Body rotating about its axis problem

Question

A rigid body composed of n particles with positions x_i, and masses m_i (i = 1, 2, . . . , n), rotates about the z-axis with constant angular speed ω. Show that the body’s kinetic energy is T = 0.5 Iω² , where you should give an expression for the moment of inertia I in terms of the particle masses and positions.

Answer 1

E_K,T = ΣE_K,i

Each particle that makes up the rigid body has E_Ki = (1/2)m_pv² = (1/2)m_p(x_iω)² (p=particle)

I for a particle at a set radius, x_i, is I_p = mx_i² (like a hoop), so E_Ki = (1/2)I_pω²

Now summing up the E_KI = (1/2) ω² ΣI_p, but the sum of the I's is the I of the entire body.