Sum of magnitudes of the two forces acting at a point is `16 N` if their resultant is normal to the smaller forces and has a magnitude `8N` then the forces are .

Options

A. `5N, 10N`
B. `8N, 8N`
C. `4N,12N`
D. `2N,14N`

Answer

To solve the problem, we need to find the magnitudes of two forces, \( F_1 \) and \( F_2 \), given that their sum is \( 16 \, \text{N} \) and their resultant is \( 8 \, \text{N} \), normal to the smaller force.

### Step-by-Step Solution:

1. **Define the Forces:**
Let \( F_1 \) be the smaller force and \( F_2 \) be the larger force. According to the problem, we have:
\[ F_1 + F_2 = 16 \, \text{N} \quad \text{(Equation 1)} \]

2. **Understanding the Resultant:**
The resultant \( R \) of the two forces is given as \( 8 \, \text{N} \) and is normal to the smaller force \( F_1 \). This means that the angle between \( F_1 \) and \( R \) is \( 90^\circ \).

3. **Using the Resultant Formula:**
The formula for the resultant of two forces is given by:
\[ R^2 = F_1^2 + F_2^2 + 2F_1F_2 \cos(\theta) \]
Since \( R \) is normal to \( F_1 \), the angle \( \theta \) between \( F_1 \) and \( F_2 \) can be found using:
\[ \cos(\theta) = -\frac{F_1}{F_2} \]
Thus, we can substitute \( R = 8 \, \text{N} \) into the resultant formula:
\[ 8^2 = F_1^2 + F_2^2 + 2F_1F_2 \left(-\frac{F_1}{F_2}\right) \]
This simplifies to:
\[ 64 = F_1^2 + F_2^2 - 2F_1^2 \]
\[ 64 = F_2^2 - F_1^2 \quad \text{(Equation 2)} \]

4. **Substituting from Equation 1:**
From Equation 1, we can express \( F_2 \) in terms of \( F_1 \):
\[ F_2 = 16 - F_1 \]
Now substitute \( F_2 \) into Equation 2:
\[ 64 = (16 - F_1)^2 - F_1^2 \]

5. **Expanding and Simplifying:**
Expanding \( (16 - F_1)^2 \):
\[ 64 = (256 - 32F_1 + F_1^2) - F_1^2 \]
This simplifies to:
\[ 64 = 256 - 32F_1 \]
Rearranging gives:
\[ 32F_1 = 256 - 64 \]
\[ 32F_1 = 192 \]
\[ F_1 = 6 \, \text{N} \]

6. **Finding \( F_2 \):**
Now substitute \( F_1 \) back into Equation 1 to find \( F_2 \):
\[ F_2 = 16 - F_1 = 16 - 6 = 10 \, \text{N} \]

### Final Answer:
The magnitudes of the two forces are:
\[ F_1 = 6 \, \text{N}, \quad F_2 = 10 \, \text{N} \]