JEE MAIN - Physics (2015 (Offline) - No. 25)
Distance of the center of mass of a solid uniform cone from its vertex is $$z{}_0$$. If the radius of its base is $$R$$ and its height is $$h$$ then $$z{}_0$$ is equal to :
$${{5h} \over 8}$$
$${{3{h^2}} \over {8R}}$$
$${{{h^2}} \over {4R}}$$
$${{3h} \over 4}$$
Explanation
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Let the density of solid cone $$\rho $$.
$$dm = \rho \pi {r^2}dy$$
$${y_{cm}} = {{\int {ydm} } \over {\int {dm} }}$$
$$ = {{\int\limits_0^h {\pi {r^2}} dy\rho \times y} \over {{1 \over 3}\pi {R^2}h\rho }}$$
$$ = {{3h} \over 4}$$
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