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Moment of inertia of cylinder with axis perpendicular to face

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Moment of inertia problem
A uniform cylinder has mass M and radius R.
a. Find by integration the moment of inertia, Io, about its center of mass
axis at center, perpendicular to the face of the cylinder.
b. Use the translation of axis theorem, 'Ip = Io + M h^2' to find the moment of inertia about an axis parallel to that above, through a point on the rim of the cylinder.

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The moment of inertia of cylinder with axis perpendicular to face are determined.

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SEE THE ATTACHMENT FOR A DIAGRAM SHOWING ELEMENT dm.
When any object is made to rotate about some axis, the net torque "T", required to give the object an angular acceleration 'alpha', is related to 'I', the moment of inertia of the object about that axis, by the expression: 'T = I alpha'. To find the moment of inertia of an object such as a thin rod, or a cylinder or a sphere, an element of mass, "dm", at a distance 'r' contributes an element 'dI= r^2 dm', and the total moment of inertia ...

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