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:sqrt(3)`
C. `3:2:1`
D. `sqrt(3):2:1`

Explanation

To solve the problem of finding the ratio of the forces P, Q, and R for equilibrium, we can use the concept of vector addition and the sine rule. Here’s a step-by-step solution:

### Step 1: Understand the Geometry of Forces
We have three forces P, Q, and R acting on a particle. The angles between the forces are given as follows:
- Angle between P and Q = 150°
- Angle between Q and R = 120°

### Step 2: Set Up the Vector Diagram
To analyze the forces, we can visualize them in a vector diagram. Place force P along the positive x-axis. Then, force Q will be at an angle of 150° from P, and force R will be at an angle of 120° from Q.

### Step 3: Use the Law of Sines
In a triangle formed by the forces, we can apply the Law of Sines. The angles in the triangle can be calculated as follows:
- The angle opposite to force P (let's call it α) can be calculated as:
α = 180° - 150° - 120° = -90° (which means we have a reflex angle, so we will take it as 90° for our calculations).

### Step 4: Apply the Law of Sines
According to the Law of Sines:
\[ \frac{P}{\sin(120°)} = \frac{Q}{\sin(90°)} = \frac{R}{\sin(150°)} \]

### Step 5: Calculate the Ratios
From the above relationships, we can express the forces in terms of a common ratio k:
- \( P = k \cdot \sin(120°) \)
- \( Q = k \cdot \sin(90°) \)
- \( R = k \cdot \sin(150°) \)

### Step 6: Calculate the Sine Values
Now we can calculate the sine values:
- \( \sin(120°) = \frac{\sqrt{3}}{2} \)
- \( \sin(90°) = 1 \)
- \( \sin(150°) = \frac{1}{2} \)

### Step 7: Substitute the Values
Substituting these values into the equations gives:
- \( P = k \cdot \frac{\sqrt{3}}{2} \)
- \( Q = k \cdot 1 \)
- \( R = k \cdot \frac{1}{2} \)

### Step 8: Find the Ratio
Now, we can find the ratio of P, Q, and R:
\[ \frac{P}{Q} : \frac{Q}{Q} : \frac{R}{Q} = \frac{\frac{\sqrt{3}}{2}}{1} : 1 : \frac{\frac{1}{2}}{1} \]
This simplifies to:
\[ \frac{\sqrt{3}}{2} : 1 : \frac{1}{2} \]

### Step 9: Clear the Fractions
To express this in a more standard form, we can multiply through by 2 to eliminate the fractions:
\[ \sqrt{3} : 2 : 1 \]

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