/sci/ - Science & Math

this rod is rotating about the point [math] O [/math] with angular velocity [math]\omega[/math]. [math]\overline{OA} = a[/math] and [math] \overline{AB} = L[/math].
For each point [math] P [/math] on the rod, its speed is given by [math]d(P)\omega[/math], where [math]d(P)[/math] is the distance from [math]O[/math] to [math] P [/math].
Let [math]\rho(P)[/math] be the linear mass density on [math] P [/math], given by
[eqn]

\rho(P) = \lim_{P_1,P_2\to P} \frac{\mu(P_1P_2)}{s(P_1P_2)}

[/eqn]
where [math]P_1P_2[/math] is a segment that contains [math] P [/math], [math]\mu(P_1P_2)[/math] is its mass and [math]s(P_1P_2)[/math] its lenght.

Find an expression for the total kinetic energy of the rod as an integral with variable [math] x = d(P) [/math] and write [math] \rho(P) [/math] as [math] \rho(x)[/math].

I'm kinda stuck on this problem. I tried doing [math]2\mathrm{d}K = v^2\mathrm{d}m = \rho(P)d^2(P)\omega^2\mathrm{d}s[/math], where [math] \mathrm{d}s = \lim_{P_1,P_2\to P}s(P_1P_2)[/math] but I don't know how to proceed. any help would be appreciated