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 solve the problem of finding 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:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Magnitude of the first force (F1) = 8 N
- Magnitude of the second force (F2) = 15 N
- Magnitude of the resultant force (R) = 17 N

2. **Use the Formula for Resultant of Two Forces**:
The formula for the resultant \( R \) of two forces \( F_1 \) and \( F_2 \) acting at an angle \( \theta \) is given by:
\[ R^2 = F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta \]

3. **Substitute the Known Values into the Formula**:
\[ 17^2 = 8^2 + 15^2 + 2 \cdot 8 \cdot 15 \cos \theta \]
Calculating the squares:
\[ 289 = 64 + 225 + 240 \cos \theta \]

4. **Combine Like Terms**:
\[ 289 = 289 + 240 \cos \theta \]

5. **Isolate the Cosine Term**:
Subtract 289 from both sides:
\[ 0 = 240 \cos \theta \]

6. **Solve for Cosine**:
Since \( 240 \cos \theta = 0 \), we find:
\[ \cos \theta = 0 \]

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

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

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