The resultant of two forces at right angles is 13N. The minimum resultant of the two forces is 7 N. The forces are

Options

A. 20 N, 6 N
B. 10 N, 20 N
C. 5 N, 12 N
D. 8 N, 15 N

Answer

To solve the problem, we need to find the magnitudes of two forces \( F_1 \) and \( F_2 \) given that their resultant at right angles is 13 N and their minimum resultant is 7 N.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two forces \( F_1 \) and \( F_2 \).
- The resultant of these forces when they are at right angles (90 degrees) is given as 13 N.
- The minimum resultant (when the forces act in opposite directions, 180 degrees) is given as 7 N.

2. **Using the Right Angle Condition**:
- When the forces are at right angles, the resultant \( R \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{F_1^2 + F_2^2} \]
- According to the problem, \( R = 13 \, \text{N} \):
\[ \sqrt{F_1^2 + F_2^2} = 13 \]
- Squaring both sides:
\[ F_1^2 + F_2^2 = 169 \quad \text{(Equation 1)} \]

3. **Using the Minimum Resultant Condition**:
- When the forces are in opposite directions, the resultant \( R \) is given by:
\[ R = |F_1 - F_2| \]
- According to the problem, \( R = 7 \, \text{N} \):
\[ |F_1 - F_2| = 7 \]
- This gives us two cases:
\[ F_1 - F_2 = 7 \quad \text{(Equation 2)} \]
or
\[ F_2 - F_1 = 7 \quad \text{(not applicable since we assume \( F_1 > F_2 \))} \]

4. **Solving the Equations**:
- From Equation 2, we can express \( F_1 \) in terms of \( F_2 \):
\[ F_1 = F_2 + 7 \]
- Substitute \( F_1 \) in Equation 1:
\[ (F_2 + 7)^2 + F_2^2 = 169 \]
- Expanding the equation:
\[ F_2^2 + 14F_2 + 49 + F_2^2 = 169 \]
\[ 2F_2^2 + 14F_2 + 49 = 169 \]
- Rearranging gives:
\[ 2F_2^2 + 14F_2 - 120 = 0 \]
- Dividing the entire equation by 2:
\[ F_2^2 + 7F_2 - 60 = 0 \]

5. **Factoring the Quadratic Equation**:
- We can factor this equation:
\[ (F_2 + 12)(F_2 - 5) = 0 \]
- Thus, \( F_2 = 5 \, \text{N} \) (since \( F_2 = -12 \, \text{N} \) is not physically meaningful).

6. **Finding \( F_1 \)**:
- Substitute \( F_2 \) back to find \( F_1 \):
\[ F_1 = F_2 + 7 = 5 + 7 = 12 \, \text{N} \]

### Final Answer:
- The forces are \( F_1 = 12 \, \text{N} \) and \( F_2 = 5 \, \text{N} \).