Radius of gyration

Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance of a point from the axis of rotation at which, if whole mass of the body is assumed to be concentrated, its moment of inertia about the given axis would be the same as with its actual distribution of mass. It is denoted by R g {displaystyle R_{g}} . Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance of a point from the axis of rotation at which, if whole mass of the body is assumed to be concentrated, its moment of inertia about the given axis would be the same as with its actual distribution of mass. It is denoted by R g {displaystyle R_{g}} . Mathematically the radius of gyration is the root mean square distance of the object's parts from either its center of mass or a given axis, depending on the relevant application. It is actually the perpendicular distance from point mass to the axis of rotation. Suppose a body consists of n {displaystyle n} particles each of mass m {displaystyle m} . Let r 1 , r 2 , r 3 , … , r n {displaystyle r_{1},r_{2},r_{3},dots ,r_{n}} be their perpendicular distances from the axis of rotation. Then, the moment of inertia I {displaystyle I} of the body about the axis of rotation is I = m 1 r 1 2 + m 2 r 2 2 + ⋯ + m n r n 2 {displaystyle I=m_{1}r_{1}^{2}+m_{2}r_{2}^{2}+cdots +m_{n}r_{n}^{2}} If all the masses are the same ( m {displaystyle m} ), then the moment of inertia is I = m ( r 1 2 + r 2 2 + ⋯ + r n 2 ) {displaystyle I=m(r_{1}^{2}+r_{2}^{2}+cdots +r_{n}^{2})} . Since m = M / n {displaystyle m=M/n} ( M {displaystyle M} being the total mass of the body), I = M ( r 1 2 + r 2 2 + ⋯ + r n 2 ) / n {displaystyle I=M(r_{1}^{2}+r_{2}^{2}+cdots +r_{n}^{2})/n}