Banked curve angle w/no friction - teacher's work differs

[1MileCrash]

Homework Statement

At what angle should the roadway on a curve with a 50m radius be banked to allow cars to make the curve at 12 m/s even if friction is 0?

Homework Equations

The Attempt at a Solution

All of the centripetal acceleration comes from normal force from the road on the car.

I choose to represent the road as being along the y-axis and Fn as being along +x axis. Then gravity is directed at angle theta to the left of the -yaxis.

Normal force must equal the component of gravity along -xaxis, which is mgsin(theta),

Since the required centripetal force is mv^2/r, this is 2.88m, mgsin(theta) = 2.88m

gsin(theta) = 2.88

theta = 17.09

My professors work seems to be similar, but at the end she says that:
gtan(theta) = 2.88

theta = 16.37

why?

 

[BruceW]

Just to check I understand your axes correctly: so the instantaneous velocity of the car is in the z-direction? And the y-direction is like the direction a pedestrian would walk along to cross the road as quick as possible? And the x-direction is the direction of the normal force?

In this case, gravity will not be directed at angle theta from the y-axis. And the normal force should not equal the component of gravity along the x-axis. I think you should draw the diagram again. I would advise making one of the axis go opposite to the direction of gravity. (Although your method should work too, but I think it makes it a bit more complicated).

 

[ehild]

The Attempt at a Solution

All of the centripetal acceleration comes from normal force from the road on the car.

I choose to represent the road as being along the y-axis and Fn as being along +x axis. Then gravity is directed at angle theta to the left of the -yaxis.

Normal force must equal the component of gravity along -xaxis, which is mgsin(theta),

Since the required centripetal force is mv^2/r, this is 2.88m, mgsin(theta) = 2.88m

You calculated N instead of the centripetal force, which is Nsin(phi). The angle gravity encloses with the negative x-axis is not the banking angle. It is phi in the figure.

ehild