Suggested languages for you:

Americas

Europe

Q6.3-25 PE

Expert-verified
Found in: Page 222

College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

What is the ideal banking angle for a gentle turn of $$1.20{\rm{ km}}$$ radius on a highway with a $$105{\rm{ km}}/{\rm{h}}$$ speed limit (about $$65{\rm{ mi}}/{\rm{h}}$$), assuming everyone travels at the limit?

The ideal working angle for gentle rotation is $${\rm{4}}{\rm{.14^\circ }}$$.

See the step by step solution

Step 1: Definition of Banking of road

To supply the centre gravity force required for a vehicle to perform a safe turn, the outside edge of curved roadways is raised above the inner edge. This is referred to as road banking.

Step 2: Calculating the angle of banking

For a gentle turn, the optimal banking angle is

$$\theta = {\tan ^{ - 1}}\left( {\frac{{{v^2}}}{{Rg}}} \right)$$

where $$v$$ is the vehicle's velocity, $$R$$ is the radius of the curved turn, and $$g$$ is the acceleration due to gravity.

Substitute $$105{\rm{ km}}/{\rm{h}}$$ for $$v$$, $$1.20{\rm{ km}}$$ for $$R$$, and $$9.8{\rm{ m}}/{{\rm{s}}^2}$$ for $$g$$,

\begin{aligned}{c}\theta = {\tan ^{ - 1}}\left[ {\frac{{{{\left( {105{\rm{ km}}/{\rm{h}}} \right)}^2}}}{{\left( {1.20{\rm{ km}}} \right) \times \left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right)}}} \right]\\ = {\tan ^{ - 1}}\left[ {\frac{{{{\left[ {\left( {105{\rm{ km}}/{\rm{h}}} \right) \times \left( {\frac{{{{10}^3}{\rm{ m}}}}{{1{\rm{ km}}}}} \right) \times \left( {\frac{{3600{\rm{ s}}}}{{1{\rm{ hr}}}}} \right)} \right]}^2}}}{{\left( {1.20{\rm{ km}}} \right) \times \left( {\frac{{{{10}^3}{\rm{ m}}}}{{1{\rm{ km}}}}} \right) \times \left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right)}}} \right]\\ = 4.14^\circ \end{aligned}

As a result, the ideal working angle for gentle rotation is $$4.14^\circ$$.