Question

Use equation I=∫r2dm to calculate the moment of inertia of a uniform, hollow sphere with mass M and radius R for an axis passing through one of its diameters. Express your answer in terms of the variables M and R.

Use equation I=∫r2dm to calculate the moment of inertia of a uniform, solid cone with mass M, radius R and height H for its axis of symmetry. Express your answer in terms of the variables M and R.

Answer 1

moment of Inertia of Hollow sphere Rcoso Take a element at & which look like Ring Rose mass per Unit Area (0) - M - M UNR2 Small Ri Ring length= R do • angle = Arc = do = Radics Are &

) if am is mars of small Ring ghen am TA (Asea of smalking) x k do poso) dm = 2XM Cosodo un dm = 11 cose do Now dI - dmg2 (moment of Intia So of small Ring = m cose do. (Rcoso) 0 2 2 I : MR?cos o do 'NO? 2:1-72 to 12 7/2 S it cover whole sphere) I= 2 MR"cos² do I = 2 M R² 2 Now How to see went we want this MI' .: I= It mR2 com - ZMR² MR? + IZ MR² L me? Il = 5 MR2 MR2

Solid cone nieght - h. Given Mass -M Radius =R We know that f11M TR²h A = volune mars density M = 3M TR2h Now Moo 1 Take e lennect from the top a y distant it make a small disk C . AO'D and AOC Are Simillan -O small disk a mass = dm. Then dm = x Ring (small

dm = 3M. (782) ody TR²h = YR from om - 3M TR²h X Y Z R2 112 am= 3M qzdy am- aam Now Small . lat dI . Tergoo I aisten 'Idiska MR² = 3 MR2 255 10