A boy travels along a circular path of radius r m. If the angle traced by the boy is `pi/3` radians, then his linear displacement is

Options

A. `rsqrt2` m
B. r m
C. `2sqrtr` m
D. `(pir)/3` m

Answer

To find the linear displacement of a boy traveling along a circular path of radius \( r \) meters, with an angle traced of \( \frac{\pi}{3} \) radians, we can follow these steps:

### Step 1: Understand the Concept of Linear Displacement
Linear displacement is the shortest distance between the initial and final position of the boy. When the boy travels along the circular path, we can visualize his movement from point A to point B on the circumference of the circle.

### Step 2: Identify the Angle and Radius
The angle \( \theta \) traced by the boy is given as \( \frac{\pi}{3} \) radians, and the radius \( r \) is given.

### Step 3: Use the Formula for Linear Displacement
The formula for linear displacement \( S \) when moving along a circular path is given by:
\[ S = 2R \sin\left(\frac{\theta}{2}\right) \]
where \( R \) is the radius of the circle and \( \theta \) is the angle in radians.

### Step 4: Substitute the Known Values
Substituting the values into the formula:
\[ S = 2r \sin\left(\frac{\frac{\pi}{3}}{2}\right) \]
This simplifies to:
\[ S = 2r \sin\left(\frac{\pi}{6}\right) \]

### Step 5: Calculate the Sine Value
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Therefore, substituting this value gives:
\[ S = 2r \cdot \frac{1}{2} \]

### Step 6: Simplify the Expression
This simplifies to:
\[ S = r \]

### Conclusion
Thus, the linear displacement of the boy traveling along the circular path is:
\[ \boxed{r} \text{ meters} \]