91Ó°ÊÓ

The coefficient of friction is defined as the ratio of frictional force to the normal reaction force. It is a scalar and dimensionless quantity with no units.

The ideal speed to take a safe turn is,

\(v = \sqrt {rg\tan \theta } \)

Here, \(r\)is the radius of the curved road, \(g\) is the acceleration due to gravity, and \(\theta \) is the ideal banking angle.

Substitute \(100{\rm{ m}}\) for \(r\), \(9.8{\rm{ m}}/{{\rm{s}}^2}\) for \(g\), and \(15.0^\circ \) for \(\theta \),

\(\begin{aligned}{}v = \sqrt {\left( {100{\rm{ m}}} \right) \times \left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right) \times \tan \left( {15.0^\circ } \right)} \\ = 16.2{\rm{ m}}/{\rm{s}}\end{aligned}\)

Hence, the ideal speed to take the turn is \(16.2{\rm{ m}}/{\rm{s}}\).

The coefficient of friction is,

\(\mu = \frac{{{v^2}}}{{rg}}\)

Substitute \(16.2{\rm{ m}}/{\rm{s}}\) for \(v\), \(100{\rm{ m}}\) for \(r\), \(9.8{\rm{ m}}/{{\rm{s}}^2}\) for \(g\),

\(\begin{aligned}{}\mu = \frac{{{{\left( {16.2{\rm{ m}}/{\rm{s}}} \right)}^2}}}{{\left( {100{\rm{ m}}} \right) \times \left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right)}}\\ = 0.27\end{aligned}\)

Hence, the minimum coefficient of friction is \(0.27\).