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Rotation + Translation

A uniform sph...

Question

A uniform sphere of mass $m$ , radius $r$ and momentum of inertia $I$ about its centre moves along the x-axis as shown in figure. Its centre of mass moves with velocity $= v_{0}$ , and it rotates about its centre of mass with angular velocity $= ω_{0}$ . Let $\to L = (I ω_{0} + m v_{0} r) (- k)$ . The angular momentum of the body about the origin $O$ is

\to L

, only if

v_{0} = ω_{0} r

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greater than

\to L

, if

v_{0} > ω_{0} r

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less than

\to L

, if

v_{0} > ω_{0} r

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\to L

, for all values of

ω_{0}

and

v_{0}

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Solution

The correct option is D $\to L$ , for all values of $ω_{0}$ and $v_{0}$
$_{O} =_{1}$ (COM frame) $+_{2}$ (COM)
$_{O} = I \to w + m (\to r \times \to v)$
$_{O} = I w_{o} (-^k) + m (r^j \times v_{o}^i)$
$_{O} = I w_{o} (-^k) + m r v_{o} (-^k)$
Thus $_{O} = \to L$ for all values of $w_{o}$ and $v_{o}$

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The correct option is D →L, for all values of ω0 and v0→LO=→L1 (COM frame) +→L2 (COM) →LO=I→w+m(→r×→v)→LO=Iwo(−^k)+m(r^j×vo^i)→LO=Iwo(−^k)+mrvo(−^k)Thus →LO=→L for all values of wo and vo