A racing car travels on a track (without banking) $ABCDEPA$. $ABC$ is a circular arc of radius $2R$. $CD$ and $FA$ are straight paths of length $R$ and $DEF$ is a circular arc of radius $R = 100 \,m$. The coefficient of friction on the road is $\mu = 0.1$. The maximum speed of the car is $50\,ms^{-1}$. Find the minimum time for completing one round.
A racing car travels on a track (without banking) $ABCDEPA$. $ABC$ is a circular arc of radius $2R$. $CD$ and $FA$ are straight paths of length $R$ and $DEF$ is a circular arc of radius $R = 100 \,m$. The coefficient of friction on the road is $\mu = 0.1$. The maximum speed of the car is $50\,ms^{-1}$. Find the minimum time for completing one round.
Frictional force provide necessary centripetal force in uniform circular motion.
For $\quad$ $AOC$ path radius $=2 \mathrm{R}$
$FED$ path radius $=\mathrm{R}$
Centripetal force $\frac{m v^{2}}{r}=f$
$f =\mu \mathrm{N}$
$\frac{m v^{2}}{r} =\mu \mathrm{N}=\mu m g$
$\therefore v=\sqrt{\mu r g}$ (where $r$ is radius of the circular track)
For path $\mathrm{ABC}$,
$\text { Path length } =\frac{3}{4}(2 \pi \times 2 \mathrm{R})=3 \pi \mathrm{R}=3 \pi \times 100$
$=300 \pi \mathrm{m}$
$v_{1}=\sqrt{\mu 2 \mathrm{Rg}} =\sqrt{0.1 \times 2 \times 100 \times 10}$
$=14.14 \mathrm{~m} / \mathrm{s}$
$\therefore \quad t_{1} =\frac{300 \pi}{14.14}=66.6 \mathrm{~s}$
For path $DEF$,
$\text { Path length } =\frac{1}{2}(2 \pi \mathrm{R})=\frac{\pi \times 100}{2}=50 \pi$
$v_{2} =\sqrt{\mu \mathrm{Rg}}=\sqrt{0.1 \times 100 \times 10}=10 \mathrm{~m} / \mathrm{s} $
$t_{2} =\frac{50 \pi}{10}=5 \pi \mathrm{s}=15.7 \mathrm{~s}$
For paths CD and FA,
Path length $=\mathrm{R}+\mathrm{R}=2 \mathrm{R}=200 \mathrm{~m}$
$t_{3}=\frac{200}{50}=4.0 \mathrm{~s}$
$\therefore$ Total time for completing one round,
$t=t_{1}+t_{2}+t_{3}=66.6+15.7+4.0=86.3 \mathrm{~s}$
Similar Questions
A coin placed on a rotating table just slips if it is placed at a distance $4r$ from the centre. On doubling the angular velocity of the table, the coin will just slip when at a distance from the centre equal to
A coin placed on a rotating table just slips if it is placed at a distance $4r$ from the centre. On doubling the angular velocity of the table, the coin will just slip when at a distance from the centre equal to
$Assertion$ : There is a stage when frictional force is not needed at all to provide the necessary centripetal force on a banked road.
$Reason$ : On a banked road, due to its inclination the vehicle tends to remain inwards without any chances of skidding.
$Assertion$ : There is a stage when frictional force is not needed at all to provide the necessary centripetal force on a banked road.
$Reason$ : On a banked road, due to its inclination the vehicle tends to remain inwards without any chances of skidding.
- [AIIMS 2016]
As shown in the figure, two strings of length $l$ each are attached with a vertical axis $AB$ of length $l$. Strings are $AC$ and $BC$. At point $C$ a point mass m is attached. Mass rotates about axis with angular velocity. Tensions in $AB$ and $BC$ are $T_1$ and $T_2$ respectively. Choose the $CORRECT$ alternative :-
As shown in the figure, two strings of length $l$ each are attached with a vertical axis $AB$ of length $l$. Strings are $AC$ and $BC$. At point $C$ a point mass m is attached. Mass rotates about axis with angular velocity. Tensions in $AB$ and $BC$ are $T_1$ and $T_2$ respectively. Choose the $CORRECT$ alternative :-
A car is moving with a constant speed of $20\,m / s$ in a circular horizontal track of radius $40\,m$. A bob is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be : (Take $g =10$ $\left.m / s ^2\right)$
A car is moving with a constant speed of $20\,m / s$ in a circular horizontal track of radius $40\,m$. A bob is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be : (Take $g =10$ $\left.m / s ^2\right)$
- [JEE MAIN 2023]
A thin circular loop of radius $R$ rotates about its vertical diameter with an angular frequency $\omega .$ Show that a small bead on the wire loop remains at its lowermost point for $\omega \leq \sqrt{g / R} .$ What is the angle made by the radius vector jotning the centre to the bead with the vertical downward direction for $\omega=\sqrt{2 g / R} ?$ Neglect friction.
A thin circular loop of radius $R$ rotates about its vertical diameter with an angular frequency $\omega .$ Show that a small bead on the wire loop remains at its lowermost point for $\omega \leq \sqrt{g / R} .$ What is the angle made by the radius vector jotning the centre to the bead with the vertical downward direction for $\omega=\sqrt{2 g / R} ?$ Neglect friction.