Book edition 1st Edition

Author(s) Paul Peter Urone

Pages 1272 pages

ISBN 9781938168000

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Short Answer

Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force), and the vertical normal force (which must equal the system’s weight).
(a) Show that \(\theta \) (as defined in the figure) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway—that is, \(\theta = {\tan ^{ - 1}}\,{v^2}/rg\)
(b) Calculate \(\theta \) for a \(12.0{\rm{ m}}/{\rm{s}}\) turn of radius \(30.0{\rm{ m}}\) (as in a race).
Figure 6.36 A bicyclist negotiating a turn on level ground must lean at the correct angle—the ability to do this becomes instinctive. The force of the ground on the wheel needs to be on a line through the center of gravity. The net external force on the system is the centripetal force. The vertical component of the force on the wheel cancels the weight of the system while its horizontal component must supply the centripetal force. This process produces a relationship among the angle \(\theta \), the speed \(v\), and the radius of curvature \(r\) of the turn similar to that for the ideal banking of roadways.

The ideal banking angle is expressed as \(\theta = {\tan ^{ - 1}}\left( {\frac{{{v^2}}}{{rg}}} \right)\)[sb1] .
The ideal banking angle is \(26.1^\circ \).

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Bending of the cyclist

To make a safe turn, the cyclist bends slightly away from the vertical axis, providing the required centripetal force.

Step 2: Determining expression for the ideal banking angle

The free-body equation along the vertical direction is,

\(F\cos \theta = W\) \({\rm{(1}}{\rm{.1)}}\)

Here, \(W\) is the weight of the cyclist and \(\theta \) is the ideal banking angle.

The weight of the cyclist is,

\(W = mg\)

Here, \(m\) is the mass of the cyclist, and \(g\) is the acceleration due to gravity.

From equation \({\rm{(1}}{\rm{.1)}}\),

\(F\cos \theta = mg\) \({\rm{(1}}{\rm{.2)}}\)

The free-body equation along the horizontal direction is,

\(F\sin \theta = {F_c}\) \({\rm{(1}}{\rm{.3)}}\)

Here, \({F_c}\) is the centripetal force.

The centripetal force is,

\({F_c} = \frac{{m{v^2}}}{r}\)

Here, \(v\) is the velocity of the cyclist, and \(r\) is the radius of the circular path.

From equation \({\rm{(1}}{\rm{.3)}}\),

\(F\sin \theta = \frac{{m{v^2}}}{r}\) \({\rm{(1}}{\rm{.4)}}\)

Dividing equation \({\rm{(1}}{\rm{.4)}}\) by equation \({\rm{(1}}{\rm{.2)}}\),

\(\begin{aligned}{}\frac{{F\sin \theta }}{{F\cos \theta }} = \frac{{\frac{{m{v^2}}}{r}}}{{mg}}\\\tan \theta = \frac{{{v^2}}}{{rg}}\end{aligned}\)

Rearranging the above equation in order to get an expression for the ideal banking angle.

\(\theta = {\tan ^{ - 1}}\left( {\frac{{{v^2}}}{{rg}}} \right)\) \({\rm{(1}}{\rm{.5)}}\)

As a result, the ideal banking angle is expressed as \(\theta = {\tan ^{ - 1}}\left( {\frac{{{v^2}}}{{rg}}} \right)\).

Step 3: Calculating Ideal banking angle

The ideal banking angle of the cyclist can be calculated using equation \({\rm{(1}}{\rm{.5)}}\).

Substitute \(12.0{\rm{ m}}/{\rm{s}}\) for \(v\), \(30.0{\rm{ m}}\) for \(r\), and \(9.8{\rm{ m}}/{{\rm{s}}^2}\) for \(g\),

\(\begin{aligned}{}\theta = {\tan ^{ - 1}}\left( {\frac{{{{\left( {12.0{\rm{ m}}/{\rm{s}}} \right)}^2}}}{{\left( {30.0{\rm{ m}}} \right) \times \left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right)}}} \right)\\ = 26.1^\circ \end{aligned}\)

Hence, the ideal banking angle of the cyclist is \(26.1^\circ \).

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College Physics (Urone)

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Short Answer

Step by Step Solution

Step 1: Definition of Bending of the cyclist

Step 2: Determining expression for the ideal banking angle

Step 3: Calculating Ideal banking angle

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College Physics (Urone)

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Short Answer

Step by Step Solution

Step 1: Definition of Bending of the cyclist

Step 2: Determining expression for the ideal banking angle

Step 3: Calculating Ideal banking angle

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