The resultant of two forces acting at an angle of `120^(@)` is 10 N. If one of the force is 10 N. The other force is

Options

A. `20//sqrt(3)` N
B. `10sqrt(3)` N
C. 10 N
D. `20sqrt(3)` N

Answer

To solve the problem, we will use the formula for the resultant of two forces acting at an angle. Let's denote the two forces as \( F_1 \) and \( F_2 \), where \( F_1 = 10 \, \text{N} \) and \( F_2 \) is the unknown force we need to find. The angle between the two forces is \( \theta = 120^\circ \), and the resultant \( R \) is given as \( 10 \, \text{N} \).

### Step-by-step Solution:

1. **Write the formula for the resultant of two forces:**
\[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} \]

2. **Substitute the known values into the formula:**
\[ 10 = \sqrt{10^2 + F_2^2 + 2 \cdot 10 \cdot F_2 \cdot \cos(120^\circ)} \]

3. **Calculate \( \cos(120^\circ) \):**
\[ \cos(120^\circ) = -\frac{1}{2} \]

4. **Substitute \( \cos(120^\circ) \) into the equation:**
\[ 10 = \sqrt{100 + F_2^2 + 2 \cdot 10 \cdot F_2 \cdot \left(-\frac{1}{2}\right)} \]
This simplifies to:
\[ 10 = \sqrt{100 + F_2^2 - 10 F_2} \]

5. **Square both sides to eliminate the square root:**
\[ 100 = 100 + F_2^2 - 10 F_2 \]

6. **Rearrange the equation:**
\[ 0 = F_2^2 - 10 F_2 \]

7. **Factor the equation:**
\[ F_2(F_2 - 10) = 0 \]

8. **Solve for \( F_2 \):**
This gives us two solutions:
\[ F_2 = 0 \quad \text{or} \quad F_2 = 10 \, \text{N} \]

9. **Select the valid solution:**
Since a force of 0 N does not make sense in this context, we conclude:
\[ F_2 = 10 \, \text{N} \]

### Final Answer:
The other force \( F_2 \) is \( 10 \, \text{N} \).