Radius of Gyration | Physics Grade XI Reference Note

Radius of Gyration

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Radius of Gyration | Physics Grade XI Reference Note

Radius of Gyration

Radius of Gyration
Radius of gyration is defined as the distance between the axis of rotation to the point where all the mass of the body is supposed to be concentrated i.e. center of mass. It is denoted by K. The moment of inertia for the body defined by the radius of gyration.
     I = MK2 ……... (i)

Expression for radius of Gyration

Radius of gyrationLet us consider a rigid body of mass M made of n particles rotate about an axis YY'.  Let r1, r2, r3,…… be the radius of gyration then, moment of inertia of particles
     I = m1r12 + m2r22 + m3r32 + …………... + mnrn2             
                                                                               
If all the particles have same mass then,
     I = m (r12+ r22+ r32+…….+rn2)
     I = mn (r12+ r22+ r32+…….+rn2)/ n                                                                                                             
  Since mn = M
     I = M (r12+ r22+ r32+…….+rn2)/ n                 
                                                                                             
From equation (i),
     MK2 = M (r12+ r22+ r32+…….+rn2)/ n
     K = √(r12+ r22+ r32+…….+rn2)/ n

Therefore, the radius of gyration is the root mean square of distance of various particles of the body from the axis of rotation.

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