Three forces P, Q and R are acting on a particel in the plane, the angle between P and Q and that between Q and R are `150^(@)` and `120^(@)` respectively. Then for equilibrium, forces P, Q and R are in the ratio

Options

A. `1 : 2 : 3`
B. `1 : 2 : sqrt3`
C. `3 : 2 : 1`
D. `sqrt3 : 2 : 1`

Explanation

To solve the problem of finding the ratio of forces P, Q, and R acting on a particle in equilibrium, we can follow these steps:

### Step 1: Understand the Angles
We are given:
- The angle between forces P and Q is \(150^\circ\).
- The angle between forces Q and R is \(120^\circ\).

### Step 2: Set Up the Coordinate System
Assume that force P is acting along the positive x-axis. Therefore:
- Force P can be represented as \(P = P\hat{i}\).
- The angle between P and Q is \(150^\circ\), so we can represent Q in terms of its components:
\[ Q_x = Q \cos(150^\circ) = -Q \cos(30^\circ) = -Q \cdot \frac{\sqrt{3}}{2} \]
\[ Q_y = Q \sin(150^\circ) = Q \sin(30^\circ) = Q \cdot \frac{1}{2} \]

### Step 3: Represent Force R
The angle between Q and R is \(120^\circ\). Since Q is at \(150^\circ\) from P, R will be at:
\[ \text{Angle of R from P} = 150^\circ + 120^\circ = 270^\circ \]
Thus, R acts downward along the negative y-axis:
\[ R_x = 0 \]
\[ R_y = -R \]

### Step 4: Apply Equilibrium Conditions
For the system to be in equilibrium, the sum of the forces in both the x and y directions must equal zero.

**In the x-direction:**
\[ P + Q_x + R_x = 0 \]
Substituting the values:
\[ P - Q \cdot \frac{\sqrt{3}}{2} + 0 = 0 \quad \Rightarrow \quad P = Q \cdot \frac{\sqrt{3}}{2} \]

**In the y-direction:**
\[ Q_y + R_y = 0 \]
Substituting the values:
\[ Q \cdot \frac{1}{2} - R = 0 \quad \Rightarrow \quad R = Q \cdot \frac{1}{2} \]

### Step 5: Find the Ratios
Now we have:
1. \(P = Q \cdot \frac{\sqrt{3}}{2}\)
2. \(R = Q \cdot \frac{1}{2}\)

To find the ratio \(P : Q : R\):
\[ P : Q : R = \left(\frac{\sqrt{3}}{2}Q\right) : Q : \left(\frac{1}{2}Q\right) \]
Dividing through by \(Q\):
\[ P : Q : R = \frac{\sqrt{3}}{2} : 1 : \frac{1}{2} \]

### Step 6: Simplify the Ratio
To eliminate the fractions, multiply through by 2:
\[ P : Q : R = \sqrt{3} : 2 : 1 \]

### Final Answer
Thus, the ratio of forces P, Q, and R for equilibrium is:
\[ \boxed{\sqrt{3} : 2 : 1} \]