Change in Velocity on a Circular Path

Introduction:
When a cyclist moves on a circular path with a constant speed, the direction of its velocity changes continuously. This change in velocity is caused by the centripetal acceleration acting towards the center of the circle. To determine the change in velocity after the cyclist has described an angle of 30 degrees, we need to analyze the components of velocity and understand how they vary.

Components of Velocity:
The velocity of the cyclist can be broken down into two components: tangential velocity (Vt) and radial velocity (Vr).

1. Tangential Velocity (Vt):
- Tangential velocity is the component of velocity that is tangent to the circular path.
- It remains constant throughout the motion because the cyclist is moving with a constant speed.
- The magnitude of tangential velocity is given by Vt = v, where v is the constant speed of the cyclist.

2. Radial Velocity (Vr):
- Radial velocity is the component of velocity that is perpendicular to the circular path.
- It changes continuously as the cyclist moves along the circular path.
- The magnitude of radial velocity can be given by Vr = Rω, where R is the radius of the circular path and ω is the angular velocity.

Change in Velocity:
To find the change in velocity after the cyclist has described an angle of 30 degrees, we need to calculate the change in the radial velocity component.

1. Change in Radial Velocity (ΔVr):
- When the cyclist moves through an angle of 30 degrees, the change in the radial velocity component can be calculated using the formula ΔVr = Vr2 - Vr1.
- As the cyclist has a constant speed, the magnitude of tangential velocity (Vt) remains the same before and after the angle change.
- Since the radial velocity (Vr) is perpendicular to the circular path, it is also the same before and after the angle change.
- Therefore, the change in radial velocity is zero, i.e., ΔVr = 0.

2. Change in Velocity (ΔV):
- The change in velocity can be calculated using the formula ΔV = √(ΔVr^2 + ΔVt^2).
- As ΔVr = 0, the change in velocity simplifies to ΔV = ΔVt = Vt2 - Vt1.
- Since the magnitude of tangential velocity (Vt) remains constant, ΔVt = 0.
- Therefore, the change in velocity is also zero, i.e., ΔV = 0.

Conclusion:
The change in velocity of the cyclist after describing an angle of 30 degrees on a circular path with constant speed is zero. This implies that the magnitude and direction of the cyclist's velocity remain unchanged. Hence, the correct answer is (d) none.

B option