Impulse-momentum relationship for planar rigid bodies
For a rigid body, the sum of external forces is equal to the time rate of change of linear momentum:
\sum \mathbf F = \frac{\mathrm d \mathbf P}{\mathrm d t}
Similarly, the sum of external moments about the center of mass equals the time rate of change of angular momentum:
\sum \mathbf M_C = \frac{\mathrm d \mathbf L_C}{\mathrm d t}
Integrating these equations over a time interval yields the impulse-momentum equations:
\begin{aligned} & \int_{t_1}^{t_2} \sum \mathbf F \, \mathrm d t = \Delta \mathbf P \\ & \int_{t_1}^{t_2} \sum \mathbf M_C \, \mathrm d t = \Delta \mathbf L_C \end{aligned}
In 2D planar motion, angular momentum about the center of mass is often simplified to:
\mathbf L_C = I_{zz}^C \omega \mathbf k
where I_{zz}^C is the moment of inertia about the z-axis and \omega is the angular velocity.
The spinning skater
Consider a skater initially spinning with angular velocity \omega_1 and moment of inertia I_1. When the skater pulls their arms in, their moment of inertia decreases to I_2 while their angular velocity increases to \omega_2. In the absence of external torques, angular momentum is conserved:
I_1 \omega_1 = I_2 \omega_2
Since I_1 > I_2, it follows that \omega_2 > \omega_1.
The rotational kinetic energy is given by:
T = \frac{1}{2} I \omega^2
Initially, the kinetic energy is:
T_1 = \frac{1}{2} I_1 \omega_1^2
After pulling in their arms:
T_2 = \frac{1}{2} I_2 \omega_2^2 = \frac{1}{2} \frac{I_1^2}{I_2} \omega_1^2
Because I_1 > I_2, it can be shown that T_2 > T_1. This increase in kinetic energy is not a violation of energy conservation. As the skater pulls their arms inward, they perform work, which converts internal energy into rotational kinetic energy.
This process demonstrates the interplay between angular momentum, moment of inertia, and rotational kinetic energy. The skater’s action of pulling their arms in results in a redistribution of mass, leading to a change in the moment of inertia and a corresponding change in angular velocity and kinetic energy.
For more insights into this topic, you can find the details here.