Parallel and Perpendicular Axis Theorem
`text(Parallel axis Theorem :)`
Used to find moment of inertia about an axis which parallel to the axis passing through C.M.
`I_(CM) ->` MI of the rigid body about an axis through CM
`I_P ->` MI of the rigid body about an axis which is parallel to the above axis through CM & is at distance d from the axis through CM
`I_(CM)=sum_im_i(x_i^2+y_i^2)`
`I_P=sum_im_i[(x_i-a)^2 + (y_i -b)^2]`
(z coordinates are not involved so `m_i,` can be replaced by sum of mass of all particles placed on z-axis with co-ordinate (`x_i,y_i`)
`=sum m_i(x_i^2+y_i^2) + sum m_i(a^2+b^2) + sum2am_ix_i - sum2bm_iy_i`
`=I_(CM) + (a^2+b^2)summ_i - 2a sum m_ix_i - 2b summ_iy_i`
`I_P = I_(CM) + Mh^2`
If M.I. of a body of mass M about an axis passing through its C.O.M. is `I_C` them M.I. of another axis which is parallel to the above central axis and is parallel to it is given by
`I_(A A^')=I_C + Mh^2`
`text(Perpendicular Axis Theorem :)`
This theorem is applicable only for the laminar bodies (i.e. plane bodies). (e.g. ring, disc, not sphere) `I_x` & `I_y` are Ml of body about a common point O in two mutually perpendicular directions in the plane of body `I_z` is MI of body about an axis perpendicular to X & Y axis & passing through point O.
`I_x = summ_iy_i^2`
`I_y = summ_ix_i^2`
`I_z=I_x+I_y=summ_i(x_i^2+y_i^2)`
The point of intersection of the three axis need not be centre of mass, it can be any point in the plane of body which lie on the body or even outside it. For relation from perpendicular axis theorem, `I_3 = I_1 +I_2` axis (3) must be perpendicular the plane of the body and axis (1) and axis (2) must be in the plane of the body.
Used to find moment of inertia about an axis which parallel to the axis passing through C.M.
`I_(CM) ->` MI of the rigid body about an axis through CM
`I_P ->` MI of the rigid body about an axis which is parallel to the above axis through CM & is at distance d from the axis through CM
`I_(CM)=sum_im_i(x_i^2+y_i^2)`
`I_P=sum_im_i[(x_i-a)^2 + (y_i -b)^2]`
(z coordinates are not involved so `m_i,` can be replaced by sum of mass of all particles placed on z-axis with co-ordinate (`x_i,y_i`)
`=sum m_i(x_i^2+y_i^2) + sum m_i(a^2+b^2) + sum2am_ix_i - sum2bm_iy_i`
`=I_(CM) + (a^2+b^2)summ_i - 2a sum m_ix_i - 2b summ_iy_i`
`I_P = I_(CM) + Mh^2`
If M.I. of a body of mass M about an axis passing through its C.O.M. is `I_C` them M.I. of another axis which is parallel to the above central axis and is parallel to it is given by
`I_(A A^')=I_C + Mh^2`
`text(Perpendicular Axis Theorem :)`
This theorem is applicable only for the laminar bodies (i.e. plane bodies). (e.g. ring, disc, not sphere) `I_x` & `I_y` are Ml of body about a common point O in two mutually perpendicular directions in the plane of body `I_z` is MI of body about an axis perpendicular to X & Y axis & passing through point O.
`I_x = summ_iy_i^2`
`I_y = summ_ix_i^2`
`I_z=I_x+I_y=summ_i(x_i^2+y_i^2)`
The point of intersection of the three axis need not be centre of mass, it can be any point in the plane of body which lie on the body or even outside it. For relation from perpendicular axis theorem, `I_3 = I_1 +I_2` axis (3) must be perpendicular the plane of the body and axis (1) and axis (2) must be in the plane of the body.