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 0.42. What is the radius of the shortest turn that the bicyclist can safely make?

To determine the radius of the shortest turn that the bicyclist can safely make, we need to consider the forces acting on the bicyclist as it goes around the curve.

First, let's identify the forces at play. The main force acting on the bicyclist is the force of friction between the tires and the road. In this case, since the bicycle is in motion, we'll be dealing with kinetic friction.

The equation for the frictional force (F_friction) can be expressed as:

F_friction = μ * N

where μ is the coefficient of friction and N is the normal force.

The normal force (N) can be defined as the force exerted by the road perpendicular to the tires, which balances the weight of the bicycle and the rider. In this case, since the road is flat and unbanked, the normal force is simply equal to the weight of the bicyclist (N = mg), where m is the mass of the bicyclist and g is the acceleration due to gravity.

Now let's calculate the radius using the equation for the maximum frictional force:

F_friction = m * a_c

where m is the mass of the bicyclist and a_c is the centripetal acceleration.

Given that the bicyclist is traveling at 8 m/s, we can calculate the centripetal acceleration using the equation:

a_c = v^2 / r

where v is the velocity of the bicyclist and r is the radius of the curve.

We also know that F_friction = μ * N = μ * m * g.

Combining these equations, we can solve for the radius (r):

μ * m * g = m * (v^2 / r)

Simplifying the equation:

μ * g = v^2 / r

Now, we can substitute the given values. The coefficient of friction (μ) is 0.42, the acceleration due to gravity (g) is approximately 9.8 m/s^2, and the velocity (v) is 8 m/s:

0.42 * 9.8 = (8^2) / r

Simplifying further:

4.116 = 64 / r

To find r, we can rearrange the equation:

r = 64 / 4.116

Calculating the result:

r ≈ 15.54 meters

Therefore, the radius of the shortest turn that the bicyclist can safely make is approximately 15.54 meters.