A car with its brakes locked will remain stationary on an inclined plane of dry concrete when the plane is at an angle of less than 45 degrees with the horizontal. What is the coefficients of static friction of rubber tires on dry concrete

Well, if the car's brakes are locked, it's not going anywhere anyway. So let's just say the coefficient of static friction between rubber tires and dry concrete is strong enough to keep the car from rolling down the incline and causing some unintentional roller derby. Let's call it "grippy enough to give Spider-Man a run for his money".

To determine the coefficient of static friction between rubber tires and dry concrete, we can use the given information about the angle of the inclined plane.

We know that the car remains stationary, indicating that the static friction force must be equal to or greater than the force pulling the car downhill due to gravity.

When the angle of the inclined plane is less than 45 degrees, the downward force component (pulling the car downhill) is given by:

Force_downhill = m * g * sin(angle)

Where:
m is the mass of the car
g is the acceleration due to gravity (approximately 9.8 m/s^2)
angle is the angle of the inclined plane

The maximum static friction force (Fs) between the tires and the concrete can be given by:

Fs = coefficient_of_static_friction * (m * g * cos(angle))

Where:
coefficient_of_static_friction is the unknown value we need to determine

Since the car remains stationary, Fs must be greater than or equal to Force_downhill:

Fs ≥ Force_downhill

Therefore, we can write the inequality:

coefficient_of_static_friction * (m * g * cos(angle)) ≥ m * g * sin(angle)

Simplifying the inequality:

coefficient_of_static_friction * cos(angle) ≥ sin(angle)

Now, we can solve for the coefficient_of_static_friction:

coefficient_of_static_friction ≥ sin(angle) / cos(angle)

cos(angle) = 1 / √(1 + tan^2(angle)) (from trigonometric identities)

sin(angle) / cos(angle) = sin(angle) * √(1 + tan^2(angle))

coefficient_of_static_friction ≥ sin(angle) * √(1 + tan^2(angle))

Therefore, the coefficient of static friction (μ) between rubber tires and dry concrete should be greater than or equal to sin(angle) * √(1 + tan^2(angle)) for the car to remain stationary on an inclined plane of dry concrete when the plane is at an angle of less than 45 degrees with the horizontal.

The coefficient of static friction (μs) is the measure of the friction between two surfaces when there is no relative motion between them. To find the coefficient of static friction of rubber tires on dry concrete, we can make use of the given condition that the car remains stationary on an inclined plane.

Here are the step-by-step instructions to find the coefficient of static friction:

1. Determine the angle of the inclined plane: We are given that the car remains stationary when the plane is at an angle of less than 45 degrees. Let's assume the angle of the inclined plane is θ.

2. Break down the forces: On an inclined plane, there are two main forces acting on the car: the force of gravity (mg) acting vertically downward and the normal force (N) acting perpendicular to the plane.

3. Calculate the components of the force of gravity: The force of gravity can be resolved into two components: one parallel to the plane (mg*sinθ) and the other perpendicular to the plane (mg*cosθ).

4. Find the maximum static friction force: The maximum static friction force (fs) can be calculated using the formula fs = μs * N, where μs is the coefficient of static friction and N is the normal force.

5. Determine the condition for the car to remain stationary: For the car to remain stationary, the component of the force of gravity parallel to the plane (mg*sinθ) must be balanced by the maximum static friction force (fs).

6. Equate the forces: Set up the equation mg*sinθ = μs * N and substitute the expression for N (which is equal to mg*cosθ) to get mg*sinθ = μs * mg*cosθ.

7. Solve for the coefficient of static friction: Simplify the equation to get sinθ = μs * cosθ and divide both sides of the equation by cosθ to obtain μs = sinθ/cosθ = tanθ.

8. Substitute the value of θ: Since we know that the angle (θ) is less than 45 degrees, we can substitute this angle in tanθ = μs to find the coefficient of static friction.

By following these steps, you can calculate the coefficient of static friction of rubber tires on dry concrete when a car with locked brakes remains stationary on an inclined plane.

forcefriction=forcegravity down street.

mu*mg*cosTheta=mg*sinTheta
mu=tanTheta