\theta}\)
• D. \(R = \sqrt{P^2 + Q^2 + 2PQ\cos2\theta}\)

Hint:

This resultant formula is a fundamental relation in vector mechanics. You can quickly verify its validity using special cases:
- For \(\theta = 0\ ^\circ\) (parallel forces): \(R = P + Q\).
- For \(\theta = 90\ ^\circ\) (perpendicular forces): \(R = \sqrt{P^2 + Q^2}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The Parallelogram Law of Forces states that if two concurrent forces acting on a particle are represented in magnitude and direction by two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through their point of intersection.
Key Formula or Approach:
Let \(P\) and \(Q\) be two forces acting at an angle \(\theta\).
By resolving the forces into perpendicular components along the direction of force \(P\):
- The horizontal component of the resultant is:
\[ \Sigma F_{\text{x}} = P + Q \cos \theta \]
- The vertical component of the resultant is:
\[ \Sigma F_{\text{y}} = Q \sin \theta \]
The magnitude of the resultant force \(R\) is calculated using the Pythagorean theorem:
\[ R = \sqrt{(\Sigma F_{\text{x}})^2 + (\Sigma F_{\text{y}})^2} \]

Step 2: Detailed Explanation:
Let us expand and simplify the expression for \(R\):
\[ R^2 = (P + Q \cos \theta)^2 + (Q \sin \theta)^2 \]
Using the algebraic identity \((a + b)^2 = a^2 + 2ab + b^2\):
\[ R^2 = P^2 + 2PQ \cos \theta + Q^2 \cos^2 \theta + Q^2 \sin^2 \theta \]
Factor out \(Q^2\) from the last two terms:
\[ R^2 = P^2 + 2PQ \cos \theta + Q^2 (\cos^2 \theta + \sin^2 \theta) \]
Since \(\cos^2 \theta + \sin^2 \theta = 1\):
\[ R^2 = P^2 + Q^2 + 2PQ \cos \theta \]
Taking the square root of both sides gives the magnitude of the resultant force:
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos \theta} \]

Step 3: Final Answer:
The resultant force \(R\) is given by \(\sqrt{P^2 + Q^2 + 2PQ \cos \theta}\). Hence, the correct option is (B).