The maximum and minimum magnitudes of the resultant of two forces are 5 N and 5N respectively. Find the magnitude of resultant force when act orthogonally to each other.

Answer

To solve the problem, we need to find the magnitude of the resultant force when two forces act orthogonally (at a right angle) to each other. The maximum and minimum magnitudes of the resultant are both given as 5 N.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have two forces, \( F_1 \) and \( F_2 \).
- The maximum resultant \( R_{\text{max}} \) is given as 5 N.
- The minimum resultant \( R_{\text{min}} \) is also given as 5 N.

2. **Using the Resultant Force Formula:**
- The resultant \( R \) of two forces can be expressed as:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta} \]
- Where \( \theta \) is the angle between the two forces.

3. **Identifying Maximum and Minimum Resultants:**
- The maximum resultant occurs when the forces are in the same direction:
\[ R_{\text{max}} = F_1 + F_2 \]
- The minimum resultant occurs when the forces are in opposite directions:
\[ R_{\text{min}} = |F_1 - F_2| \]

4. **Setting Up Equations:**
- From the problem, we know:
\[ F_1 + F_2 = 5 \quad \text{(1)} \]
\[ |F_1 - F_2| = 5 \quad \text{(2)} \]

5. **Solving the Equations:**
- From equation (1), we can express \( F_2 \) in terms of \( F_1 \):
\[ F_2 = 5 - F_1 \]
- Substitute \( F_2 \) into equation (2):
\[ |F_1 - (5 - F_1)| = 5 \]
\[ |2F_1 - 5| = 5 \]
- This gives us two cases to solve:
- Case 1: \( 2F_1 - 5 = 5 \)
\[ 2F_1 = 10 \implies F_1 = 5 \]
- Case 2: \( 2F_1 - 5 = -5 \)
\[ 2F_1 = 0 \implies F_1 = 0 \]

6. **Finding \( F_2 \):**
- For \( F_1 = 5 \):
\[ F_2 = 5 - 5 = 0 \]
- For \( F_1 = 0 \):
\[ F_2 = 5 - 0 = 5 \]

7. **Resultant Force When Acting Orthogonally:**
- We need to find the resultant when \( F_1 = 5 \, \text{N} \) and \( F_2 = 0 \, \text{N} \) acting at \( 90^\circ \):
\[ R = \sqrt{F_1^2 + F_2^2} \]
- Substitute the values:
\[ R = \sqrt{5^2 + 0^2} = \sqrt{25} = 5 \, \text{N} \]

### Final Answer:
The magnitude of the resultant force when the two forces act orthogonally is **5 N**.