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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 4C / Solution

Writing

x_{i} = (x_{i}, y_{i}, z_{i})

, rotation about the

z

-axis gives particle

i

the speed

ω x_{i}^{2} + y_{i}^{2}

. Its kinetic energy is therefore

m_{i} ω^{2} (x_{i}^{2} + y_{i}^{2}) /2

, so

T = \frac{1}{2} I ω^{2}, I = \sum_{i = 1}^{N} m_{i} (x_{i}^{2} + y_{i}^{2}) .

This is the defining mass sum for the moment of inertia about the

z

-axis.

For the uniform cuboid, the mass density is

ρ = M / (8 ab c)

and

I = ρ \int_{- a}^{a} \int_{- b}^{b} \int_{- c}^{c} (x^{2} + y^{2}) d z d y d x = \frac{M}{3} (a^{2} + b^{2}) .

Solved by gpt-5.6-sol high.

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