# A splid sphere and cylinder of identical radii approah an incline with the same liner velocity (see figure ). Both roll without slipping all throughout. The two climb maximum hrights $h_{sph}$ and $h_{cyl}$ on the incline. The ratio $\frac{h_{sph}}{h_{cyl}}$ is given by:Option 1)$\frac{2}{\sqrt{5}}$Option 2)1Option 3)$\frac{14}{15}$Option 4)$\frac{4}{5}$

$V=wR$

For sphere

$mgh_{1}=k_{1}=\frac{1}{2}Iw^{2}+\frac{1}{2}mv^{2}$

$=\frac{1}{2}\times \frac{2}{5}mR^{2}w^{2}+\frac{1}{2}mv^{2}$

$=\left ( \frac{1}{5}+\frac{1}{2} \right )mv^{2}=\frac{7}{10}mv^{2}$

For solid cylinder

$mgh_{2}=k_{2}=\frac{1}{2}mv^{2}+\frac{1}{2}\frac{mR^{2}}{2}w^{2}= mv^{2}\left ( \frac{1}{2}+\frac{1}{4} \right )=\frac{3}{4}mv^{2}$

$\frac{mgh_{1}}{mgh_{2}}=\frac{k_{1}}{k_{2}}=\frac{\frac{7}{10}mv^{2}}{\frac{3}{4}mv^{2}}=\frac{28}{30}=\frac{14}{15}$

Option 1)

$\frac{2}{\sqrt{5}}$

Option 2)

1

Option 3)

$\frac{14}{15}$

Option 4)

$\frac{4}{5}$

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