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. (a)`1 : 2 : 3`
B. (b)`1 : 2 : sqrt(3)`
C. (c)`3:2:1`
D. (d)`sqrt(3) : 2 : 1`

Answer

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

### Step 1: Understand the Problem
We have three forces P, Q, and R acting on a particle. 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 2: Set Up the Forces
Assume the forces are represented in the plane. We can place force P along the positive x-axis for simplicity. Thus:
- Force P is along the x-axis.
- Force Q makes an angle of \(150^\circ\) with P.
- Force R makes an angle of \(120^\circ\) with Q.

### Step 3: Use the Law of Sines
In equilibrium, the sum of the forces in both the x and y directions must be zero. We can use the sine rule to relate the forces based on the angles between them.

1. **For the force Q:**
- The angle between P and Q is \(150^\circ\).
- The angle between Q and R is \(120^\circ\), which means the angle between P and R can be calculated as:
\[ \text{Angle between P and R} = 360^\circ - (150^\circ + 120^\circ) = 90^\circ \]

2. **Applying the Law of Sines:**
\[ \frac{P}{\sin(120^\circ)} = \frac{Q}{\sin(90^\circ)} = \frac{R}{\sin(150^\circ)} \]

### Step 4: Calculate the Sine Values
Using known sine values:
- \(\sin(120^\circ) = \frac{\sqrt{3}}{2}\)
- \(\sin(90^\circ) = 1\)
- \(\sin(150^\circ) = \frac{1}{2}\)

### Step 5: Set Up the Ratios
From the law of sines, we can set up the following equations:
1. \(\frac{P}{\frac{\sqrt{3}}{2}} = \frac{Q}{1}\)
2. \(\frac{Q}{1} = \frac{R}{\frac{1}{2}}\)

### Step 6: Express Forces in Terms of a Common Variable
From the first equation:
\[ Q = P \cdot \frac{2}{\sqrt{3}} \]

From the second equation:
\[ R = Q \cdot \frac{1}{2} = \left(P \cdot \frac{2}{\sqrt{3}}\right) \cdot \frac{1}{2} = \frac{P}{\sqrt{3}} \]

### Step 7: Find the Ratios
Now we have:
- \(P = P\)
- \(Q = \frac{2P}{\sqrt{3}}\)
- \(R = \frac{P}{\sqrt{3}}\)

Thus, the ratio of the forces P, Q, and R is:
\[ P : Q : R = P : \frac{2P}{\sqrt{3}} : \frac{P}{\sqrt{3}} \]
Dividing through by \(P\):
\[ 1 : \frac{2}{\sqrt{3}} : \frac{1}{\sqrt{3}} \]

To express this in a more standard form, we can multiply through by \(\sqrt{3}\):
\[ \sqrt{3} : 2 : 1 \]

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