Two forces 12 N and 16 N are acting upon a body, what can be the maximum and minimum resultant force on the body ?

Answer

To find the maximum and minimum resultant force when two forces of 12 N and 16 N are acting upon a body, we can use the following steps:

### Step-by-Step Solution:

1. **Identify the Forces**:
- Let \( F_1 = 12 \, \text{N} \) and \( F_2 = 16 \, \text{N} \).

2. **Use the Formula for Resultant Force**:
The resultant force \( R \) when two forces are acting at an angle \( \theta \) is given by:
\[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} \]

3. **Determine Maximum Resultant Force**:
- The maximum resultant occurs when the forces are in the same direction, i.e., \( \theta = 0^\circ \).
- Substitute \( \theta = 0 \) into the formula:
\[ R_{\text{max}} = \sqrt{12^2 + 16^2 + 2 \cdot 12 \cdot 16 \cdot \cos(0)} \]
\[ R_{\text{max}} = \sqrt{12^2 + 16^2 + 2 \cdot 12 \cdot 16 \cdot 1} \]
\[ = \sqrt{144 + 256 + 384} \]
\[ = \sqrt{784} = 28 \, \text{N} \]

4. **Determine Minimum Resultant Force**:
- The minimum resultant occurs when the forces are in opposite directions, i.e., \( \theta = 180^\circ \).
- Substitute \( \theta = 180 \) into the formula:
\[ R_{\text{min}} = \sqrt{12^2 + 16^2 + 2 \cdot 12 \cdot 16 \cdot \cos(180)} \]
\[ R_{\text{min}} = \sqrt{12^2 + 16^2 + 2 \cdot 12 \cdot 16 \cdot (-1)} \]
\[ = \sqrt{144 + 256 - 384} \]
\[ = \sqrt{16} = 4 \, \text{N} \]

5. **Conclusion**:
- The maximum resultant force is \( 28 \, \text{N} \) when the forces are aligned (0 degrees).
- The minimum resultant force is \( 4 \, \text{N} \) when the forces are opposite (180 degrees).

### Final Answer:
- Maximum Resultant Force: \( 28 \, \text{N} \)
- Minimum Resultant Force: \( 4 \, \text{N} \)

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