Saved Bookmarks
| 1. |
Consider a sphere of mass M and radius R centered at origin. The density of material of the sphere is rho = Ar ^(alpha), where r is the radial distance, alpha andA are constants. If the moment of inertia of the sphere about the axis passing through centre is 6/7MR ^(2), then the value of alpha is |
|
Answer» Solution :`(**)` Given, density of sphere, `rho = Ar ^(alpha )` (where, r= radial distance and A and `alpha` are constants) Consider an elemental spherical shell of radius r and thickness dr  Mass of elemental spherical shell, dm= Volume `XX` Density `dm = (4pi r ^(2)) dr. Ar ^(alpha ) = 4pi a r ^(2+ alpha ) dr""...(i)` Mass of entire solid sphere, `M = 4pi A int _(0)^(R) r ^(2+ alpha ) dr` `M = 4piA [(R ^(3+alpha ))/(3 + alpha )]_(0 ^(R)) = (4piA)/(3 + alpha ) . R ^(3 + alpha)""...(ii)` Now, moment of inertia of elemental spherical shell is `dI = 2/3 (dm). r ^(2) = 2/3 (4pi A r ^(2+alpha ) dr) r ^(2)` Moment of inertia of entire solid sphere, `I = int _(0) ^(R) dI = 2/3 4 pi A int _(0) ^(R) r^(A + alpha). dr` `I = 2/3 4piA [(r ^(5+ alpha ))/(5 + alpha )]_(0)^(R)implies I = 2/3 4pi A [(R ^(5 + alpha ))/(5 + alpha ) ]` `I = 2/3 ((4piA)/(3 + alpha ) . R ^(3 + alpha )) . (R ^(2) (3 + alpha ))/(5 + alpha )` From Eq. (ii), we get `I = 2/3 MR ^(2) . ((3+ alpha )/( 5 + alpha )) ""...(iii)` It is given in the question. `I = 6/7 MR^(2) ""(iv)` On comparing EQS (iii) and (ov), we get `2/3 MR^(2). ((3 + alpha )/(5 + alpha ))= 6/7MR^(2) implies (3 +alpha)/(5 + alpha ) = 9/7` `21+ 7 alpha = 45 + 9 alpha` `implies alpha =-12` |
|