Question

three identical uniform spheres each of mass M and radius R are placed as shown in the figure the moment of inertia of the system about the axis ab is
Its ans is OPTION 2 but how?
can anyone help as soon as possible.​

Answer 1

Answer:

Given:

Mass of each sphere = M

Radius of each sphere = R

To find:

Total Moment Of Inertia of the system.

Calculation:

Let the spheres be named A , B , C as shown in the attached photo.

Moment of Inertia of A along AB axis :

$I_ {1} = \dfrac{2}{5} M {R}^{2}$

Perpendicular distance of sphere B and C from the AB axis can be found using Trigonometry.

$x = \sqrt{ 4 {R}^{2} - {R}^{2} }$

$x = R \sqrt{ 3 }$

Moment of Inertia of B along AB axis :

$I_ {2} = \dfrac{2}{5} M {R}^{2} + M {x }^{2}$

$= > I_ {2} = \dfrac{2}{5} M {R}^{2} + 3M {R}^{2}$

$= > I_ {2} = \dfrac{17}{5} M {R}^{2}$

Moment of Inertia for C along AB axis:

Similarly we can say :

$= > I_ {3} = \dfrac{17}{5} M {R}^{2}$

Total Moment of Inertia will be :

$= > I_ {total} = \{( \dfrac{17}{5} M {R}^{2} ) \times 2 \} + \dfrac{2}{5} M {R}^{2}$

$= > I_ {total} = \dfrac{36}{5} M {R}^{2}$

Answer 2

answer is 36MR²/5

refer attachment

hope it helps