Question

Moment of inertia of a uniform hollow hemisphere of mass m and radius r about various axes

Answer 1

Answer:

The moment of inertia of a hollow sphere of

mass M having internal and external radii R

and 2R about an azis passing through its centre and perpendicular to its plane is

A)3/2 MR² B)13/32 MR² C)31/35 MR² D) 62/35 MR²

rho equals m over V

V equals 4 over 3 pi space left parenthesis left parenthesis 2 R right parenthesis cubed minus R cubed right parenthesis space

space equals space fraction numerator 4 x 7 over denominator 3 end fraction pi space R cubed

rho equals m over V equals space fraction numerator 3 m over denominator 4 x 7 x pi space R cubed end fraction

c o n s i d e r i n g space e l e m e n t space o f space t h i c k n e s s space d x space w i t h space m a s s space d m space space left parenthesis d x space i s space a t space a space d i s tan c e space x space f r o m space t h e space c e n t e r right parenthesis

m a s s equals d m equals rho 4 pi x squared d x

equals fraction numerator 3 m space over denominator 4 x 7 x pi space R cubed end fraction space X space 4 pi x squared d x space equals fraction numerator 3 over denominator 7 R cubed end fraction m x squared d x space

f o r space t h i s space e l e m e n t a l space h o l l o w space s p h e r e space m o m e n t space o f space i n e r t i a space w i l l space b e

d I equals 2 over 3 space d m space x squared

d I equals 2 over 7 space open parentheses M over R cubed close parentheses space x to the power of 4 d x

a f t e r space i n t e g r a t i n g

I equals integral subscript R superscript 2 R end superscript 2 over 35 open parentheses M over R cubed close parentheses x to the power of 5

equals 62 over 35 M space R squared