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workshop 4 problem 1

workshop 4 problem 1 - -1 Problem Moment of inertia The...

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1 Problem. Moment of inertia The purpose of this project is to calculate the angular momentum of the Earth. We will model the Earth as a solid uniform sphere with a mass density of ρ = 5500 kg / m 3 that rotates about its north–south axis once per day. The average radius of the Earth is a = 6378 km . The angular momentum of a rotating object is where I is the moment of inertia about the rotation axis and ω is the angular frequency in rad / s . Computing ω is straightforward and involves no calculus. The moment of inertia is computed relative to the axis of rotation: it measures how difficult it is to change the angular frequency of the object. The moment of inertia of a hollow cylinder, having mass M and radius r , and rotating about its axis of symmetry, is Mr 2 . (We assume the cylinder is empty, with no top or bottom, and with its thickness very small relative to r .) If an object lies within a distance R from the axis, and the differential dM describes the distribution of the object’s mass with respect to distance from the axis, then the moment of inertia (of the object about the axis, to be precise about what we’re measuring) is I = R 0 r 2 dM.
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