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

Solve by difficulty

A solid hemisphere (A) and a hollow hemisphere (B) Each having mass M are placed as shown in diagram.What is the y-coordinate
of the centre of mass of system

$\frac{R}{16}$

- $\frac{R}{16}$

$\frac{R}{5}$

- $\frac{R}{5}$

View Solution

As shown in figure $A$ solid hemisphere is given of Mass $M$ and Radius $R$ and its centre at origin.

So on increasing its volume uniformly, its COM will

Shift toward origin

Shift away from origin

Remains same

None of these

View Solution

Concepts Covered - 1

Position of centre of mass for solid Hemisphere

Have a look at the figure of solid 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 $y_{cm}$ which is given by

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

So Take a small elemental hollow hemisphere of mass dm of radius r as shown in figure.

Now have a look at an elemental hollow hemisphere of mass dm of radius r

Since our element mass is hollow hemisphere its C.O.M is at (r/2)

Now $d m = ρ d v = ρ (2 π r^{2}) d r$

Where,

$ρ = \frac{M}{\frac{2}{3} π R^{3}}$

$y_{c m} = \frac{\int \frac{r}{2} d m}{M} = \frac{\int_{0}^{R} \frac{r}{2} * \frac{3 M}{2 π R^{3}} * 2 π r^{2} d r}{M} = \frac{3}{2 R^{3}} * \int_{0}^{R} r^{3} d r = \frac{3 R}{8}$

so $y_{c m} = {\frac{3 R}{8}}_{from base}$

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