Forces on an Inclined Plane Question

[catenn]

Hi I have a problem that reads: A car drives downhill on a road that is inclined 20 degrees to the horizontal. At a speed on 30 m/s the driver suddenly brakes until the car stops (coefficent of kinetic friction = .8, coefficent of static friction = .9).
A. When all of the wheels are locked, how far will the car skid over the road until it stops?
B. What is the steepest slope of a road on which a car can rest (with locked wheels) w/out slipping? Find the equation and maximum angle of the incline.

I began the problem and am not quite sure what to do. Once I drew the free body diagram I tried multiplying the 30 m/s by the .8 kinetic friction to get 24N of friction while the car still moves. It doesn't tell me how to find distance traveled though and there is no mass of the car given. I don't really know what to do with the coefficents and haven't used them yet in class. Also, for part B do we solve with an inverse cos or sin? Thanks.

 

[Nate810]

A. To find the distance the car will skid over the road until it stops, you need to use the equation for kinetic friction force (FK = μk*mg) and the equation for kinetic energy (K = ½mv^2). The mass of the car (m) can be calculated using the kinetic friction force equation, since you know the coefficient of kinetic friction (μk) and the acceleration due to gravity (g). Once you have the mass of the car, you can use the equation for kinetic energy to solve for the initial velocity (v) of the car before braking. With the initial velocity and the coefficient of kinetic friction, you can then calculate the total distance traveled by the car until it stops using the equation d = vt - (μk * g * t^2)/2.B. To find the maximum angle of the incline of a road on which a car can rest without slipping, you need to use the equation for static friction (FS = μs*mg). You know both the coefficient of static friction (μs) and the acceleration due to gravity (g), so you can calculate the maximum static friction force (FSmax) that the car can generate. With the maximum static friction force, you can then calculate the maximum angle of the incline (θ) using the equation θ = arctan (FSmax/ mg).

 

[kyle8921]

Hello,

Thank you for reaching out with your question. The problem you have described is a classic example of using forces on an inclined plane. I will provide some guidance and steps to help you solve this problem.

First, let's start with a free body diagram. This diagram will show all the forces acting on the car as it moves downhill on the inclined plane. The forces acting on the car are gravity (mg), normal force (N), and friction (F). Since the car is moving downhill, the force of gravity will be pulling the car down the slope. The normal force will be perpendicular to the surface of the incline and will counteract the force of gravity. Finally, the force of friction will act in the opposite direction of the car's motion to slow it down.

Next, we need to use Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F=ma). Since the car is moving at a constant speed, the acceleration is equal to zero, meaning that the net force acting on the car is also equal to zero. This allows us to set up an equation:

Fnet = ma = 0

We can then substitute in the forces acting on the car:

Fnet = mg - F - N = 0

We know that the force of friction (F) is equal to the coefficient of kinetic friction (μk) multiplied by the normal force (N). We also know that the normal force is equal to the force of gravity (mg) multiplied by the cosine of the angle of inclination (20 degrees in this case). We can then rewrite our equation as:

mg - μkN - N = 0

Now, we can solve for the distance traveled by the car by using the formula for work (W = Fd). We know that the work done by the force of friction (F) is equal to the force multiplied by the distance traveled. We can then set up an equation:

W = Fd = μkN * d

We can then substitute in our values for the force of friction (F), the normal force (N), and the distance traveled (d). This will give us an equation to solve for d, the distance traveled by the car.

For part B, we can use this same equation to find the steepest slope of a road on which a car can rest without slipping. We know that when a car is at