Understanding the Problem
When analyzing the motion of a particle on a vertical circular path, the radius (r) plays a crucial role in determining the conditions under which the particle leaves the surface. The particle experiences gravitational forces and centripetal acceleration as it moves along the path.
Vertical Distance at the Highest Point
To find the vertical distance (h) between the highest point of the circular path and the point where the particle leaves the surface, we can apply the concepts of energy conservation and circular motion.
Key Factors to Consider
- Gravitational Force: The weight of the particle acts downwards and influences its motion.
- Centripetal Force: At the highest point, the centripetal force is provided by the gravitational force acting on the particle.
- Critical Speed: The particle will leave the surface when the centripetal force required to keep it in circular motion is greater than the gravitational force acting on it.
Condition for Leaving the Surface
- As the particle reaches the highest point, the centripetal force (mv²/r) must equal the gravitational force (mg) at the minimum speed for the particle to maintain contact.
- Setting these equal gives the critical speed, which is derived from the balance of forces.
Conclusion: Vertical Distance Calculation
The vertical distance (h) between the highest point and the point where the particle leaves can be expressed as:
- h = r/2
This indicates that the particle will leave the surface when it has fallen a vertical distance equal to half the radius of the circular path, demonstrating the relationship between radius and the dynamics of circular motion.