If the angle between two forces increases, the magnitude of their resultant

Options

A. decreases
B. increases
C. remains unchanged
D. first decreases and then increases

Answer

To solve the problem of how the magnitude of the resultant force changes as the angle between two forces increases, we can follow these steps:

### Step 1: Understand the Resultant of Two Forces
When two forces, \( \vec{A} \) and \( \vec{B} \), act at an angle \( \theta \) to each other, the magnitude of the resultant force \( R \) can be calculated using the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
where \( A \) and \( B \) are the magnitudes of the two forces.

### Step 2: Analyze the Effect of Increasing the Angle
As the angle \( \theta \) increases from \( 0^\circ \) to \( 180^\circ \), the value of \( \cos \theta \) decreases. Specifically:
- At \( \theta = 0^\circ \), \( \cos 0 = 1 \) (forces are in the same direction).
- At \( \theta = 90^\circ \), \( \cos 90 = 0 \) (forces are perpendicular).
- At \( \theta = 180^\circ \), \( \cos 180 = -1 \) (forces are in opposite directions).

### Step 3: Substitute Values into the Resultant Formula
As \( \theta \) increases:
- When \( \theta = 0^\circ \):
\[ R = \sqrt{A^2 + B^2 + 2AB \cdot 1} = A + B \]
- When \( \theta = 90^\circ \):
\[ R = \sqrt{A^2 + B^2 + 2AB \cdot 0} = \sqrt{A^2 + B^2} \]
- When \( \theta = 180^\circ \):
\[ R = \sqrt{A^2 + B^2 + 2AB \cdot (-1)} = |A - B| \]

### Step 4: Conclusion
From the analysis, as \( \theta \) increases from \( 0^\circ \) to \( 180^\circ \), the resultant \( R \) decreases. Therefore, we can conclude that if the angle between two forces increases, the magnitude of their resultant decreases.

### Final Answer
The magnitude of the resultant decreases as the angle between the two forces increases.

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