Question

Hello, I have this question that I have been trying to solve for days. I am always able to solve the first part of the question, but I am never able to get the second part. Please help me understand how to set up this problem so I am able to solve it!

Q: Let's revisit the banked curves from earlier to see another reason they are useful.We are building a road, and at one place we need to make a turn with a radius of 30m.  The coefficient of static friction between rubber and the material we are using is 0.6.

Part A: Assuming that the road is flat through the turn, what is the maximum speed a car could have and safely navigate the turn without sliding off the road?

Part B: Instead of keeping the road flat, let's bank the turn a little bit, giving the road an angle of 20∘ above the horizontal. With the banked turn, what is the maximum speed a car could have and still make it around the turn without sliding?

The answer for part A is 13.3 m/s and the answer for part B is 19 m/s but I don't know how to get the answer for part B.

Answer 1

There are two directions possible for static friction.

If the speed is less, there won't be enough centrifugal force to balance the component of gravity along the slope. In this case, friction needs to act up along the slope so as to keep the car from falling.

In the second case, as we have been asked here, the car has a maximum speed, just small enough to prevent it from being thrown away due to centrifugal force. In this case, the friction needs to help gravity keep the car in place by acting down the slope. Its value will be the maximum value static friction can reach, that is .

N is perpendicular to the slope, and will be equal to the external forces acting perpendicular to the slope.

This along with component of gravity must balance the component of centrifugal force parallel to the slope.