Friction provides the force needed for a car to travel around a flat, circular race track. What is the maximum speed at which a car can safely travel if the radius of the track is 82.0 m and the coefficient of friction is 0.39?

You must be refering to the coefficient of STATIC friction, Us, which determines the force at which slipping begins.

Solve this equation for Vmax:

M*Vmax^2/R = M*g*Us

Note that the mass M cancels out.

Vmax = sqrt(g*Us*R)

To determine the maximum speed at which a car can safely travel around a flat, circular race track, we can use the centripetal force formula. The centripetal force required to keep the car moving in a circle is equal to the product of the car's mass (m), the velocity of the car (v), and the acceleration due to friction (a).

The centripetal force (F) is given by the formula:

F = m * v^2 / r

Where:
F = Centripetal force
m = Mass of the car
v = Velocity of the car
r = Radius of the race track

In this case, we need to solve for the velocity (v).

To do so, we need to find the maximum value of the frictional force, which can be calculated using:

Frictional force (Ff) = coefficient of friction (μ) * Normal force (N)

The normal force (N) is the force exerted by the car on the track, which is equal and opposite to the force exerted by the track on the car.

N = m * g

Where:
N = Normal force
m = Mass of the car
g = Acceleration due to gravity

To determine the maximum value of the frictional force, we use:

Ff(max) = μ * N

Substituting the equations together, we have:

Ff(max) = μ * m * g

We can equate the frictional force to the centripetal force to find the maximum speed:

Ff(max) = m * v^2 / r

Now, we can substitute:

μ * m * g = m * v^2 / r

Canceling out the mass (m) and rearranging the equation, we can solve for v:

v^2 = μ * r * g

Taking the square root of both sides, we get:

v = √(μ * r * g)

Now, we can substitute the given values into the equation. The coefficient of friction (μ) is 0.39, and the radius (r) is 82.0 m. The acceleration due to gravity (g) is approximately 9.8 m/s^2.

v = √(0.39 * 82.0 * 9.8)

Calculating this expression, we find:

v ≈ 26.43 m/s

Therefore, the maximum speed at which the car can safely travel on the circular race track is approximately 26.43 m/s.