A circular race course track has a radius of $500 \mathrm{~m}$ and is banked at $10^{\circ}$. The coefficient of static friction between the tyres of a vehicle and the road surface is 0.25 . Compute
(i) the maximum speed to avoid slipping
(ii) the optimum speed to avoid wear and tear of the tyres.
(i) the maximum speed to avoid slipping
(ii) the optimum speed to avoid wear and tear of the tyres.
Q 47.13
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$
\text { Data : } r=500 \mathrm{~m}, \theta=10^{\circ}, \mu_{\mathrm{s}}=0.25, \mathrm{~g}=9.8 \mathrm{~m} / \mathrm{s}^2, \tan 10^{\circ}=0.1763
$
(i) On the banked track, the maximum speed of the vehicle without slipping (skidding) is
$
\begin{aligned}
& v_{\max }=\sqrt{\frac{r g\left(\mu_{\mathrm{s}}+\tan \theta\right)}{1-\mu_{\mathrm{s}} \tan \theta}} \\
& =\sqrt{\frac{500 \times 10(0.25+0.1763)}{1-(0.25 \times 0.1763)}} \\
& \text { MaharashtraBoardSolutions. Guru } \\
& =\sqrt{\frac{500 \times 10 \times 0.4263}{0.9559}}=\sqrt{2230} \\
& =47.22 \mathrm{~m} / \mathrm{s} \\
&
\end{aligned}
$
(ii) The optimum speed of the vehicle on the track is
$
\begin{aligned}
v_{\text {opt }} & =\sqrt{r g \tan \theta} \\
& =\sqrt{500 \times 10 \times 0.1763} \\
& =\sqrt{881.5}=29.69 \mathrm{~m} / \mathrm{s}
\end{aligned}
$
\text { Data : } r=500 \mathrm{~m}, \theta=10^{\circ}, \mu_{\mathrm{s}}=0.25, \mathrm{~g}=9.8 \mathrm{~m} / \mathrm{s}^2, \tan 10^{\circ}=0.1763
$
(i) On the banked track, the maximum speed of the vehicle without slipping (skidding) is
$
\begin{aligned}
& v_{\max }=\sqrt{\frac{r g\left(\mu_{\mathrm{s}}+\tan \theta\right)}{1-\mu_{\mathrm{s}} \tan \theta}} \\
& =\sqrt{\frac{500 \times 10(0.25+0.1763)}{1-(0.25 \times 0.1763)}} \\
& \text { MaharashtraBoardSolutions. Guru } \\
& =\sqrt{\frac{500 \times 10 \times 0.4263}{0.9559}}=\sqrt{2230} \\
& =47.22 \mathrm{~m} / \mathrm{s} \\
&
\end{aligned}
$
(ii) The optimum speed of the vehicle on the track is
$
\begin{aligned}
v_{\text {opt }} & =\sqrt{r g \tan \theta} \\
& =\sqrt{500 \times 10 \times 0.1763} \\
& =\sqrt{881.5}=29.69 \mathrm{~m} / \mathrm{s}
\end{aligned}
$
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