If the resultant of two forces of magnitudes P and Q acting at a point at an angle of `60^(@)` is `sqrt(7) Q` , find the value of ratio P/Q ?

Explanation

To solve the problem, we need to find the ratio \( \frac{P}{Q} \) given that the resultant of two forces \( P \) and \( Q \) acting at an angle of \( 60^\circ \) is \( \sqrt{7} Q \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two forces \( P \) and \( Q \) acting at an angle of \( 60^\circ \). The resultant \( R \) of these forces is given as \( R = \sqrt{7} Q \).

2. **Using the Formula for Resultant of Two Forces**:
The formula for the resultant \( R \) of two forces \( P \) and \( Q \) acting at an angle \( \theta \) is given by:
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos(\theta)} \]
Here, \( \theta = 60^\circ \), and \( \cos(60^\circ) = \frac{1}{2} \).

3. **Substituting Values into the Formula**:
Substitute \( R \) and \( \cos(60^\circ) \) into the resultant formula:
\[ \sqrt{7} Q = \sqrt{P^2 + Q^2 + 2PQ \cdot \frac{1}{2}} \]
This simplifies to:
\[ \sqrt{7} Q = \sqrt{P^2 + Q^2 + PQ} \]

4. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ 7Q^2 = P^2 + Q^2 + PQ \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ P^2 + Q^2 + PQ - 7Q^2 = 0 \]
This simplifies to:
\[ P^2 + PQ - 6Q^2 = 0 \]

6. **Letting \( y = \frac{P}{Q} \)**:
We can express \( P \) in terms of \( Q \):
\[ P = yQ \]
Substituting \( P \) in the equation gives:
\[ (yQ)^2 + (yQ)Q - 6Q^2 = 0 \]
This simplifies to:
\[ y^2Q^2 + yQ^2 - 6Q^2 = 0 \]

7. **Dividing by \( Q^2 \)**:
Assuming \( Q \neq 0 \), we can divide the entire equation by \( Q^2 \):
\[ y^2 + y - 6 = 0 \]

8. **Factoring the Quadratic Equation**:
The quadratic equation \( y^2 + y - 6 = 0 \) can be factored as:
\[ (y - 2)(y + 3) = 0 \]

9. **Finding the Roots**:
The roots of the equation are:
\[ y = 2 \quad \text{and} \quad y = -3 \]

10. **Selecting the Valid Root**:
Since \( y = \frac{P}{Q} \) must be positive, we reject \( y = -3 \). Therefore, we have:
\[ \frac{P}{Q} = 2 \]

### Final Answer:
The ratio \( \frac{P}{Q} \) is \( 2:1 \).