Explain radius of gyrations with derivation
Answers
Question;
Explain radius of gyrations with derivation ?
Answer;
The moment of inertia of a body about an axis is sometimes represented using the radius of gyration. Now, what do you mean by the radius of gyration? We can define the radius of gyration as the imaginary distance from the centroid at which the area of cross-section is imagined to be focused at a point in order to obtain the same moment of inertia. It is denoted by k.
Derivation ;
Radius of Gyration Formula
The formula of moment inertia in terms of the radius of gyration is given as follows:
I = mk2 (1)
where I is the moment of inertia and m is the mass of the body
Accordingly, the radius of gyration is given as follows
k=Im−−√ (2)
The unit of the radius of gyration is mm. By knowing the radius of gyration, one can find the moment of inertia of any complex body equation (1) without any hassle.
I hope it helps you mate ✨
Explanation:
Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there.
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. One can represent a trajectory of a moving point as a body. Then radius of gyration can be used to characterize the typical distance travelled by this point.
Suppose a body consists of {\displaystyle n} n particles each of mass {\displaystyle m} m. Let {\displaystyle r_{1},r_{2},r_{3},\dots ,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 {\displaystyle I} I 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}} {\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} m), then the moment of inertia is {\displaystyle I=m(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})} {\displaystyle I=m(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})}.
Since {\displaystyle m=M/n} {\displaystyle m=M/n} ( {\displaystyle M} M being the total mass of the body),
{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n} {\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} {\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} {\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. It is also known as a measure of the way in which the mass of a rotating rigid body is distributed about its axis of rotation.