Unreasonable Results (a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked \({\bf{50}}.{\bf{0}}{\rm{ }}{\bf{m}}\) radius curve at\({\bf{30}}.{\bf{0}}{\rm{ }}{\bf{m}}/{\bf{s}}\). (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

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.

Radius of curve \({\rm{r = 50 m}}\)

Velocity of the car \({\rm{v = 30 m/s}}\)

The minimum coefficient of friction needed\({{\rm{\mu }}_{\rm{s}}}{\rm{ = ?}}\)

\({{\rm{\mu }}_{\rm{s}}}{\rm{ = }}{{\rm{v}}^{\rm{2}}}{\rm{/rg}}\)

Here, v is the velocity of the car, r is the radius of the curve and µ_s is the coefficient of the static friction.

The minimum coefficient of the friction needed is

\(\begin{aligned}{l}{{\rm{\mu }}_{\rm{s}}}{\rm{ = }}{{\rm{v}}^{\rm{2}}}{\rm{/rg}}\\{{\rm{\mu }}_{\rm{s}}}{\rm{ = 3}}{{\rm{0}}^{\rm{2}}}{\rm{/(50*9}}{\rm{.8)}}\\{{\rm{\mu }}_{\rm{s}}}{\rm{ = 900/490}}\\{{\rm{\mu }}_{\rm{s}}}{\rm{ = 1}}{\rm{.84}}\end{aligned}\)

For concrete/asphalt roads, the coefficient of static friction reaches a maximum of \({\rm{0}}{\rm{.90}}\). The result that we have obtained for the minimum coefficient of the friction is which is almost twice the normal value. This is unreasonable in part (a).

The statement ‘road is unbanked’ is unreasonable. Due to the opening of the road to the banks, the car will be disposed off. The velocity of the car should be less or the radius of the curve should be made greater.