Radius of Curvature

A bicyclist traveling at 8 m/s rides around an unbanked curve. The coefficient of friction (is this static or kinetic friction?) between the tires and the road is 0.36. What is the radius of the shortest turn that the bicyclist can safely make?

Please, answer
Thank you

force centripetal=force friction

mv^2/r=mg*mu
solve for r

To find the radius of the shortest turn that the bicyclist can safely make, we need to consider the forces acting on the bicycle. In this case, the friction between the tires and the road is the limiting factor, as it prevents the bicycle from sliding or skidding.

The friction force can be calculated using the coefficient of friction (μ) and the normal force (N). The normal force is the force exerted by the road on the bicycle and is equal to the weight of the bicycle since it is not accelerating vertically. The weight of the bicycle can be found using the formula:

Weight (W) = mass (m) * gravity (g)

The friction force (f) can be calculated using the formula:

f = μN

Now, let's break down the steps to get the answer:

Step 1: Calculate the weight of the bicycle.
Given:
Speed of the bicyclist (v) = 8 m/s
Acceleration due to gravity (g) = 9.8 m/s²

Using the formula: W = mg
Substituting the values, we get:
W = m * g = m * 9.8

Step 2: Calculate the friction force (f).
Given:
Coefficient of friction (μ) = 0.36

Using the formula: f = μN
Substituting the values, we get:
f = 0.36 * W = 0.36 * m * 9.8

Step 3: Calculate the normal force (N).
Since the bicyclist is not accelerating vertically, the normal force is equal to the weight of the bicycle (N = W).

Step 4: Equate the centrifugal force and the friction force.
The centrifugal force acting on the bicycle is given by the formula:

Centrifugal force (Fc) = (m * v²) / r

Where:
m = mass of the bicycle
v = velocity of the bicycle in m/s
r = radius of the turn

The friction force f must be equal to the centrifugal force Fc to prevent the bicycle from sliding. Therefore, we can equate the two:

f = Fc
0.36 * m * 9.8 = (m * v²) / r

Step 5: Solve for the radius (r).
Rearranging the equation, we can solve for r:

r = v² / (0.36 * 9.8)

Substituting the given values:
r = 8² / (0.36 * 9.8)

Calculating this expression will give you the radius of the shortest turn that the bicyclist can safely make.