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Standard X

Physics

Torque

A bullet of m...

Question

A bullet of mass $m$ is fired horizontally into a large sphere of mass $M$ and radius $R$ resting on a smooth horizontal table.
The bullet hits the sphere at a height $h$ from the table and sticks to its surface. If the sphere starts rolling without slipping immediately on impact, then

\frac{h}{R} = \frac{4 m + 3 M}{2 (m + M)}

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\frac{h}{R} = \frac{m + M}{m + 2 M}

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\frac{h}{R} = \frac{10 m + 7 M}{5 (m + M)}

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\frac{h}{R} = \frac{4 m + 3 M}{m + M}

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Solution

The correct option is B $\frac{h}{R} = \frac{10 m + 7 M}{5 (m + M)}$
Using momentum conservation
$V = ω R$
$m v_{o} = (m + M) V = (m + M) ω R$
Conservation of angular momentum about center
$m v_{o} (h - R) = (\frac{2 M R^{2}}{5} + m R^{2}) ω$
on solving we get
$h = \frac{10 m + 7 M}{5 (m + M)}$

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A bullet of mass m is fired horizontally into a large sphere of mass M and radius R resting on a smooth horizontal table.The bullet hits the sphere at a height h from the table and sticks to its surface. If the sphere starts rolling without slipping immediately on impact, then

The correct option is B hR=10m+7M5(m+M)Using momentum conservation V=ωRmvo=(m+M)V=(m+M)ωRConservation of angular momentum about centermvo(h−R)=(2MR25+mR2)ωon solving we get h=10m+7M5(m+M)

A bullet of mass $m$ is fired horizontally into a large sphere of mass $M$ and radius $R$ resting on a smooth horizontal table.
The bullet hits the sphere at a height $h$ from the table and sticks to its surface. If the sphere starts rolling without slipping immediately on impact, then

The correct option is B $\frac{h}{R} = \frac{10 m + 7 M}{5 (m + M)}$
Using momentum conservation
$V = ω R$
$m v_{o} = (m + M) V = (m + M) ω R$
Conservation of angular momentum about center
$m v_{o} (h - R) = (\frac{2 M R^{2}}{5} + m R^{2}) ω$
on solving we get
$h = \frac{10 m + 7 M}{5 (m + M)}$