what is the radius of gyration k for a rigid body about a given rotation axis ?
Answers
Radius of gyration is the distance from the axis where the body can be assumed to be concentrated such that the moment of inertia about that axis of the concentrated body(point mass) is equal to the original moment of inertia (of the body)about the axis.
The radius of gyration k for a rigid body about a given rotation axis is √ ( ( r1² + r2² + r3² + ... + rn² ) / n ).
Given: A rigid body with a given axis of rotation.
To Find: The radius of gyration k for a rigid body about a given rotation axis
Solution:
- The radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distribution of a point from the axis of rotation at which, if the whole mass of the body is assumed to be concentrated, its moment of inertia about the given axis would be same as with its actual distribution of mass.
- A body can consist of 'n' number of particles and hence they will have 'n' number of radii from the axis of rotation. To avoid miscalculation, we define the radius of gyration.
Suppose we have a body that contains 'n' number of particles with mass m1, m2, m3, m4, ..., mn and radii r1, r2, r3, r4, ..., rn.
So, the moment of inertia about an axis can be given as,
Moment of Inertia (I) = m1.r1² + m2.r2² + m3.r3² + ... + mn.rn² ....(1)
Now, for a rigid body, we know that,
m1 = m2 = m3 = m4 = ... = mn = m, so (1) becomes,
Moment of Inertia (I) = m.r1² + m.r2² + m.r3² + ... + m.rn²
= m×n × ( r1² + r2² + r3² + ... + rn² ) / n
Now, m×n = total mass of body = M
Moment of Inertia (I) = m.r1² + m.r2² + m.r3² + ... + m.rn²
= M × ( r1² + r2² + r3² + ... + rn² ) / n
= M×k²
where, k² = ( r1² + r2² + r3² + ... + rn² ) / n
again, k = √ ( ( r1² + r2² + r3² + ... + rn² ) / n )
And this k here is the radius of gyration.
Hence, the radius of gyration k for a rigid body about a given rotation axis is √ ( ( r1² + r2² + r3² + ... + rn² ) / n ).
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