Earlier we derived Newton’s second law for rotation of a rigid body about a fixed axis to be
\begin{equation}
\mathcal{T}_{\text{net,ext}} = I\,\alpha,\tag{9.81}
\end{equation}
(Calculus step) Now, let us integrate both sides over an interval of time, say from \(t=t_i\) to \(t=t_f\text{.}\) We will get
\begin{equation}
\int_{t_i}^{t_f} \mathcal{T}_{\text{net,ext}}\, dt = \int_{t_i}^{t_f} I\,\alpha\, dt.\tag{9.82}
\end{equation}
We recognize the left side as just the net rotational impulse. Now, on the right side, suppose we replace \(\alpha\) on the right side by \(d\omega/dt\text{,}\) cancel out \(dt\text{,}\) and then complete the integral. We get
\begin{equation}
J_{\text{rot,net}} = I \omega_f - I \omega_i,\tag{9.83}
\end{equation}
where \(\omega_i\) and \(\omega_f\) are angular velocities at the initial and final instants. Since \(I\omega\) is angular momentum for rotation about a principal axis, find that net impulse equals change in angular momentum.
\begin{equation}
J_{\text{rot,net}} = \Delta L,\tag{9.84}
\end{equation}
where
\begin{equation}
\Delta L = L_f - L_i = I \omega_f - I \omega_i.\tag{9.85}
\end{equation}
Another way of arriving at result in Eq.
(9.84) is by looking at plots. If we plot both sides of Eq.
(9.81) versus time and compare the areas under the two curves, the left side gives the change in angular momentum and the right side gives the net rotational impulse by external forces as shown in
Figure 9.112.
Remark 9.113.
We have derived Eq.
(9.84) from Eq.
(9.81), which is, strictly speaking, only correct if we assume
\(I \) is constant in time. So, it would appear that we should not use Eq.
(9.84) if
\(I \) is changing.
However, more general considerations in a different derivation shows that if moment of inertia varies during the time interval, say from being
\(I_i\) at
\(t_i\) to
\(I_f\) at
\(t_f\text{,}\) then, you can still use Eq.
(9.84), but now, we will have to use a different formula for
\(\Delta L\text{.}\)
\begin{equation}
\Delta L = I_f\omega_f - I_i\omega_i\tag{9.86}
\end{equation}