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I found this question in my physics textbook:
From a certain height a cat is dropped back-side down. The cat rotates his body while falling and lands on his four legs. Does the cat's angular momentum change during the fall?
The answer is no, but I said yes, because I thought the gravitational force will change the angular momentum? Am I missing something?
To change angular momentum, a torque must be applied. Since gravity pulls every part of the cat with a force proportional to its mass (that is, with the same acceleration), there is no net torque on the free falling cat, and thus no change in angular momentum.
This is true for any free falling object, but not necessarily if it is supported at any point. The support together with the gravitational force can apply a torque and therefore change angular momentum.
As to how the cat manages to turn around even with no net torque, this is known as the Falling cat problem, and is visualized nicely in this very disturbing animation from Wikipedia
The rotation is based on the fact that the cat is not a rigid body, and can thus bend in a way that results in its reorientation.
An external force like the gravitational force acts on an object like if it acts on its center of mass. Since the cat's center of mass is on its rotational axis, this would mean that the gravitational force doesn't give any angular momentum to the cat.
In general, a force does not always give angular momentum to an object. It will if the force is applied at a certain distance from the rotation axis.
The gravitational force can indeed give angular momentum to a system. Think of a pendulum that you drop after raising it from it's rest position. In that scenario, the rotation axis is the pendulum holding point and the center of mass would be close to the end of the pendulum. Thus, the gravitational force acts away from the rotational axis and the pendulum will start to rotate around its holding point, thus gaining angular momentum.
The problem is flawed because it does not specify which axis we are supposed to be measuring the angular momentum around.
Angular momentum is always specified relative to an axis of rotation. However, you can always split the total angular momentum of an object into two parts, $L_{total} = L_{external}+L_{CM}$.
The first part is the angular momentum of the whole object around some external axis of rotation. Specifically, $L_{external} = \vec{R}_{CM}\times \vec{P}_{CM}$ where $\vec{R}_{CM}$ is the position vector of the center of mass of the object, and $\vec{P}_{CM}$ is the momentum of the center of mass.
The second part is the internal angular momentum, or the angular momentum of the object measured around its own center of mass.
Gravity can change the external angular momentum, depending on the axis of rotation you choose. The torque from gravity around an external center of mass is just the weight of the object times the horizontal distance of the center of mass from the axis.
What the question is probably trying to get, however, at is the fact that gravity cannot change the internal angular momentum of an object. Since gravity "acts" at the center of mass, the torque of gravity around the center of mass is always zero!
