A particle of mass m is taken from position A to position B along the circular path by a constant horizontal force `vecF`. What will be the work done by the force `vecF`? Here radius of Track is R.
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Options

A. a.FR
B. b.`F((piR)/(2))`
C. `F(sqrt(2)R)`
D. None

Answer

To find the work done by the constant horizontal force \( \vec{F} \) when moving a particle of mass \( m \) from position A to position B along a circular path of radius \( R \), we can follow these steps:

### Step 1: Understand the Motion
The particle moves along a circular path from point A to point B. The radius of the circular path is \( R \).

### Step 2: Identify the Displacement
As the particle moves from A to B, it rises vertically to a height equal to the radius \( R \). Therefore, the vertical displacement \( h \) is \( R \).

### Step 3: Determine the Work Done Formula
The work done \( W \) by a force is given by the formula:
\[ W = F \cdot s \cdot \cos(\theta) \]
where:
- \( F \) is the magnitude of the force,
- \( s \) is the displacement,
- \( \theta \) is the angle between the force and the direction of displacement.

### Step 4: Analyze the Angle
In this case, the force \( \vec{F} \) is applied horizontally, while the displacement in the vertical direction is \( R \). Therefore, the angle \( \theta \) between the force and the direction of displacement is \( 0^\circ \) (since the force is horizontal and the displacement is vertical).

### Step 5: Calculate the Work Done
Since the angle \( \theta = 0^\circ \), we have:
\[ \cos(0^\circ) = 1 \]
Thus, the work done can be simplified to:
\[ W = F \cdot R \cdot 1 = F \cdot R \]

### Conclusion
The work done by the force \( \vec{F} \) in moving the particle from position A to position B along the circular path is:
\[ W = F \cdot R \]

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