The four pairs of force vectors are given, which pairs of force vectors cannot be added to give a resultant vector of magnitude 10N ?

Options

A. 2N,13N
B. 5N,16N
C. 7N,8N
D. 100N,105N

Answer

To solve the problem of determining which pairs of force vectors cannot be added to give a resultant vector of magnitude 10N, we will follow these steps:

### Step 1: Understand the Range of Resultant Forces
The resultant force \( F_R \) from two force vectors \( F_1 \) and \( F_2 \) can vary based on their magnitudes and the angle between them. The minimum and maximum resultant forces can be calculated using the following formulas:
- Maximum Resultant Force:
\[ F_{R_{max}} = F_1 + F_2 \]
- Minimum Resultant Force:
\[ F_{R_{min}} = |F_1 - F_2| \]

### Step 2: Analyze Each Pair of Forces
We will apply the above formulas to each of the given pairs of forces to find their minimum and maximum resultant forces.

#### Pair A: \( F_1 = 2N, F_2 = 13N \)
- Minimum:
\[ F_{R_{min}} = |2 - 13| = 11N \]
- Maximum:
\[ F_{R_{max}} = 2 + 13 = 15N \]
- Resultant Range: \( [11N, 15N] \)
- Since 10N is not in this range, this pair cannot give a resultant of 10N.

#### Pair B: \( F_1 = 5N, F_2 = 16N \)
- Minimum:
\[ F_{R_{min}} = |5 - 16| = 11N \]
- Maximum:
\[ F_{R_{max}} = 5 + 16 = 21N \]
- Resultant Range: \( [11N, 21N] \)
- Since 10N is not in this range, this pair cannot give a resultant of 10N.

#### Pair C: \( F_1 = 7N, F_2 = 8N \)
- Minimum:
\[ F_{R_{min}} = |7 - 8| = 1N \]
- Maximum:
\[ F_{R_{max}} = 7 + 8 = 15N \]
- Resultant Range: \( [1N, 15N] \)
- Since 10N is in this range, this pair can give a resultant of 10N.

#### Pair D: \( F_1 = 100N, F_2 = 105N \)
- Minimum:
\[ F_{R_{min}} = |100 - 105| = 5N \]
- Maximum:
\[ F_{R_{max}} = 100 + 105 = 205N \]
- Resultant Range: \( [5N, 205N] \)
- Since 10N is in this range, this pair can give a resultant of 10N.

### Step 3: Conclusion
The pairs of forces that cannot be added to give a resultant vector of magnitude 10N are:
- Pair A: \( (2N, 13N) \)
- Pair B: \( (5N, 16N) \)

### Final Answer
**The pairs of force vectors that cannot give a resultant vector of magnitude 10N are:**
- Pair A: \( (2N, 13N) \)
- Pair B: \( (5N, 16N) \)