A resultant of 60 N is produced due to two forces having magnitudes in the ratio `4:5` Calculate the magnitude of each if the angle between them is `30^(@)`

Answer

To solve the problem, we need to find the magnitudes of two forces \( P \) and \( Q \) that produce a resultant force of 60 N when they are at an angle of \( 30^\circ \) to each other and are in the ratio \( 4:5 \).

### Step-by-Step Solution:

1. **Define the Forces**:
Let the magnitudes of the two forces be \( P \) and \( Q \). Given the ratio \( P:Q = 4:5 \), we can express \( Q \) in terms of \( P \):
\[ Q = \frac{5}{4}P \]

2. **Use the Law of Cosines**:
The magnitude of the resultant \( R \) of two vectors \( P \) and \( Q \) at an angle \( \theta \) is given by the formula:
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos(\theta)} \]
Here, \( R = 60 \, \text{N} \) and \( \theta = 30^\circ \).

3. **Substitute \( Q \) into the Equation**:
Substitute \( Q \) into the resultant formula:
\[ 60 = \sqrt{P^2 + \left(\frac{5}{4}P\right)^2 + 2P\left(\frac{5}{4}P\right) \cos(30^\circ)} \]

4. **Calculate \( \cos(30^\circ) \)**:
The value of \( \cos(30^\circ) \) is \( \frac{\sqrt{3}}{2} \). Substitute this value into the equation:
\[ 60 = \sqrt{P^2 + \left(\frac{25}{16}P^2\right) + 2P\left(\frac{5}{4}P\right) \cdot \frac{\sqrt{3}}{2}} \]

5. **Simplify the Equation**:
Simplifying the terms inside the square root:
\[ 60 = \sqrt{P^2 + \frac{25}{16}P^2 + \frac{5\sqrt{3}}{4}P^2} \]
Combine the terms:
\[ 60 = \sqrt{P^2 \left(1 + \frac{25}{16} + \frac{5\sqrt{3}}{4}\right)} \]

6. **Calculate the Coefficient**:
Convert \( \frac{5\sqrt{3}}{4} \) into a fraction with a common denominator:
\[ \frac{5\sqrt{3}}{4} = \frac{20\sqrt{3}}{16} \]
Thus, the expression becomes:
\[ 60 = \sqrt{P^2 \left(1 + \frac{25}{16} + \frac{20\sqrt{3}}{16}\right)} \]
Combine the terms:
\[ 60 = \sqrt{P^2 \left(\frac{16 + 25 + 20\sqrt{3}}{16}\right)} \]

7. **Square Both Sides**:
Squaring both sides gives:
\[ 3600 = P^2 \left(\frac{41 + 20\sqrt{3}}{16}\right) \]

8. **Solve for \( P^2 \)**:
Rearranging gives:
\[ P^2 = \frac{3600 \cdot 16}{41 + 20\sqrt{3}} \]

9. **Calculate \( P \)**:
Calculate \( P \) using a calculator:
\[ P \approx 27.6 \, \text{N} \]

10. **Calculate \( Q \)**:
Now substitute \( P \) back to find \( Q \):
\[ Q = \frac{5}{4}P = \frac{5}{4} \times 27.6 \approx 34.5 \, \text{N} \]

### Final Answer:
- Magnitude of \( P \) is approximately \( 27.6 \, \text{N} \)
- Magnitude of \( Q \) is approximately \( 34.5 \, \text{N} \)