Final answer:
The smallest radius of an unbanked track for a bicyclist traveling at 19.5 km/h with a μs of 0.535 is calculated using the centripetal force and frictional force relationship, resulting in a minimum radius of approximately 5.56 meters.
Step-by-step explanation:
The smallest radius of an unbanked (flat) track around which a bicyclist can travel at a speed of 19.5 km/h with a coefficient of static friction (μs) between tires and track of 0.535, can be calculated using the centripetal force equation. The centripetal force needed to keep the cyclist on the track is provided by the frictional force, which can be expressed as F = μs × N, where N is the normal force which, in this case, is equal to the weight of the cyclist. Since the centripetal force is also equal to (m × v²)/r, where m is the mass of the cyclist, v is the velocity, and r is the radius of the curve, we can equate the two forces to solve for the radius r.
To find the radius, we'll use the following relationship:
- F = (m × v²)/r = μs × m × g
Where g is the acceleration due to gravity (9.8 m/s²). Canceling out the mass m from both sides of the equation and rearranging to solve for r:
First, we need to convert the speed from km/h to m/s:
- 19.5 km/h × (1000 m/km) × (1 h/3600 s) = 5.42 m/s
Now, plugging the values into the radius equation gives us:
- r = (5.42 m/s)²/(0.535 × 9.8 m/s²) ≈ 5.56 m
Therefore, the smallest radius of the track is approximately 5.56 meters.