Direction of Angular Momentum

Rotational motion is one of the most commonly observed motions in the universe. The movement of planets around a star and the movement of electrons in an atom are examples of rotational motion. One of the fundamental attributes of a rotating body is its angular momentum.

Much like linear momentum that opposes the change in the body’s state from rest to motion and vice versa, angular momentum is the innate virtue of a rotating body that keeps it in motion. 

The conservation of angular momentum in a rotating body forms the basis of many spectacular phenomena. A demonstration of angular momentum is a skater controlling the rate of their rotation while spinning. Angular momentum is a vector quantity. Therefore, the study of its direction is also important.

What is Angular Momentum?

Newton’s second law states that any change in the state of a body, either to take it from rest to motion or from motion to rest, is brought about by applying force. But what opposes force when it tries to change the state of a body? The answer is momentum.

Momentum is equal to the force required to bring a body in motion to rest in unit length or make a body move from rest. Similarly, in rotational kinematics, the analogue to linear momentum is angular momentum. What are angular momentum and linear momentum?

Linear momentum is the product of mass and velocity, whereas angular momentum is the product of the moment of inertia and the angular velocity of the body. The moment of inertia is the resistance by a body to angular acceleration. 

So what is angular momentum? Angular momentum is a vector quantity and depends on both magnitude and direction. Similar to linear momentum, angular momentum also follows the law of conservation of momentum, which states that the initial and the final momentum of a changing system must be equal. Therefore, the angular momentum of any rotating body is always conserved, provided that there is no external torque acting on the body.

Types of Angular Momentum

Similar to angular velocity, there are two ways of differentiating angular momentum for an object. 

1. Spin Angular Momentum

The spin angular momentum is the angular momentum about an object’s centre of mass. Our planet has orbital angular momentum due to its revolution around the sun.

2. Orbital Angular Momentum

The orbital angular momentum is the angular momentum around a given centre. Our planet possesses spin angular momentum due to rotation about its polar axis.

3. Cumulative Angular Momentum

The cumulative angular momentum is the sum of the spin angular momentum and the orbital angular momentum. On the topic of our planet, the primary value that is conserved is the cumulative angular momentum of the solar system due to the angular momentum being replaced by a small but significant stretch in between the other planets and the sun.

Examples of the Use of Direction of Angular Momentum

A person sitting on the rotating chair holds a spinning bicycle wheel such that the wheel is rotating perpendicular to the direction in which the person is. If the wheel is spinning in the clockwise direction, due to angular momentum, the person sitting on the chair will start spinning too. The direction of the spin will be along the direction in which the bicycle wheel is spinning.

But if the person were to turn the spinning bicycle wheel such that the wheel is now spinning in the counterclockwise direction, then the rotating chair would also start spinning in the counterclockwise direction. This happens because angular momentum is a vector quantity, and by turning the direction of the spinning wheel, the direction in which the chair rotates also changes.

Right-hand rule of Conservation of Angular Momentum

A simple rule can be used to find the direction of angular momentum. This rule is called the right-hand rule of conservation of angular momentum. According to this rule, if a body is rotating, then by holding your right-hand parallel to the body and curling the fingers in the direction of rotation, the thumb of the right hand will give the direction of the angular momentum of the body. This is because the angular momentum is perpendicular to the rotation of the body. When we curl the fingers in the direction of the rotation, the finger that is perpendicular to the curled fingers is the thumb.

Conclusion

Angular momentum is a vector quantity and depends on both magnitude and direction. Similar to linear momentum, angular momentum also follows the law of conservation. Therefore the angular momentum of any rotating body is always conserved provided that there is no external torque acting on the body. The rule to find the direction of angular momentum is the right-hand rule of conservation of angular momentum.

The right-hand rule states that if you position your right hand such that the fingers are in the direction r, which is the rotation of the body, then curl them around your palm such that they point towards the direction of linear momentum (p). The outstretched thumb gives the direction of angular momentum (L).