Circular Motion - Banked Curve

[Iser]

Homework Statement

Designing an on ramp for the 401 the engineer wants cars to be able to make the turn with a radius of 50 m while traveling 40km/hr in conditions with no friction. What angle must he bank the curve at to make this possible?

Homework Equations

F_R = (m)(a_R)
F_R = (m)(v²)/(r)

The Attempt at a Solution

I first started by drawing a diagram and noting the values I have for the question which are the velocity (40 km/h) and the radius (50 m). The value I we're looking for in the question is the angle the bank makes.

I first know F_N(Sin θ) = (m)(v²)/R

which I can then isolate F_N through which becomes

F_N = (m*g)/(cos θ)Since in the actual equation has no movement in the y-axis it is:
F_Nsinθ = m * v²/r

Then I sub in the F_N I got from the previous equation and sub it into this one getting:

(m*g/cos θ)(sin θ) = m * v²/r

which then simplifies into:

tan θ = v²/r*g

I then put in the original values I had into the equation and got 3.26 then I put 3.26 into the tan^-1 thing in the calculator and got 73^o degrees.

Is my answer and thought process correct?

 

[Doc Al]

I then put in the original values I had into the equation and got 3.26 then I put 3.26 into the tan^-1 thing in the calculator and got 73^o degrees.

Is my answer and thought process correct?

Your thought process is perfect but your answer is not. (Convert the speed to standard units. )

 

[Iser]

Your thought process is perfect but your answer is not. (Convert the speed to standard units. )

Converted 40 km/h to 11.111 m/s.

Now I input the following into the formula:

tan θ = (11.1 m/s)²/(50)(9.8)

Which equals 0.2514489...

I then input that into the tan^-1 function into the calculator and get:

14.11^o degrees

Is this correct?

 

[Doc Al]

Converted 40 km/h to 11.111 m/s.

Now I input the following into the formula:

tan θ = (11.1 m/s)²/(50)(9.8)

Which equals 0.2514489...

I then input that into the tan^-1 function into the calculator and get:

14.11^o degrees

Is this correct?

Looks good!