How do curve speed and incline angle affect the downward force on a car?

[NA19]

Homework Statement

The question is attached!

Homework Equations

mgsintheta = downward force on incline

The Attempt at a Solution

I was thinking that this has something to do with the downward force on the incline and making sure that the car doesn't slide down the incline. Changing the mass or curve speed limits may cause the car to slide down the incline but I'm not sure how this makes sense mathematically. And so, I'm trying to understand the mathematical explanation.

My answer key says the answer is C.

 

[CWatters]

1) The car of mass M is going around a curve of radius R with velocity V. What is the equation for the centripetal force required to make it turn?

2) What's the equation for the component of the force due to gravity acting towards the center of the curve?

If the force calculated in 2) matches that calculated in 1) then there is no tendency to slide down or up the slope.

Equate the two forces. Something will cancel.

 

[NA19]

So you would equate mgsintheta = mv^2/r?
And then since speed increased, you would also need to increase theta?

What is the conceptual explanation? Why would the angle need to be increased if the speed increases?

 

[CWatters]

Draw the free body diagram.

Work out the rough direction of the net force..

a) When the car is going very slowly or is stationary
b) When the car is going very fast

Have a think about what that means for the friction/grip required.

 

[Nickie]

C. The car's speed and the incline's angle affect the downward force and friction acting on the car, which can impact its ability to maintain traction and stay on the incline. As the speed increases, the downward force and friction also increase, making it easier for the car to stay on the incline. However, if the incline becomes too steep, the downward force and friction may not be enough to keep the car from sliding down. Therefore, it is important to consider the balance between speed and incline angle to ensure the car can safely navigate the curve. Mathematically, this can be represented by the equation mgsinθ = Ff, where m is the mass of the car, g is the acceleration due to gravity, θ is the incline angle, and Ff is the friction force. By manipulating this equation, we can determine the maximum speed the car can travel at a given incline angle without sliding down.