MCQ
A solid hemisphere of weight $P$ rests with its curved surface in contact with a rough inclined plane. A weight $Q$ is placed at some point on the rim of the hemisphere to keep its plane surface horizontal then its minimum coefficient of friction is
  • $\mu  = \,\frac{Q}{{\sqrt {P(P + 2Q)} }}$
  • B
    $\mu  = \,\frac{Q}{{\sqrt {P(Q + 2P)} }}$
  • C
    $\mu  = \,\frac{P+Q}{{\sqrt {P(P + 2Q)} }}$
  • D
    $\mu  = \,\frac{P-Q}{{\sqrt {P(P + 2Q)} }}$

Answer

Correct option: A.
$\mu  = \,\frac{Q}{{\sqrt {P(P + 2Q)} }}$
a
At $O$

$\mathrm{P}(\mathrm{R} \sin \phi)=\mathrm{Q}(1-\sin \phi) \mathrm{R}$

$\sin \phi=Q /(P+Q)$

$\mu=\tan \phi=\frac{\mathrm{Q}}{\sqrt{\mathrm{P}(\mathrm{P}+2 \mathrm{Q})}}$

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