A particle is moving on a circular path of radius r with uniform speed v. What is the displacement of the particle after it has described an angle of `60^(@)`?

Options

A. `rsqrt(2)`
B. `rsqrt(3)`
C. r
D. 2r

Explanation

To find the displacement of a particle moving along a circular path of radius \( r \) after it has described an angle of \( 60^\circ \), we can follow these steps:

### Step 1: Understand the Geometry
The particle starts at point A on the circumference of the circle and moves to point B after subtending an angle of \( 60^\circ \). The displacement is the straight-line distance between points A and B.

### Step 2: Draw the Circle and Points
Draw a circle with radius \( r \). Mark point A at the starting position and point B at the position after moving \( 60^\circ \) around the circle.

### Step 3: Use the Cosine Rule
To find the displacement \( x \) (the straight line distance between points A and B), we can use the cosine rule in triangle OAB, where O is the center of the circle:
\[ x^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cdot \cos(\theta) \]
Here, \( OA = OB = r \) (the radius of the circle) and \( \theta = 60^\circ \).

### Step 4: Substitute Values into the Cosine Rule
Substituting the values into the cosine rule:
\[ x^2 = r^2 + r^2 - 2 \cdot r \cdot r \cdot \cos(60^\circ) \]
Since \( \cos(60^\circ) = \frac{1}{2} \):
\[ x^2 = r^2 + r^2 - 2 \cdot r^2 \cdot \frac{1}{2} \]
\[ x^2 = r^2 + r^2 - r^2 \]
\[ x^2 = r^2 \]

### Step 5: Solve for Displacement
Taking the square root of both sides gives:
\[ x = r \]

### Conclusion
The displacement of the particle after it has described an angle of \( 60^\circ \) is \( r \).

### Final Answer
The displacement is \( r \).

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