Moment of solid sphere about axis, Physics

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Q. Moment of Solid Sphere About Axis?

The instant of enertia for a homogeneous solid sphere about an axis can be found by integrating spherical shells by integrating disks or by solving the equations for the moment of an element of mass throughout the volume. We will utilize the latter.

The instant of a mass element is I = ∫V l2dm, where dm = ρ dV l is the distance of the element from the axis and ρ is the mass density. Note that the distance to an element from the centre of the sphere is r and l = r sin θ where θ is the angle between the axis and the radius, The volume element dV is

dV = (2π l)(dr)(r dθ) = 2πr2 sinθ dr dθ.

ρ =m/V =m/(4/3(π R3))

And dm = ρdv the moment of inertia is        

I = ∫l2dm = ∫ (r sin θ)2ρdV.

2169_Moment of Solid Sphere About Axis.png

2182_Moment of Solid Sphere About Axis1.png

It is convenient to alter the variable of integration from θ to - cos θ.

Let x = - cos θ Then the limits of the above integral become

1681_Moment of Solid Sphere About Axis2.png

The corresponding integral is

1286_Moment of Solid Sphere About Axis3.png

Finally,

I =2/5(mR2).


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