Maximum and minimum values of the resultant of two forces acting at a point are 7 N and 3N respectively. The smaller force will be equal to x N. Find x

Explanation

To solve the problem, we need to find the smaller force \( x \) given the maximum and minimum resultant forces of two forces acting at a point.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two forces \( F_1 \) and \( F_2 \).
- The maximum resultant \( R_{\text{max}} = F_1 + F_2 = 7 \, \text{N} \).
- The minimum resultant \( R_{\text{min}} = |F_1 - F_2| = 3 \, \text{N} \).
- We need to find the smaller force, which we denote as \( x \).

2. **Setting Up the Equations**:
- From the maximum resultant, we can write:
\[ F_1 + F_2 = 7 \quad \text{(1)} \]
- From the minimum resultant, we can write:
\[ |F_1 - F_2| = 3 \quad \text{(2)} \]
- Since we are looking for the smaller force, we can assume \( F_1 > F_2 \) (without loss of generality). Therefore, we can rewrite equation (2) as:
\[ F_1 - F_2 = 3 \quad \text{(3)} \]

3. **Solving the Equations**:
- Now we have two equations:
1. \( F_1 + F_2 = 7 \) (equation 1)
2. \( F_1 - F_2 = 3 \) (equation 3)
- We can add equations (1) and (3):
\[ (F_1 + F_2) + (F_1 - F_2) = 7 + 3 \]
\[ 2F_1 = 10 \]
\[ F_1 = 5 \, \text{N} \]

4. **Finding \( F_2 \)**:
- Now, substitute \( F_1 \) back into equation (1):
\[ 5 + F_2 = 7 \]
\[ F_2 = 7 - 5 = 2 \, \text{N} \]

5. **Conclusion**:
- The smaller force \( x = F_2 = 2 \, \text{N} \).

### Final Answer:
The smaller force \( x \) is \( 2 \, \text{N} \).

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