the moment of inertia of a solid cylinder about it's own axis is the same as the moment of inertia about as axis passing through the center of gravity and perpendicular to its length. what is the relation between L and R
Question
the moment of inertia of a solid cylinder about it's own axis is the same as the moment of inertia about as axis passing through the center of gravity and perpendicular to its length. what is the relation between L and R
Solution
The moment of inertia of a solid cylinder about its own axis (which is along its length) is given by the formula:
I = 0.5 * M * R^2
where: I is the moment of inertia, M is the mass of the cylinder, and R is the radius of the cylinder.
The moment of inertia of the same cylinder about an axis passing through its center of gravity and perpendicular to its length (which is along its radius) is given by the formula:
I = M * (R^2 + L^2) / 12
where: L is the length of the cylinder.
Since the two moments of inertia are the same, we can equate the two formulas:
0.5 * M * R^2 = M * (R^2 + L^2) / 12
Solving for L, we get:
L = sqrt[24 * R^2 - 12 * R^2] L = sqrt[12 * R^2] L = sqrt[12] * R L = 3.46 * R
So, the length of the cylinder is approximately 3.46 times its radius.
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