The greater and least resultant of two forces are 9 N and 5 N respectively. If they are applied at 60°. The magnitude of the resultant is

Options

A. 100N
B. 119N
C. `sqrt(119)` N
D. `sqrt(67)` N

Answer

To solve the problem, we need to find the magnitude of the resultant of two forces, \( F_1 \) and \( F_2 \), given that the greatest resultant is 9 N, the least resultant is 5 N, and the angle between the forces is 60°.

### Step-by-Step Solution:

1. **Define the Forces**:
Let \( F_1 \) be the greater force and \( F_2 \) be the lesser force. According to the problem, we have:
\[ F_1 + F_2 = 9 \, \text{N} \quad \text{(1)} \]
\[ F_1 - F_2 = 5 \, \text{N} \quad \text{(2)} \]

2. **Solve the Equations**:
To find \( F_1 \) and \( F_2 \), we can add equations (1) and (2):
\[ (F_1 + F_2) + (F_1 - F_2) = 9 + 5 \]
This simplifies to:
\[ 2F_1 = 14 \implies F_1 = 7 \, \text{N} \]

Now, substitute \( F_1 \) back into equation (1) to find \( F_2 \):
\[ 7 + F_2 = 9 \implies F_2 = 2 \, \text{N} \]

3. **Calculate the Resultant at 60°**:
Now that we have \( F_1 = 7 \, \text{N} \) and \( F_2 = 2 \, \text{N} \), we can calculate the resultant \( R \) when the angle \( \theta = 60° \):
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(60°)} \]
Substitute the values:
\[ R = \sqrt{7^2 + 2^2 + 2 \cdot 7 \cdot 2 \cdot \frac{1}{2}} \]
Simplifying further:
\[ R = \sqrt{49 + 4 + 14} \]
\[ R = \sqrt{67} \]

4. **Final Result**:
Therefore, the magnitude of the resultant is:
\[ R = \sqrt{67} \, \text{N} \]