Respuesta :
By direct integration, the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end, is ML²/3 and at the centre ML²/12.
Given that, mass of the rod is M
Length is L
Rotating around an axis perpendicular to its length and passing at a distance d from one end.
Consider a very small strip of length dx and having a mass dm.
Moment of inertia is given by I = ∫ x² dm
So, mass per unit length λ = M/L
The length of the small strip dx can be written as
dm = λ dx = M/L dx
From the above two equations, we get, I = ∫ M/L x² dx
The limits of the integral are from -d to L, the value of I is M/3 (L²-3Ld+3d²)
For calculating moment of inertia on the end substitute d = 0, we get I = ML²/3
For calculating moment of inertia at the centre substitute d =L/2 , we get I = ML²/12.
To know more about moment of inertia:
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