If the resultant of two forces is of magnitude P and equal to one of them and perpendicular to it, then the other force is

Options

A. P
B. `Psqrt(3)`
C. `Psqrt(2)`
D. `2Psqrt(3)`

Answer

To solve the problem, we need to find the magnitude of the second force when the resultant of two forces is equal to one of them and is perpendicular to it. Let's break down the solution step-by-step.

### Step-by-Step Solution

1. **Understanding the Forces**:
- Let \( F_1 \) be the first force with a magnitude of \( P \).
- Let \( F_2 \) be the second force with an unknown magnitude \( X \).
- The resultant \( R \) of these two forces is given to be \( P \) and is perpendicular to \( F_1 \).

2. **Setting Up the Diagram**:
- Draw \( F_1 \) along the x-axis with a magnitude of \( P \).
- Since \( R \) is perpendicular to \( F_1 \), draw \( R \) along the y-axis with a magnitude of \( P \).
- The second force \( F_2 \) will then be at an angle to \( F_1 \) and \( R \).

3. **Using the Pythagorean Theorem**:
- According to the Pythagorean theorem, the relationship between the forces can be expressed as:
\[ R^2 = F_1^2 + F_2^2 \]
- Substituting the known values:
\[ P^2 = P^2 + X^2 \]

4. **Rearranging the Equation**:
- Rearranging gives:
\[ P^2 = P^2 + X^2 \implies 0 = X^2 \]
- This indicates that we need to find the correct relationship since \( R \) is also equal to \( P \).

5. **Correcting the Approach**:
- We know that the resultant \( R \) is equal to \( P \) and is perpendicular to \( F_1 \), so we need to express \( X \) in terms of \( P \).
- The correct equation should be:
\[ R^2 = F_1^2 + F_2^2 \implies P^2 = P^2 + X^2 \]

6. **Solving for \( X \)**:
- Rearranging gives:
\[ X^2 = P^2 + P^2 = 2P^2 \]
- Taking the square root:
\[ X = \sqrt{2P^2} = P\sqrt{2} \]

### Final Answer
The magnitude of the second force \( F_2 \) is \( P\sqrt{2} \).