Understanding the Problem
The cyclist is navigating a circular path with a radius of 50 meters while leaning at an angle of 30 degrees from the vertical. To maintain balance and avoid falling, the cyclist must achieve a certain speed.
Forces Acting on the Cyclist
- Gravitational Force (Weight): Acts vertically downward.
- Normal Force: Acts perpendicular to the surface.
- Centripetal Force: Required to keep the cyclist moving in a circular path, directed towards the center of the circle.
Components of Forces
- The gravitational force can be resolved into two components:
- Vertical Component: Balances the weight of the cyclist.
- Horizontal Component: Provides the necessary centripetal force.
Calculating Centripetal Force
- The cyclist leans at 30 degrees, creating a balance between gravitational force and centripetal force.
- The centripetal force can be expressed as:
- F_c = m * v^2 / r, where:
- m = mass of the cyclist
- v = speed of the cyclist
- r = radius of the circular path (50 m)
Finding the Speed
- The angle of lean (30 degrees) affects the ratio of forces:
- tan(30) = F_c / Weight, leading to:
- F_c = m * g * sin(30), where g is the acceleration due to gravity (9.81 m/s^2).
- Using the relationship:
- m * (v^2 / r) = m * g * sin(30)
- Simplifying gives:
- v^2 = g * r * sin(30)
- Substituting values:
- v^2 = 9.81 * 50 * 0.5
- v ≈ 7.0 m/s
Conclusion
The cyclist must achieve a speed of approximately 7.0 m/s to successfully navigate the turn while leaning at an angle of 30 degrees from the vertical.