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Centre Of Mass Of Hollow Hemisphere - Practice Questions & MCQ
Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main
Quick Facts
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3 Questions around this concept.
Solve by difficulty
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?
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
Concepts Covered - 1
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, $r=R \sin \theta, \quad y=R \cos \theta$
$
\sigma=\frac{M}{2 \pi R^2}
$
$\mathrm{So}_{\mathrm{o}} d m=\sigma d A=\sigma(2 \pi R \cos \theta) R d \theta$
$\mathrm{So}_{\mathrm{cm}}=\frac{\int y \cdot d m}{\int d m}$
$y_{c m}=\int_0^{\frac{\pi}{2}} R \sin \theta \sigma(2 \pi R \cos \theta) R d \theta=\frac{R}{2}$
${ }_{\mathrm{So}} \boldsymbol{y}_{\mathrm{cm}}=\frac{\boldsymbol{R}}{2}$ from base
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