Question

Two forces of magnitude A and \(\frac{A}{2}\) act perpendicular to each other. The magnitude of the resultant force is equal to:

• A. \(\frac{A}{2}\)
• B. \(\frac{\sqrt{5A}}{{2}}\)
• C. \(\frac{3A}{2}\)
• D. \(\frac{5A}{2}\)

Answer 1

The Correct Option is B

To find the magnitude of the resultant force when two forces act perpendicularly, we can use the Pythagorean theorem. Let's go through the solution step-by-step:

1. Identify the forces involved:
   • Force 1 (\(F_1\)) = \(A\)
   • Force 2 (\(F_2\)) = \(\frac{A}{2}\)
2. Use the Pythagorean theorem to calculate the resultant force:

When two forces acting at a right angle to each other, the magnitude of the resultant force (\(R\)) is given by:

1. \(R = \sqrt{F_1^2 + F_2^2}\)
2. Substitute the values of \(F_1\) and \(F_2\):

Here, \(F_1 = A\) and \(F_2 = \frac{A}{2}\), substitute these values into the formula:

1. \(R = \sqrt{A^2 + \left(\frac{A}{2}\right)^2}\)
2. Simplify the equation:

First, calculate \(\left(\frac{A}{2}\right)^2\):

1. \(= \frac{A^2}{4}\)

Now, substitute back to the equation:

1. \(R = \sqrt{A^2 + \frac{A^2}{4}}\)

Combine the terms:

1. \(R = \sqrt{\frac{4A^2}{4} + \frac{A^2}{4}}\)

This simplifies to:

1. \(R = \sqrt{\frac{5A^2}{4}}\)

Further simplify:

1. \(R = \frac{\sqrt{5A^2}}{2}\) \(R = \frac{\sqrt{5}A}{2}\)
2. Conclusion:

The magnitude of the resultant force is:

1. \(R = \frac{\sqrt{5}A}{2}\)

This matches the correct answer option: \(\frac{\sqrt{5A}}{2}\)