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

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Curve speed and incline angle significantly influence the downward force on a car, particularly in maintaining stability on an incline. The downward force can be expressed mathematically as mgsin(theta), where theta is the incline angle. As the car's speed increases while navigating a curve, the centripetal force required also increases, necessitating a larger incline angle to prevent sliding. If the gravitational force component acting towards the center of the curve matches the centripetal force, the car will remain stable. Understanding these dynamics is essential for analyzing the relationship between speed, incline, and the forces at play.
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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.
 

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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.
 
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?
 
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.
 

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