If the resultant of the two forces has a magnitude smaller than the magnitude of larger force ,then two forces must be

Options

A. Different both in magnitude and direction
B. Mutually perpendicular to one direction
C. Posses extremely small magnitude
D. Point in opposite directions

Answer

To solve the problem of determining the conditions under which the resultant of two forces is smaller than the magnitude of the larger force, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces**: Let \( F \) be the larger force and \( f \) be the smaller force. We assume \( F > f \).

2. **Use the Parallelogram Law of Forces**: According to the parallelogram law, the magnitude of the resultant \( R \) of two forces \( F \) and \( f \) acting at an angle \( \theta \) is given by:
\[ R = \sqrt{F^2 + f^2 + 2Ff \cos(\theta)} \]

3. **Set Up the Condition**: We need to find when the resultant \( R \) is smaller than the magnitude of the larger force \( F \):
\[ R < F \]

4. **Square Both Sides**: To eliminate the square root, we square both sides:
\[ R^2 < F^2 \]
This leads to:
\[ F^2 + f^2 + 2Ff \cos(\theta) < F^2 \]

5. **Simplify the Inequality**: Subtract \( F^2 \) from both sides:
\[ f^2 + 2Ff \cos(\theta) < 0 \]

6. **Analyze the Inequality**: The left side consists of two terms: \( f^2 \) (which is always positive) and \( 2Ff \cos(\theta) \). For the sum to be negative:
\[ 2Ff \cos(\theta) < -f^2 \]
This implies that \( \cos(\theta) \) must be negative, which occurs when:
\[ \theta > 90^\circ \text{ and } \theta < 270^\circ \]
This means the forces must be acting in opposite directions.

7. **Conclusion**: Therefore, for the resultant of the two forces to be smaller than the magnitude of the larger force, the two forces must be acting in opposite directions.