Two forces whose magnitudes are in the ratio `3:5` give a resultant of `35N`.If the angle between them is `60^(@)`, the magnitude of each force is

Options

A. `3N, 5N`
B. `9N, 25N`
C. `15N, 25N`
D. `21N, 35N`

Explanation

To solve the problem, we need to find the magnitudes of two forces \( P \) and \( Q \) that are in the ratio \( 3:5 \) and produce a resultant force of \( 35N \) when acting at an angle of \( 60^\circ \) to each other.

### Step-by-Step Solution:

1. **Define the Forces**:
Let the magnitudes of the forces \( P \) and \( Q \) be represented as:
\[ P = 3x \quad \text{and} \quad Q = 5x \]
where \( x \) is a common multiplier.

2. **Use the Resultant Force Formula**:
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, \( R = 35N \) and \( \theta = 60^\circ \).

3. **Substituting the Values**:
Substitute \( P \), \( Q \), and \( \cos 60^\circ = \frac{1}{2} \) into the resultant formula:
\[ 35 = \sqrt{(3x)^2 + (5x)^2 + 2(3x)(5x) \cdot \frac{1}{2}} \]

4. **Simplifying the Equation**:
Calculate \( (3x)^2 \) and \( (5x)^2 \):
\[ (3x)^2 = 9x^2 \quad \text{and} \quad (5x)^2 = 25x^2 \]
Thus, the equation becomes:
\[ 35 = \sqrt{9x^2 + 25x^2 + 15x^2} \]
Combine the terms:
\[ 35 = \sqrt{49x^2} \]

5. **Solving for \( x \)**:
Taking the square root:
\[ 35 = 7x \]
Therefore, solving for \( x \):
\[ x = \frac{35}{7} = 5 \]

6. **Finding the Magnitudes of Each Force**:
Now substitute \( x \) back to find \( P \) and \( Q \):
\[ P = 3x = 3 \times 5 = 15N \]
\[ Q = 5x = 5 \times 5 = 25N \]

### Final Answer:
The magnitudes of the forces are:
- \( P = 15N \)
- \( Q = 25N \)