Four solid spheres each of mass M and radius R are placed at four corners of a light square frame of side length 4R such that centres of spheres coincide with corners of the square. What will be the moment of inertia of the system about an axis normal to the plane of the frame and passing through its centre?

Question

Four solid spheres each of mass M and radius R are placed at four corners of a light square frame of side length 4R such that centres of spheres coincide with corners of the square. What will be the moment of inertia of the system about an axis normal to the plane of the frame and passing through its centre? (A) (21 M R2/5) (B) (42 M R2/5) (C) (84 M R2/5) (D) (168 M R2/5)

Tardigrade Admin · Accepted Answer

I=4I1 Where I1 is MI of each sphere I1=Ic+Md2 And Ic=(2/5)MR2 ; d=(L/√2) ; L=4R

$\frac{21 M R ^{2}}{5}$

$\frac{42 M R ^{2}}{5}$

$\frac{84 M R ^{2}}{5}$

$\frac{168 M R ^{2}}{5}$

$I = 4 I_{1}$
Where $I_{1}$ is MI of each sphere
$I_{1} = I_{c} + M d^{2}$
And $I_{c} = \frac{2}{5} M R^{2}$ ; $d = \frac{L}{2}$ ; $L = 4 R$

521MR2​

542MR2​

584MR2​

5168MR2​

I=4I1​ Where I1​ is MI of each sphere I1​=Ic​+Md2 And Ic​=52​MR2 ; d=2​L​ ; L=4R

$\frac{21 M R ^{2}}{5}$

$\frac{42 M R ^{2}}{5}$

$\frac{84 M R ^{2}}{5}$

$\frac{168 M R ^{2}}{5}$

$I = 4 I_{1}$
Where $I_{1}$ is MI of each sphere
$I_{1} = I_{c} + M d^{2}$
And $I_{c} = \frac{2}{5} M R^{2}$ ; $d = \frac{L}{2}$ ; $L = 4 R$