Maximum and minimum velocity of a moving object on a bank angle

[Violagirl]

Homework Statement

A curve of radius 300 m is banked at an angle of 10°. A) At what speed is no friction required? B) If the coefficient of friction is 0.8, what are the maximum and minimum speeds at which the curve can be traveled?

Homework Equations

F=m*a
a_c=mv²/r

The Attempt at a Solution

For part A) I found the velocity needed for no friction is 18.9 m/s. It's B that I am stumped on. I have no idea if I'm even setting it up correctly as determining the angles and perpendicular forces that are related to one another has been problematic for me.

Here's what I've gotten so far:

F<sub>x[/SUB=max, ax=mv²/r

Fx=f cos 10°+N sin10°=ax=mv²/r

Fy=0

Fy=-N cos 10°-mg=0

f = μN

Is this the correct setup?

Also, in drawing out the free body diagram, I had a question regarding angle setup. In determining the x coordinate system, it was easy to visually see that friction had an angle of cos θ. In determining the angle for N however, I saw that the angle next to the x coordinate system by f, it corresponded to 90-θ. Since this angle of 90- is next to the angle of for f, someone told me it was opposite sin θ since cos θ was found for f and I did not fully understand that. Otherwise, what is next after setting it up? Any suggestions? Thank you for your time!</sub>

 

[anathema]

For part B, you are on the right track with your setup. You correctly identified the forces acting on the object (friction, normal force, and weight) and used the equations for Newton's second law and centripetal acceleration. However, there are a few things to consider when solving this problem.

First, you need to define the direction of positive and negative for your x and y coordinate systems. It is important to be consistent with this throughout your problem. In this case, you can define the positive x direction as pointing towards the center of the curve and the positive y direction as pointing perpendicular to the surface of the curve. This way, the negative y direction will be pointing towards the center of the circle.

Next, you can use the equations you have correctly set up to find the maximum and minimum speeds at which the curve can be traveled. To do this, you will need to use the fact that the net force in the x direction must equal zero for the object to remain on the curve. This means that the sum of the forces in the x direction (friction and the component of the normal force in the x direction) must equal zero.

You can also use the fact that the net force in the y direction must equal the centripetal force for the object to remain on the curve. This means that the sum of the forces in the y direction (the component of the normal force in the y direction and the weight) must equal the centripetal force.

Using these equations and the given coefficient of friction, you can solve for the maximum and minimum speeds at which the curve can be traveled.

In terms of the angles, it is important to remember that the angles of the forces are measured with respect to the positive x direction. So, for example, the angle for the component of the normal force in the x direction would be 90-10=80 degrees, as you correctly identified.

I hope this helps and good luck with your problem!