Two rods of equal mass m and length ℓ lie along the x axis and y axis with their centres origin. What is the moment of inertia of both about the line x=y :
Question
Two rods of equal mass m and length ℓ lie along the x axis and y axis with their centres origin. What is the moment of inertia of both about the line x=y :
Solution
The moment of inertia of a rod about an axis through its center and perpendicular to the length of the rod is given by (1/12)ml².
For the rod along the x-axis, the axis of rotation (line x=y) is at an angle of 45 degrees. Using the perpendicular axis theorem, the moment of inertia Ix for this rod about the line x=y is given by Ix = (1/12)mℓ² + (1/2)mℓ² = (7/12)mℓ².
Similarly, for the rod along the y-axis, the moment of inertia Iy about the line x=y is also (7/12)mℓ².
The total moment of inertia I for both rods about the line x=y is then I = Ix + Iy = (7/12)mℓ² + (7/12)mℓ² = (7/6)mℓ².
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