The figure shows a rigid structure consisting of a circular

The figure shows a rigid structure consisting of a circular hoop of radius R and mass m, and a square made of four thin bars, each of length R and mass m. The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of 1.2 s. If R = 0.7 m and m = 1.9 kg, calculate the angular momentum about that axis.

Solution

I₁ = I_com + m h² = 1/2 m R² + m R² = 3/2 m R²

I₂ = I_com + m h² = 0 + m R² = m R²

Now the two members ofthe square perpendicular to the axis have the same rotational inertia

I₃ = I_com + m h² = 1/12 m R² + m (R/2)² = 1/3 m R²

Total rotational inertia is I₁ + I₂ + I₃ + I₄ = 19/6 m R²

= 19/6 * 1.9 * 0.7²

= 2.95 kg.m²

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w = delta theta / delta t = 2 pi / 1.2

= 5.24 rad/s

L = I_total * w = 2.95 * 5.24

= 15.46 kg.m²/s