If the greatest and the least resultants of two forces are P and Q, respectively, then the resultant of these forces, when acting at right angles, will be

Options

A. `sqrt""[(P^(2)-Q^(2))//2]`
B. `(P-Q)//2`
C. `(P+Q)//2`
D. `sqrt""(P^(2) +Q^(2))//2`

Answer

To solve the problem, we need to determine the resultant of two forces acting at right angles, given that the greatest resultant is \( P \) and the least resultant is \( Q \).

### Step-by-Step Solution:

1. **Understanding Resultants**:
- When two forces \( F_1 \) and \( F_2 \) act at an angle, the resultant \( R \) can be calculated using the formula:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta} \]
- For the case where the forces are acting at right angles, \( \theta = 90^\circ \) and \( \cos 90^\circ = 0 \). Thus, the formula simplifies to:
\[ R = \sqrt{F_1^2 + F_2^2} \]

2. **Identifying Forces**:
- The greatest resultant \( P \) occurs when the two forces are in the same direction:
\[ P = F_1 + F_2 \]
- The least resultant \( Q \) occurs when the two forces are in opposite directions:
\[ Q = |F_1 - F_2| \]

3. **Expressing Forces in Terms of \( P \) and \( Q \)**:
- From the equations for \( P \) and \( Q \), we can express \( F_1 \) and \( F_2 \):
- Let \( F_1 = a \) and \( F_2 = b \).
- Then, we have:
\[ a + b = P \quad \text{(1)} \]
\[ |a - b| = Q \quad \text{(2)} \]

4. **Solving for \( a \) and \( b \)**:
- From equation (2), we can write two cases:
- Case 1: \( a - b = Q \)
- Case 2: \( b - a = Q \)
- Solving Case 1:
- From \( a - b = Q \) and \( a + b = P \):
\[ 2a = P + Q \implies a = \frac{P + Q}{2} \]
\[ 2b = P - Q \implies b = \frac{P - Q}{2} \]
- Solving Case 2:
- From \( b - a = Q \) and \( a + b = P \):
\[ 2b = P + Q \implies b = \frac{P + Q}{2} \]
\[ 2a = P - Q \implies a = \frac{P - Q}{2} \]

5. **Calculating the Resultant at Right Angles**:
- Now substituting \( a \) and \( b \) back into the resultant formula:
\[ R = \sqrt{a^2 + b^2} \]
- Substituting \( a = \frac{P + Q}{2} \) and \( b = \frac{P - Q}{2} \):
\[ R = \sqrt{\left(\frac{P + Q}{2}\right)^2 + \left(\frac{P - Q}{2}\right)^2} \]
- Simplifying:
\[ R = \sqrt{\frac{(P + Q)^2 + (P - Q)^2}{4}} = \sqrt{\frac{P^2 + 2PQ + Q^2 + P^2 - 2PQ + Q^2}{4}} = \sqrt{\frac{2P^2 + 2Q^2}{4}} = \frac{1}{2} \sqrt{2(P^2 + Q^2)} \]

### Final Result:
Thus, the resultant of the two forces when acting at right angles is:
\[ R = \frac{1}{2} \sqrt{2(P^2 + Q^2)} \]