Give an example of physical situation in which the angular momentum is zero yet the translational and rotational angular momenta are both non-zero.

The rotating inertia of an object or system of objects in motion about an axis that may or may not pass through the object or system is described by angular momentum.

The rotating analog of linear momentum is angular momentum (also known as moment of momentum or rotational momentum). A closed system's total angular momentum remains constant.

A spacecraft carries a gyroscope that is not spinning as shown in the following figure.

A gyroscope aboard a spacecraft is the greatest example of this physical scenario. Assume that the spaceship has a non-rotating gyroscope, as shown in the diagram. In this situation, the spacecraft's angular momentum around its center of mass is zero. If the gyroscope is rotated, it has an angular momentum greater than zero. Because the isolated system (spacecraft + gyroscope) has no external torque, the angular momentum of the system must remain zero according to the principle of conservation of angular momentum.

This principle can only be satisfied if the spacecraft rotates in the opposite direction as the gyroscope, causing the angular momentum vectors of the gyroscope and spacecraft to cancel, leaving the system with no angular momentum. The spacecraft turns around as a result of rotating the gyroscope, as seen in the diagram above.

As a result, the spacecraft's gyroscope is an excellent example of this physical scenario.