Question

Use calculus to derive the moment of inertia of a uniform rod of length L and mass M rotating about an axis at 0.25L.

Addn Problem: Use calculus to derive the moment of inertia of a uniform rod of length L and mass M rotating about an axis at 0.251

Answer 1

Solution:- Let AB be a uniform rod of length L rotating about an axis YY' passing through the rod at O. Y -0.25* 07= - B - axis of rotation ly! element of length dr' at a from o towards B as shown in Taking an distance the figure. Total mass of the rod = M (given) : :: Mass per unit length = Mass of the element do (d

We know that moment of inertia continuous body is defined as of a I = (822m Here Is Moment of Inertia dm = mass of an element 8 = distance of element of mass dm from the axis Hence, the of required moment of inertial the rod is given by 0.75L 182dm Is -0.25L

0.75L = I=/82 Ido [dm = m 2 do] -0.25L 0.75L 11 -0.25L SL J-0.25L : [ 71° 764 = oped . touse"] - [ 0975°. (-001593)] = x 0.4218754?+ 0.01562523 sMx 043752 0.42) 0.4218 0.01562 3 0.4 = 0.146 ML? Hence, the required moment of I = 0.146 M2? inertia Answer