Two force in the ratio `1:2` act simultaneously on a particle. The resultant of these forces is three times the first force. The angle between them is

Options

A. `0^(@)`
B. `60^(@)`
C. `90^(@)`
D. `45^(@)`

Answer

To solve the problem, we need to determine the angle between two forces \( F_1 \) and \( F_2 \) that act simultaneously on a particle. We know the following:

1. The forces are in the ratio \( 1:2 \).
2. The resultant of these forces is three times the first force.

### Step-by-Step Solution:

**Step 1: Define the Forces**
Let \( F_1 = F \) (the first force) and \( F_2 = 2F \) (the second force, since it is twice the first force).

**Step 2: Define the Resultant Force**
According to the problem, the resultant force \( F_R \) is three times the first force:
\[ F_R = 3F \]

**Step 3: Use the Formula for Resultant of Two Forces**
The formula for the resultant \( F_R \) of two forces \( F_1 \) and \( F_2 \) acting at an angle \( \theta \) is given by:
\[ F_R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} \]

**Step 4: Substitute the Known Values**
Substituting \( F_1 = F \) and \( F_2 = 2F \) into the resultant formula:
\[ 3F = \sqrt{F^2 + (2F)^2 + 2 \cdot F \cdot 2F \cdot \cos \theta} \]
This simplifies to:
\[ 3F = \sqrt{F^2 + 4F^2 + 4F^2 \cos \theta} \]
\[ 3F = \sqrt{5F^2 + 4F^2 \cos \theta} \]

**Step 5: Square Both Sides**
Squaring both sides to eliminate the square root gives:
\[ (3F)^2 = 5F^2 + 4F^2 \cos \theta \]
\[ 9F^2 = 5F^2 + 4F^2 \cos \theta \]

**Step 6: Rearrange the Equation**
Rearranging the equation:
\[ 9F^2 - 5F^2 = 4F^2 \cos \theta \]
\[ 4F^2 = 4F^2 \cos \theta \]

**Step 7: Simplify**
Dividing both sides by \( 4F^2 \) (assuming \( F \neq 0 \)):
\[ 1 = \cos \theta \]

**Step 8: Find the Angle**
The value \( \cos \theta = 1 \) implies:
\[ \theta = 0^\circ \]

### Conclusion
The angle between the two forces \( F_1 \) and \( F_2 \) is \( 0^\circ \).