What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is 31 km/h and the coefficient of static friction between the tires and the road is 0.29?
To find the smallest radius of an unbanked track, we can first determine the maximum speed at which the bicyclist can travel around the track without slipping. This is the speed at which the maximum static friction force is equal to the centripetal force required to keep the bicyclist moving in a circle.
The maximum static friction force can be calculated using the formula: F_friction = μ * F_normal, where μ is the coefficient of static friction and F_normal is the normal force.
The normal force is equal to the weight of the bicyclist, which can be calculated as: F_normal = m * g, where m is the mass of the bicyclist and g is the acceleration due to gravity.
Once we have the maximum static friction force, we can equate it to the centripetal force to find the maximum speed. The centripetal force can be calculated using the formula: F_centripetal = m * v^2 / r, where m is the mass of the bicyclist, v is the speed, and r is the radius of the track.
Rearranging the equation, we can express the radius in terms of the mass, speed, and coefficients:
r = (m * v^2) / (μ * m * g)
Given:
v = 31 km/h (convert to m/s: 31 km/h * 1000 m/1 km * 1 h/3600 s = 31 * 1000 / 3600 m/s)
μ = 0.29 (coefficient of static friction)
The mass of the bicyclist and the acceleration due to gravity are not given in the question, so we cannot calculate the exact value of the radius without this information.
However, if we assume a typical mass for a bicyclist of 70 kg and an acceleration due to gravity of 9.8 m/s^2, we can substitute these values into the equation to calculate an approximate value for the radius:
r = (70 kg * (31 * 1000 / 3600 m/s)^2) / (0.29 * 70 kg * 9.8 m/s^2)
Simplifying the equation, we can find an approximate value for the radius.
To find the smallest radius of an unbanked (flat) track, we need to determine the maximum speed at which the bicyclist can travel without slipping.
The maximum speed without slipping can be found using the formula:
v = √(g * r * μs)
Where:
v = velocity (31 km/h in this case)
g = acceleration due to gravity (9.8 m/s²)
r = radius of the track
μs = coefficient of static friction (0.29 in this case)
Let's first convert the velocity from km/h to m/s:
v = 31 km/h = 31 * (1000 m / 3600 s) = 8.61 m/s
Substituting the given values into the formula, we have:
8.61 = √(9.8 * r * 0.29)
Now, let's solve for the radius (r):
8.61 = √(2.842 * r)
Squaring both sides of the equation to eliminate the square root, we get:
74.1021 = 2.842 * r
Dividing both sides by 2.842:
r = 74.1021 / 2.842 ≈ 26.01 meters
Therefore, the smallest radius of an unbanked track around which the bicyclist can travel at a speed of 31 km/h with a coefficient of static friction of 0.29 is approximately 26.01 meters.