Question

What is the distance from axis about which a uniform, balsa-wood sphere will have the same moment of inertia as does a thin-walled, hollow, lead sphere of the same mass and radius R, with the axis along a diameter, to the center of the balsa-wood sphere?

Answer 1

Answer:

$D_{s}$ ≈ 2.1 R

Explanation:

The moment of inertia of the bodies can be calculated by the equation

     I = ∫ r² dm

For bodies with symmetry this tabulated, the moment of inertia of the center of mass

Sphere               $Is_{cm}$ = 2/5 M R²

Spherical shell   $Ic_{cm}$ = 2/3 M R²

The parallel axes theorem allows us to calculate the moment of inertia with respect to different axes, without knowing the moment of inertia of the center of mass

    I = $I_{cm}$ + M D²

Where M is the mass of the body and D is the distance from the center of mass to the axis of rotation

Let's start with the spherical shell, axis is along a diameter

     D = 2R

    Ic = $Ic_{cm}$ + M D²

    Ic = 2/3 MR² + M (2R)²

    Ic = M R² (2/3 + 4)

    Ic = 14/3 M R²

The sphere

    Is =$Is_{cm}$ + M [$D_{s}$²

    Is = Ic

    2/5 MR² + M $D_{s}$² = 14/3 MR²

    $D_{s}$² = R² (14/3 - 2/5)

    $D_{s}$ = √ (R² (64/15)

    $D_{s}$ = 2,066 R