* Derive the equation of angular momentum in the cabe of rotat

Question

* Derive the equation of angular momentum in the case of rotational motion about a fixid axis 

Answer 1

Angular momentum of a particle of rigid body is in the direction of a fixed Z-axis

∴ \(\vec{l_1}=\vec{r_1}\times\vec{p_1}\)

If the rigid body consisting of n particle, then the vector sum of angular momentum of all particle will be obtain in the direction of Z-axis.

but for all particle Σ I_i = I,

\(\vec{L_Z}\) = Iω\(\hat k\) where \(\hat k\) is a unit vector in Z-axis. But angular momentum \(\vec{L_1}\) perpendicular to the axis in rigid body, then total angular momentum

\(\vec L\) = \(\vec{L_Z}\) + \(\vec{L_1}\)

but \(\vec{L_1}\) = 0

∴ \(\vec L\) = \(\vec{L_Z}\)

\(\vec L\) = Iω\(\hat k\)

∴ L = Iω similar to p = mv in linear motion.