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)

    1

  • Option 3)

    \frac{14}{15}

  • Option 4)

    \frac{4}{5}

Answers (1)

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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