From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is
Question
From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is
Solution
The maximum volume of a cube that can be cut from a sphere is when the diameter of the sphere is equal to the side of the cube.
Step 1: Find the side of the cube (a)
The diameter of the sphere is 2R, which is equal to the side of the cube (a). So, a = 2R.
Step 2: Find the volume of the cube (V)
The volume of a cube is given by the formula V = a^3. Substituting a = 2R, we get V = (2R)^3 = 8R^3.
Step 3: Find the mass of the cube (m)
The mass of the cube is proportional to its volume. Since the cube is cut from the sphere, the densities of the sphere and the cube are the same. Therefore, the mass of the cube (m) is given by m = M * (V of cube / V of sphere) = M * (8R^3 / (4/3)πR^3) = 6M/π.
Step 4: Find the moment of inertia of the cube (I)
The moment of inertia of a cube about an axis passing through its center and perpendicular to one of its faces is given by the formula I = (m*a^2)/6. Substituting m = 6M/π and a = 2R, we get I = (6M/π * (2R)^2) / 6 = 4MR^2/π.
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