Question

The resultant of two forces with a magnitude of \( \frac{a}{4} \) acting at right angle is :

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

Hint:

For two equal forces \( F \) acting at \( 90^\circ \), the resultant is always \( \sqrt{2} F \).
Here, \( R = \sqrt{2} \left(\frac{a}{4}\right) = \frac{a}{2\sqrt{2}} \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
When two forces act on a body at an angle to each other, their combined effect can be represented by a single resultant force.
The resultant of two concurrent forces can be calculated using the parallelogram law of forces.
Key Formula or Approach:
The magnitude of the resultant force \( R \) of two forces \( F_1 \) and \( F_2 \) acting at an angle \( \theta \) is:
\[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta} \]

Step 2: Detailed Explanation:
Let the two forces be \( F_1 = \frac{a}{4} \) and \( F_2 = \frac{a}{4} \).
They are acting at a right angle, so \( \theta = 90^\circ \).
Since \( \cos 90^\circ = 0 \), the formula simplifies to:
\[ R = \sqrt{F_1^2 + F_2^2} \]
Substitute the values into the equation:
\[ R = \sqrt{\left(\frac{a}{4}\right)^2 + \left(\frac{a}{4}\right)^2} = \sqrt{2 \left(\frac{a}{4}\right)^2} \]
\[ R = \sqrt{2} \times \frac{a}{4} \]
We can simplify this expression by rationalizing the denominator:
\[ R = \frac{\sqrt{2} a}{2 \times 2} = \frac{\sqrt{2} a}{\sqrt{2} \times \sqrt{2} \times 2} = \frac{a}{2\sqrt{2}} \]
Therefore, the magnitude of the resultant force is \( \frac{a}{2\sqrt{2}} \).

Step 3: Final Answer:
The resultant of the two forces is \( \frac{a}{2\sqrt{2}} \).