Two forces of magnitude 8 N and 15 N respectively act at a point . If the resultant force is 17 N , the angle between the forces has to be

Options

A. `60^(@)`
B. `45^(@)`
C. `90^(@)`
D. `30^(@)`

Answer

To find the angle between two forces of magnitudes 8 N and 15 N, given that their resultant is 17 N, we can use the formula for the resultant of two vectors. The formula is:

\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]

where:
- \( R \) is the resultant force,
- \( A \) and \( B \) are the magnitudes of the two forces,
- \( \theta \) is the angle between the two forces.

### Step-by-step Solution:

1. **Identify the given values**:
- Magnitude of the first force \( A = 8 \, \text{N} \)
- Magnitude of the second force \( B = 15 \, \text{N} \)
- Resultant force \( R = 17 \, \text{N} \)

2. **Substitute the values into the resultant formula**:
\[ 17 = \sqrt{8^2 + 15^2 + 2 \cdot 8 \cdot 15 \cdot \cos \theta} \]

3. **Square both sides to eliminate the square root**:
\[ 17^2 = 8^2 + 15^2 + 2 \cdot 8 \cdot 15 \cdot \cos \theta \]
\[ 289 = 64 + 225 + 240 \cos \theta \]

4. **Combine the constants on the right side**:
\[ 289 = 289 + 240 \cos \theta \]

5. **Rearrange the equation to isolate \( \cos \theta \)**:
\[ 289 - 289 = 240 \cos \theta \]
\[ 0 = 240 \cos \theta \]

6. **Solve for \( \cos \theta \)**:
\[ \cos \theta = 0 \]

7. **Determine the angle \( \theta \)**:
The angle \( \theta \) for which \( \cos \theta = 0 \) is:
\[ \theta = 90^\circ \]

### Final Answer:
The angle between the two forces is \( 90^\circ \).