Centre Of Mass Of Hollow Hemisphere MCQ - Practice Questions & Answers

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As shown in the figure a Hollow hemisphere of mass $M$ and radius $R$ is given with its centre at the origin 0 . If the point $P$ is on $y -$ axis and is at a distance $(3 R / 4)$ from $O$ .

Then what is the distance from C.O.M of Hollow hemisphere?

$\frac{R}{2}$

$\frac{R}{4}$

$R / 3$

$3 R / 4$

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A hollow hemisphere and a hollow cone of same mass are arranged as shown in figure. find the position of center of mass from center of hemisphere

$\frac{11}{13} R$

$\frac{14}{9} R$

$\frac{11}{12} R$

$\frac{5}{7} R$

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Position of centre of mass for Hollow Hemisphere

Have a look at the figure of Hollow Hemisphere

Since it is symmetrical about $y$ -axis
So we can say that its $x_{c m} = 0$ and $z_{c m} = 0$
Now we will calculate its $y_{c m}$ which is given by

$y_{c m} = \frac{\int y \cdot d m}{\int d m}$

So, Take a small elemental ring of mass dm of radius r at a height y from the origin as shown in the figure.

And, $r = R \sin θ, y = R \cos θ$

$σ = \frac{M}{2 π R^{2}}$

${So}_{o} d m = σ d A = σ (2 π R \cos θ) R d θ$
${So}_{cm} = \frac{\int y \cdot d m}{\int d m}$
$y_{c m} = \int_{0}^{\frac{π}{2}} R \sin θ σ (2 π R \cos θ) R d θ = \frac{R}{2}$
$_{So} y_{cm} = \frac{R}{2}$ from base

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