Which pair of the following forces will never give resultant force of `2 N`?

Options

A. 2 N and 2 N
B. 1 N and 1 N
C. 1 N and 3 N
D. 1 N and 4 N

Answer

To solve the problem of which pair of forces will never give a resultant force of 2 N, we need to analyze the possible resultant forces that can be obtained from different pairs of forces. The resultant force \( F_R \) can be determined using the following relationships:

1. The minimum resultant force is given by \( |F_1 - F_2| \).
2. The maximum resultant force is given by \( F_1 + F_2 \).

Thus, the resultant force can vary from \( |F_1 - F_2| \) to \( F_1 + F_2 \).

Now, let's analyze each pair of forces provided in the question:

### Step 1: Analyze the first pair of forces
Assume \( F_1 = 2 \, \text{N} \) and \( F_2 = 2 \, \text{N} \).
- Minimum resultant force:
\[ |F_1 - F_2| = |2 - 2| = 0 \, \text{N} \]
- Maximum resultant force:
\[ F_1 + F_2 = 2 + 2 = 4 \, \text{N} \]
- Range of resultant force: \( 0 \, \text{N} \) to \( 4 \, \text{N} \)

Since \( 2 \, \text{N} \) lies within this range, this pair can give a resultant force of \( 2 \, \text{N} \).

### Step 2: Analyze the second pair of forces
Assume \( F_1 = 1 \, \text{N} \) and \( F_2 = 1 \, \text{N} \).
- Minimum resultant force:
\[ |F_1 - F_2| = |1 - 1| = 0 \, \text{N} \]
- Maximum resultant force:
\[ F_1 + F_2 = 1 + 1 = 2 \, \text{N} \]
- Range of resultant force: \( 0 \, \text{N} \) to \( 2 \, \text{N} \)

Since \( 2 \, \text{N} \) is the maximum and lies within this range, this pair can also give a resultant force of \( 2 \, \text{N} \).

### Step 3: Analyze the third pair of forces
Assume \( F_1 = 3 \, \text{N} \) and \( F_2 = 1 \, \text{N} \).
- Minimum resultant force:
\[ |F_1 - F_2| = |3 - 1| = 2 \, \text{N} \]
- Maximum resultant force:
\[ F_1 + F_2 = 3 + 1 = 4 \, \text{N} \]
- Range of resultant force: \( 2 \, \text{N} \) to \( 4 \, \text{N} \)

Since \( 2 \, \text{N} \) is the minimum and lies within this range, this pair can also give a resultant force of \( 2 \, \text{N} \).

### Step 4: Analyze the fourth pair of forces
Assume \( F_1 = 1 \, \text{N} \) and \( F_2 = 4 \, \text{N} \).
- Minimum resultant force:
\[ |F_1 - F_2| = |1 - 4| = 3 \, \text{N} \]
- Maximum resultant force:
\[ F_1 + F_2 = 1 + 4 = 5 \, \text{N} \]
- Range of resultant force: \( 3 \, \text{N} \) to \( 5 \, \text{N} \)

Since \( 2 \, \text{N} \) does not lie within this range, this pair will never give a resultant force of \( 2 \, \text{N} \).

### Conclusion
The pair of forces that will never give a resultant force of \( 2 \, \text{N} \) is \( F_1 = 1 \, \text{N} \) and \( F_2 = 4 \, \text{N} \).

**Final Answer: D (1 N and 4 N)**