Three forces P, Q, R are acting at a point in a plane. The angle between P and Q and Q and R are `150^(@) and 120^(@)` respectively, then for equilibrium, forces P, Q, R are in the ratio

Options

A. `1:2:3`
B. `1:2:3^(1//2)`
C. `3:2:1`
D. `3^(1//2):2:1`

Answer

To solve the problem of finding the ratio of the forces P, Q, and R acting at a point in a plane, we can use the law of sines. The angles between the forces are given as follows:

- The angle between forces P and Q is \(150^\circ\).
- The angle between forces Q and R is \(120^\circ\).

### Step-by-Step Solution:

1. **Identify the Angles**:
- Let the angle between forces P and Q be \( \theta_{PQ} = 150^\circ \).
- Let the angle between forces Q and R be \( \theta_{QR} = 120^\circ \).
- The angle between forces P and R can be calculated as:
\[ \theta_{PR} = 360^\circ - \theta_{PQ} - \theta_{QR} = 360^\circ - 150^\circ - 120^\circ = 90^\circ \]

2. **Apply the Law of Sines**:
According to the law of sines, for forces in equilibrium:
\[ \frac{P}{\sin(\theta_{PR})} = \frac{Q}{\sin(\theta_{RQ})} = \frac{R}{\sin(\theta_{PQ})} \]
Substituting the angles:
\[ \frac{P}{\sin(90^\circ)} = \frac{Q}{\sin(120^\circ)} = \frac{R}{\sin(150^\circ)} \]

3. **Calculate the Sine Values**:
- \( \sin(90^\circ) = 1 \)
- \( \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(150^\circ) = \sin(180^\circ - 30^\circ) = \sin(30^\circ) = \frac{1}{2} \)

4. **Set Up the Ratios**:
Now we can set up the ratios:
\[ \frac{P}{1} = \frac{Q}{\frac{\sqrt{3}}{2}} = \frac{R}{\frac{1}{2}} \]

5. **Expressing the Forces in Terms of a Common Variable**:
Let \( P = k \), then:
\[ Q = k \cdot \frac{\sqrt{3}}{2} \quad \text{and} \quad R = k \cdot \frac{1}{2} \]

6. **Finding the Ratio**:
Now we can express the ratio of the forces:
\[ P : Q : R = k : k \cdot \frac{\sqrt{3}}{2} : k \cdot \frac{1}{2} \]
Dividing through by \( k \):
\[ 1 : \frac{\sqrt{3}}{2} : \frac{1}{2} \]
To eliminate the fractions, multiply through by 2:
\[ 2 : \sqrt{3} : 1 \]

7. **Final Ratio**:
Thus, the final ratio of the forces P, Q, and R is:
\[ P : Q : R = 2 : \sqrt{3} : 1 \]

### Conclusion:
The forces P, Q, and R are in the ratio \( 2 : \sqrt{3} : 1 \).