If the line of action of the resultant of two forces P and Q divides the angle between them in the ratio `1:2` then the magnitude of the resultant is

Options

A. `(P^(2) + Q^(2))//P`
B. `(P^(2) + Q^(2))//Q`
C. `(P^(2)- Q^(2))//P`
D. `(P^(2)-Q^(2))//Q`

Answer

To find the magnitude of the resultant of two forces \( P \) and \( Q \) when the line of action of the resultant divides the angle between them in the ratio \( 1:2 \), we can follow these steps:

### Step 1: Understand the Geometry
Let the angle between the two forces \( P \) and \( Q \) be \( \theta \). According to the problem, the angle is divided in the ratio \( 1:2 \). This means that if we let the smaller angle be \( \alpha \), then the larger angle will be \( 2\alpha \). Therefore, we can express the total angle as:
\[ \theta = \alpha + 2\alpha = 3\alpha \]
This implies:
\[ \alpha = \frac{\theta}{3} \quad \text{and} \quad 2\alpha = \frac{2\theta}{3} \]

### Step 2: Apply the Law of Sines
The resultant \( R \) of two forces \( P \) and \( Q \) can be found using the law of sines in the triangle formed by the two forces and the resultant. The law of sines states:
\[ \frac{R}{\sin(\theta)} = \frac{P}{\sin(2\alpha)} = \frac{Q}{\sin(\alpha)} \]

### Step 3: Calculate the Sines
Using the values of \( \alpha \) and \( 2\alpha \):
- \( \sin(2\alpha) = \sin\left(\frac{2\theta}{3}\right) \)
- \( \sin(\alpha) = \sin\left(\frac{\theta}{3}\right) \)

### Step 4: Substitute into the Law of Sines
From the law of sines, we can express \( R \) as:
\[ R = \frac{P \cdot \sin\left(\frac{2\theta}{3}\right)}{\sin\theta} = \frac{Q \cdot \sin\left(\frac{\theta}{3}\right)}{\sin\theta} \]

### Step 5: Use the Resultant Formula
Using the law of cosines, we can also express the resultant \( R \) in terms of \( P \), \( Q \), and \( \theta \):
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos(\theta)} \]

### Step 6: Combine Results
To find the magnitude of the resultant, we can equate the two expressions we derived for \( R \) and simplify accordingly.

### Final Result
The magnitude of the resultant \( R \) can be calculated using the above relationships and substituting the known values of \( P \), \( Q \), and \( \theta \).