Banked Curve

Table Of Contents

Banked Curve

A banked curve looks something like such: It is incline at 3 axis, which makes this only truly visualized with 3D.

Visualization

How we tend to visualize this is by making 2 views.

Assumptions

• No rotated set of axis in FBDs

Banked Curve Without Friction(UCM)

Key Concepts

• Normal force is what allows the car to turn
• $F_{Norm}$ > $F_g$
• If you are undergoing UCM, you don’t need friction to turn

Banked Curve With Friction

The friction acting on the banked curve is static friction $F_s$ and uses ${\mu }_s$ Friction will only act to assist in turning if the car moves quicker or faster than $v_{ideal}$ which is the speed required for UCM ($v=\sqrt{rg\tan \theta }$)

Analysis

Banked Curve Bigger Velocity Proofs Banked Curve Smaller Velocity Proofs

Formulas

Angle Of Banking

$v>v_{ideal}$

$\frac{v^2}{rg}=\frac{\mu \cos \theta +\sin \theta }{\cos \theta -\mu \sin \theta }$

$v<v_{ideal}$

$\frac{v^2}{rg}=\frac{\sin \theta -\mu \cos \theta }{\cos \theta +\mu \sin \theta }$

UCM

where $\mu =0$, $\frac{v^2}{rg}=\tan \theta$

Acceleration

$a_c=g\tan \theta$

Velocity

$v=\sqrt{rg\tan \theta }$

Radius

$r=\frac{v^2}{g\tan \theta }$