Banked Curve With Friction

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Unit: Uniform Circular Motion and Gravitation

Prerequisites

Banked Curves Inclines with Friction

Later Topics

None

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Part 1

A highway curve has a radius of $R = 80\,\mathrm{m}$ and is banked at an angle of $\theta = 21^\circ$ .

Derive the design speed (in both m/s and km/h) for this banked curve, assuming no friction is needed. It may be helpful to draw a free body diagram.

Correct!

Solution:

Solution

For a banked curve with no friction, the horizontal component of the normal force provides the centripetal force. We construct a free-body diagram with the weight $mg$ pointing downward and the normal force $N$ perpendicular to the banked surface.

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Resolving forces perpendicular to the road (no acceleration in this direction):

N \cos \theta = mg

Resolving forces parallel to the road (horizontal, toward the center of the curve):

N \sin \theta = \frac{m v^2}{R}

Dividing the second equation by the first:

\frac{N \sin \theta}{N \cos \theta} = \frac{m v^2 / R}{mg}

\tan \theta = \frac{v^2}{gR}

Solving for the design speed:

v_{\text{design}} = \sqrt{gR \tan \theta}

Substituting $R = 80\,\mathrm{m}$ , $\theta = 21^\circ$ , and $g = 9.81\,\mathrm{m/s^2}$ :

v_{\text{design}} = \sqrt{9.81 \cdot 80 \cdot \tan(21^\circ)} = 17.36\,\mathrm{m/s}

Converting to km/h by multiplying by

$\frac{60s}{1min}\times\frac{60min}{1h}\times\frac{1km}{1000m}=\frac{60\cdot 60}{1000}=\frac{3600}{1000}=3.6$

v_{\text{design}} = 17.36 \cdot 3.6 = 62.48\,\mathrm{km/h}

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