Understanding Displacement along a Circular Arc
When a particle moves along a circular arc, it covers a certain distance while also changing its position with respect to the center. The displacement is the shortest distance between the initial and final positions of the particle.
Key Concepts
- Circular Arc Definition: A circular arc is a segment of the circumference of a circle, defined by a radius (r) and an angle (theta).
- Displacement vs. Distance:
- Distance is the length of the path traveled along the arc.
- Displacement is a straight line connecting the start and end points.
Calculating Displacement
- Displacement Formula: For a circular arc with radius r and central angle theta in radians, the displacement (D) can be calculated using the chord length formula.
- D = 2 * r * sin(theta/2)
- Explanation of the Formula:
- The angle theta is halved because the displacement forms an isosceles triangle with the radius lines as equal sides.
- The sin function is used to find the length of the chord that directly connects the two endpoints of the arc.
Example Illustration
- If r = 5 units and theta = 60 degrees (which is pi/3 radians):
- D = 2 * 5 * sin(pi/6) = 2 * 5 * 0.5 = 5 units.
This means the particle's displacement is 5 units, regardless of the arc length it traversed.
Conclusion
Understanding the concepts of circular motion and displacement is crucial for various applications in physics and engineering, particularly in mechanics and kinematics.