Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

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Solution

Centre of mass of a body is defined as the point where the whole mass of a body is supposed to be concentrated.

For the given geometric shapes, such as sphere, cylinder, ring and cube, each of uniform mass density, the centre of mass of each body will lie in their respective geometric centers. It is because the given geometric shapes are geometrically symmetrical about their centers. Hence, the centre of mass of these bodies will lie in their respective geometric centers.

The centre of mass of a body may or may not lie within it, depending on their geometry. As in the case of a hollow sphere the centre of mass lies outside the body while for a solid sphere, centre of mass lies within its body.

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Q. Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

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Q. Column I of the following table gives a list of bodies and column II gives their moments of inertia about the axis of symmetry passing through their centre of mass. Assume mass of each of the bodies is

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Column-I	Column-II
i. Thin uniform hollow cylinder of radius $R$	p. $\frac{2}{5} M R^{2}$
ii. Uniform solid sphere of radius $R$	q. $\frac{2}{3} M R^{2}$
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