Question

I am now stuck on the impacts of choice of the coordinate system on the change in angular momentum and I would really appreciate it if someone can give me some help on this.

Consider the following example, there is a stick with mass m
and length l
which is initially at rest. Suppose a mass travels perpendicular to the stick and hits the stick at one of its ends in a very short time (then it disappears) and then the stick will have both translational and angular momentum.

However, I am very confused that if I choose the fixed point which coincides with the point of hit as the origin of our coordinate system, then the torque will be zero because the lever is 0
, which implies that the angular momentum will be conserved. Then after the collision, there is no external forces, and the angular momentum will be also conserved. In other words, the angular momentum doesn't change because the impact is so fast and I choose the point of impact as origin. I am wondering why this argument is not correct?

Answer 1

The conservation of angular momentum is not violated when the correct system is considered, including any external torques. Angular momentum is conserved only in an isolated system with no external torque, and the choice of origin impacts calculations of angular momentum when external forces are present.

Step-by-step explanation:

The confusion regarding the change in angular momentum due to the choice of coordinate system arises when considering the point of impact as the origin. Even though choosing the point of impact as the origin may intuitively suggest that torque is zero (since the lever arm is zero), this is a misconception.

What actually happens is a change in the system's angular momentum due to the external force at the point of collision, which is not evident when looking at the torque about the impact point. However, the law of conservation of angular momentum states that the total angular momentum is conserved only when there's no external torque on the system.

As such, if there's an external force present, one should consider the entire system along with any external torques that might affect the conservation of angular momentum.

The key point here is the definition of the system. If the system is defined to include the nail and the surfaces it connects with (which are not frictionless), external forces are present, and thus there's an external torque, meaning angular momentum is not conserved.

The proper way to apply conservation of angular momentum is to include all elements of the system where no external torques are acting. In the example given, if only the stick and the impacting mass are considered, without the nail or surface providing friction, one could argue for conservation of angular momentum about the center of mass of the stick-mass system.

This is because the external forces (assuming no gravitational effects or other forces) act through the center of mass, and they do not exert a net torque on the system.

In essence, this demonstrates that angular momentum depends on the choice of origin and on how the system is defined. When external forces (and resultant external torques) are acting on the system of interest, then the angular momentum of that system is subject to change.