Answer
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Hint: We can use the idea for a circular motion some force to be providing as a centripetal force to act towards the centre.
We can consider all forces acting on the car which can provide sufficient centripetal force.
Complete step by step answer:
We know the banking of roads \[-\] To avoid risk of skidding the road surface at a bend is tatted inwards, i.e. the outer side of road is raised above its inner ride. This is called ‘Banking of roads’
We can consider taking a left turn along a road or banked at an angle $\theta $ for a designed optimum speed V. Let M be the man of the car. In general, the force is aching on the car one.
(a) Its weight$\overrightarrow {mg} $, acting vertically drawn.
(b) The functional reactor of the road$\overrightarrow N $, perpendicular to the road surface.
(c) The frictional force ${\overrightarrow f _1}$ along the inclined surface of the road.
Resolve $\overrightarrow N $ and $\overrightarrow {{f_1}} $ into two perpendicular components.
$N\cos \theta $ Vertically up and ${\overrightarrow f _s}\sin \theta $ vertically down, $N\sin \theta $ and ${\overrightarrow f _s}\cos \theta $ horizontally towards the center of the circular path.
If ${V_{\max }}$ is the maximum safe speed without skidding, then
$
\dfrac{{mv_{\max }^2}}{t} = N\sin \theta + {f_s}\cos \theta \\
= N\sin \theta + {\mu _s}N\cos \theta \\
$
$\dfrac{{mv_{\max }^2}}{{{t_0}}} = N\left( {\sin \theta + {\mu _s}\cos \theta } \right)...(1)$
And,
$
N\cos \theta = mg + {f_s}\sin \theta \\
= mg + {\mu _s}N\sin \theta \\
mg = N\left( {\cos \theta - {\mu _s}\sin \theta } \right)...(2) \\
$
Dividing ($1$) by ($2$)
$\dfrac{{mv_{\max }^2}}{{{V_0}mg}} = \dfrac{{N\left( {\sin \theta + {\mu _s}\cos \theta } \right)}}{{N\left( {\cos \theta - {\mu _s}\sin \theta } \right)}}$
$\therefore \dfrac{{v_{\max }^2}}{{mg}} = \dfrac{{\sin \theta + {\mu _s}\cos \theta }}{{\cos \theta - {\mu _s}\sin \theta }} = \dfrac{{\tan \theta + {\mu _s}}}{{1 - {\mu _s}\tan \theta }}$
${V_{\max }} = \sqrt {\dfrac{{mg\left( {\tan \theta + {\mu _s}} \right)}}{{1 - {\mu _s}\tan \theta }}} $
This is the expression for the maximum safety speed on a curved horizontal road.
Note: The friction reaction of the road $\overrightarrow N $ perpendicular to the road surface.
And the frictional force ${\overrightarrow d _s}$ along the inclined surface of the road.
Other than banked roads in formula races we can see they bend the bike at some angle.
We can consider all forces acting on the car which can provide sufficient centripetal force.
Complete step by step answer:
We know the banking of roads \[-\] To avoid risk of skidding the road surface at a bend is tatted inwards, i.e. the outer side of road is raised above its inner ride. This is called ‘Banking of roads’
We can consider taking a left turn along a road or banked at an angle $\theta $ for a designed optimum speed V. Let M be the man of the car. In general, the force is aching on the car one.
(a) Its weight$\overrightarrow {mg} $, acting vertically drawn.
(b) The functional reactor of the road$\overrightarrow N $, perpendicular to the road surface.
(c) The frictional force ${\overrightarrow f _1}$ along the inclined surface of the road.
Resolve $\overrightarrow N $ and $\overrightarrow {{f_1}} $ into two perpendicular components.
$N\cos \theta $ Vertically up and ${\overrightarrow f _s}\sin \theta $ vertically down, $N\sin \theta $ and ${\overrightarrow f _s}\cos \theta $ horizontally towards the center of the circular path.
If ${V_{\max }}$ is the maximum safe speed without skidding, then
$
\dfrac{{mv_{\max }^2}}{t} = N\sin \theta + {f_s}\cos \theta \\
= N\sin \theta + {\mu _s}N\cos \theta \\
$
$\dfrac{{mv_{\max }^2}}{{{t_0}}} = N\left( {\sin \theta + {\mu _s}\cos \theta } \right)...(1)$
And,
$
N\cos \theta = mg + {f_s}\sin \theta \\
= mg + {\mu _s}N\sin \theta \\
mg = N\left( {\cos \theta - {\mu _s}\sin \theta } \right)...(2) \\
$
Dividing ($1$) by ($2$)
$\dfrac{{mv_{\max }^2}}{{{V_0}mg}} = \dfrac{{N\left( {\sin \theta + {\mu _s}\cos \theta } \right)}}{{N\left( {\cos \theta - {\mu _s}\sin \theta } \right)}}$
$\therefore \dfrac{{v_{\max }^2}}{{mg}} = \dfrac{{\sin \theta + {\mu _s}\cos \theta }}{{\cos \theta - {\mu _s}\sin \theta }} = \dfrac{{\tan \theta + {\mu _s}}}{{1 - {\mu _s}\tan \theta }}$
${V_{\max }} = \sqrt {\dfrac{{mg\left( {\tan \theta + {\mu _s}} \right)}}{{1 - {\mu _s}\tan \theta }}} $
This is the expression for the maximum safety speed on a curved horizontal road.
Note: The friction reaction of the road $\overrightarrow N $ perpendicular to the road surface.
And the frictional force ${\overrightarrow d _s}$ along the inclined surface of the road.
Other than banked roads in formula races we can see they bend the bike at some angle.
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