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In this problem we seek to compute the rotational inertia of a disk of mass M and radius R about an axis through its center and perpendicular to its surface. Consider a mass element dm in the shape of a ring of radius r and width dr (see Fig). (a) What is the mass dill of this element, expressed as a fraction of the total mass M of the disk? (b) What is the rotational inertia dI of this element? (c) Integrate the result of part (b) to find the rotational inertia of the entire disk.
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

Simran Bhatia , 11 Years ago
Grade 11
anser 1 Answers
Aditi Chauhan
236-1180_1.PNG
236-306_1.PNG
236-2423_1.PNG
Therefore, the rotational inertia of the circular disc about the axis of rotation passing through the center and perpendicular to the plane of disk is236-2484_1.PNG
Last Activity: 11 Years ago
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