What do you understand by 'radius of gyration'? Calculate the radius of gyration of a hoop of radius 5.00 cm rotating about an axis passing through its centre and normal to its plane.
Answers
Explanation:
Radius of gyration is defined as the distance from the axis of rotation to a point where the total mass of the body is supposed to be concentrated, so that the moment of inertia about the axis may remain the same. Simply, gyration is the distribution of the components of an object.
Explanation:
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.
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.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.
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.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 {\displaystyle n} particles each of mass {\displaystyle m}. Let {\displaystyle r_{1},r_{2},r_{3},\dots ,r_{n}} be their perpendicular distances from the axis of rotation. Then, the moment of inertia {\displaystyle I} of the body about the axis of rotation is
of the body about the axis of rotation is{\displaystyle I=m_{1}r_{1}^{2}+m_{2}r_{2}^{2}+\cdots +m_{n}r_{n}^{2}}
of the body about the axis of rotation is{\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 ({\displaystyle m}), then the moment of inertia is {\displaystyle I=m(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})}.
of the body about the axis of rotation is{\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 ({\displaystyle m}), then the moment of inertia is {\displaystyle I=m(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})}.Since {\displaystyle m=M/n} ({\displaystyle M} being the total mass of the body),
being the total mass of the body),{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
being the total mass of the body),{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}From the above equations, we have
being the total mass of the body),{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}From the above equations, we have{\displaystyle MR_{g}^{2}=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
being the total mass of the body),{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}From the above equations, we have{\displaystyle MR_{g}^{2}=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}Radius of gyration is the root mean square distance of particles from axis formula
being the total mass of the body),{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}From the above equations, we have{\displaystyle MR_{g}^{2}=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}Radius of gyration is the root mean square distance of particles from axis formula{\displaystyle R_{g}^{2}=(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
being the total mass of the body),{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}From the above equations, we have{\displaystyle MR_{g}^{2}=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}Radius of gyration is the root mean square distance of particles from axis formula{\displaystyle R_{g}^{2}=(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation.