( $1$ ) Centre of gravity of a body is the point at which the weight of the body acts.
( $2$ ) Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius.
( $3$ ) To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its centre of gravity.
( $4$ ) The radius of gyration of any body rotating about an axis is the length of the perpendicular drawn from the centre of gravity of the body to the axis.
Which one of the following pairs of statements is correct?

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$(1)$ and $(2)$

$(1)$ and $(3)$

$(2)$ and $(3)$

$(3)$ and $(4)$

answer is A.

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Detailed Solution

$(1)$ Center of gravity is the point where weight acts.

$(2)$ Center of mass coincides with center of gravity in the case where mass density is uniform.

$(3)$ Gravitational field at any extreme point can be calculated, assuming it to be a point mass.

$(4)$ $I_{about any axis} = {mk}^{2}$ -radius of gyration

$I_{C . G} + {mx}^{2} = {mk}^{2} k^{2} = \frac{I_{C . G}}{m} + x^{2}$

$(1)$ and $(3)$ are correct.

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