Parallel axis theorem: Difference between revisions

Revision as of 11:29, 21 August 2020

Revision as of 17:42, 22 August 2020

Line 19:

:<math>I_\mathrm{cm} = \int (x^2 + y^2) \, dm.</math>

The moment of inertia relative to the axis {{math|''z&prime;''}}, which is a perpendicular distance {{math|''dD''}} along the ''x''-axis from the centre of mass, is

:<math>I = \int \left[(x + D)^2 + y^2\right] \, dm.</math>

Line 27:

:<math>I = \int (x^2 + y^2) \, dm + D^2 \int dm + 2D\int x\, dm.</math>

The first term is {{math|''I''_cm}} and the second term becomes {{math|''mdmD''²}}. The integral in the final term is a multiple of the x-coordinate of the [[center of mass]]{{snd}}which is zero since the center of mass lies at the origin. So, the equation becomes:

:<math> I = I_\mathrm{cm} + mD^2.</math>