JEE Main & Advanced Mathematics Statistics Position of Centre of Gravity in Some Special Cases

(1) C.G. of a uniform rod : The C.G. of a uniform rod lies at its mid-point.

(2) C.G. of a uniform parallelogram : The C.G. of a uniform parallelogram is the point of inter-section of the diagonals.

(3) C.G. of a uniform triangular lamina : The C.G. of a triangle lies on a median at a distance from the base equal to one third of the medians.

(4) Some Important points to remember (i) The C.G. of a uniform tetrahedron lies on the line joining a vertex to the C.G. of the opposite face, dividing this line in the ratio 3 : 1.

(5) The C.G. of a right circular solid cone lies at a distance \[\frac{h}{4}\] from the base on the axis and divides it in the ratio 3 : 1.

(6) The C.G. of the curved surface of a right circular hollow cone lies at a distance \[\frac{h}{3}\] from the base on the axis and divides it in the ratio 2 : 1

(7) The C.G. of a hemispherical shell at a distance \[a/2\] from the centre on the symmetrical radius.

(8) The C.G. of a solid hemisphere lies on the central radius at a distance \[\frac{3a}{8}\] from the centre where a is the radius.

(9) The C.G. of a circular arc subtending an angle \[2\alpha \] at the centre is at a distance \[\frac{a\sin \alpha }{\alpha }\] from the centre on the symmetrical radius, a being the radius, and \[\alpha \] in radians.

(10) The C.G. of a sector of a circle subtending an angle \[2\alpha \] at the centre is at a distance \[\frac{2a}{3}\frac{\sin \alpha }{\alpha }\]from the centre on the symmetrical radius, a being the radius and \[\alpha \]  in radians.

(11) The C.G. of the semi circular arc lies on the central radius at a distance of \[\frac{2a}{\pi }\] from the boundary diameter, where a is the radius of the arc.