Two forces are such that the sum of their magnitudes is 18N and their resultant is 12 N which is perpendicular to the smaller force. Then the magnitude of the forces are

Options

A. 12 N ,6 N
B. 13 N , 5 N
C. 10 N, 8 N
D. 16 N , 2 N

Explanation

To solve the problem, we need to find the magnitudes of two forces \( F_1 \) and \( F_2 \) given the following conditions:

1. The sum of their magnitudes is \( F_1 + F_2 = 18 \, \text{N} \).
2. The magnitude of their resultant is \( R = 12 \, \text{N} \).
3. The resultant \( R \) is perpendicular to the smaller force \( F_2 \).

### Step-by-Step Solution:

**Step 1: Set up the equations.**
We know from the problem that:
- \( F_1 + F_2 = 18 \, \text{N} \) (Equation 1)
- \( R = 12 \, \text{N} \)

Since \( R \) is perpendicular to \( F_2 \), we can use the Pythagorean theorem:
\[ R^2 = F_1^2 - F_2^2 \]
Substituting \( R \):
\[ 12^2 = F_1^2 - F_2^2 \]
This simplifies to:
\[ 144 = F_1^2 - F_2^2 \quad \text{(Equation 2)} \]

**Step 2: Express \( F_1 \) in terms of \( F_2 \).**
From Equation 1, we can express \( F_1 \):
\[ F_1 = 18 - F_2 \quad \text{(Equation 3)} \]

**Step 3: Substitute \( F_1 \) into Equation 2.**
Now, substitute Equation 3 into Equation 2:
\[ 144 = (18 - F_2)^2 - F_2^2 \]
Expanding the square:
\[ 144 = (324 - 36F_2 + F_2^2) - F_2^2 \]
This simplifies to:
\[ 144 = 324 - 36F_2 \]

**Step 4: Solve for \( F_2 \).**
Rearranging gives:
\[ 36F_2 = 324 - 144 \]
\[ 36F_2 = 180 \]
\[ F_2 = \frac{180}{36} = 5 \, \text{N} \]

**Step 5: Find \( F_1 \).**
Now, substitute \( F_2 \) back into Equation 3 to find \( F_1 \):
\[ F_1 = 18 - F_2 = 18 - 5 = 13 \, \text{N} \]

### Final Answer:
The magnitudes of the forces are:
- \( F_1 = 13 \, \text{N} \)
- \( F_2 = 5 \, \text{N} \)