An aggregate of objects may have combination of motions. Some of which may be translating, others rotating and remaining may be undergoing a mix of motions. Conservation of angular momentum encompasses to analyze such complexity in motion. The generality of the conservation law is actually the reason why angular momentum has been defined about a point against an axis. This provides flexibility to combine all motion types. Had the angular quantities been defined only about an axis, then it would not have been possible to associate motions of objects with a common axis.

Conservation of angular momentum, like conservation of linear momentum and energy, is fundamental to the nature and laws governing it. It is more fundamental than classical laws as it holds where Newton's law breaks down. It holds at sub-atomic level and also in the realm of motion, when it nears the speed of light.

We have studied that the time rate of change of angular momentum of a system of particles is equal to net external torque on the system. This is what is known as Newton's second law in angular form for a system of particles. It, then, follows that there will not be any change in the angular momentum, if net external torque on the system is zero. Though, there is no external torque on the system, the particles inside the system may still be subjected to forces (torques) and, therefore, may undergo multiple change in velocity (angular velocity). Evidently, we can analyze resulting motions of the system with the help of the conservation of angular momentum.

It may appear that conservation law is limited in scope to situation when external torque is zero. However, it all depends how do we define our isolated system (no exchange of force and mass with the surrounding is permitted in isolated system). It is not very difficult to imagine that it is always possible to include constituents in the system such that there is no external torque by simply extending the boundary of the isolated system. We must, however, realize that this preposition is a basically a theoretical consideration for understanding purpose and may not have any practical value.

Figure 1: Particles move randomly or may rotate about an axis.

System of particles

There are many ways or forms in which this law can be stated. Mathematically,

t net = 0 t net = 0

⇒ t net = ⅆ L ⅆ t = 0 ⇒ t net = ⅆ L ⅆ t = 0

⇒ ⅆ L = 0 ⇒ ⅆ L = 0

From this result, we can state conservation of angular momentum in following equivalent ways :

Definition 1: Conservation of angular momentum

If net external torque on a system is zero, then the angular momentum of the system can not change.

⇒ Δ L = 0 ⇒ Δ L = 0

Definition 2: Conservation of angular momentum

If net external torque on a system is zero, then the angular momentum of the system remains same.

⇒ L i = L f ⇒ L i = L f

Application of the law of conservation of angular momentum is not as straight forward as it may appear. Theoretically, though, it is possible to conceive or define a system such that there is no external torque, but in real time situation it is not advisable to extend the system. If we also consider the fact that we have to consider angular momentum of all objects within the system about a common point, then we realize that it is not actually possible to apply this law in most of the real time situation - unless when we have some comples algorithm with a powerful computer at our disposal.

However, there is the fact that motions along mutually perpendicular axes are independent of each other. This is an experimental fact, which has been described in the module on projectile motion. This independence is a characteristic feature of motion and comes to our rescue in analyzing motion of an isolated system in the context of conservation of angular momentum.

The angular momentum is a vector quantity having direction as well. As such, we can express conservation of angular momentum along three mutually perpendicular axes of a rectangular coordinate system.

L xi = L xf , when t x = 0 L yi = L yf , when t y = 0 L zi = L zf , when t z = 0 L xi = L xf , when t x = 0 L yi = L yf , when t y = 0 L zi = L zf , when t z = 0

Looking at the above formulations, we realize that application of conservation law in three mutually perpendicular directions is a powerful paradigm. Even if external torque is not zero on a system, it is likely and possible that component of torque in a particular direction is zero. The component form of the conservation law allows us to apply conservation in that particular direction, irrespective of consideration in other directions. For, we can always orient our coordinate system such that one of the axes coincides or becomes parallel to the direction in which net external torque is zero. This is a great improvisation as far as application of the conservation of angular momentum is concerned. We, therefore, can state the component form of the law :

If the net external torque on a system along a certain direction is zero, then the component of angular momentum of the system in that direction can not change.

Isolated body system is a special case of general conservation law. It may occur to us that the rotational description can very well be analyzed in terms of Newton's second law in angular form. Why do we need to consider such eventuality? As a matter of fact, we are considering isolated bodies, which are capable to change their mass distribution. It would be very difficult to analyze motion in terms of second law of motion of a rigid body system, which changes mass distribution internally. Conservation law, on the other hand, can elegantly provide the solution.

Humans are one such body. We can change our body configuration by manipulating arms and legs. This is what dancers, skaters and spring board divers do. They change their body configuration. This changes their moment of inertia about the axis of rotation. However as there is no external torque involved, there is corresponding change in their angular velocity to conserve the angular momentum of the body system.

There is yet another situation, when conservation of angular momentum for body system can be helpful in analyzing motion. We can consider multiple parts of the body system which may selectively undergo rotation about a common axis of rotation. Like motions of two discs along a common spindle can be analyzed by considering conservation of angular momentum of the isolated body system.

Figure 2: The disks rotate about a common axis.

System of two disks

The statement of conservation of angular momentum for isolated body system can take advantage of the relation valid for the rigid body. Here,

L i = L f I i ω i = I f ω f L i = L f I i ω i = I f ω f

Here, some examples are given to illustrate conservation of angular momentum, as applicable to the system of rigid bodies.

1: The revolution of planets around Sun

The planet like the Earth moves around Sun along an elliptical orbit. No external torque is applied to the Earth system. We can, therefore, apply conservation of angular momentum to the system. It must be noted here that distribution of mass within the Earth about its own axis of rotation does not change. However, distribution of Earth's mass about an axis passing through the center of mass of Sun changes as the Earth transverses along the elliptical path.

Figure 3: The Earth moves around Sun in an elliptical path.

Earth revolving around Sun

When the Earth comes closer to the Sun, the moment of inertia of the Earth about an axis through the center of mass of the Sun decreases. In order to conserve its angular momentum, it begins to orbit the Sun faster. Similarly, the Earth rotates slower when it is away from the Sun. All through Earth's revolution,

I i ω i = I f ω f I i ω i = I f ω f

2: A person sitting on a turn table

A person sitting on a turn table can manipulate angular speed by changing moment of inertia about the axis of rotation without any external aid. In order to accentuate the effect, we consider that person is holding some weights in his outstretched hands. Let us consider that the system of turntable and person is rotating about vertical axis at certain angular velocity in the beginning. The moment of inertia of the body and weights about the axis of rotation is :

Figure 4

A person sitting on a turn table
(a) The person with extended arms (b) The person with folded arms

I = ∑ m i r i 2 I = ∑ m i r i 2

When the person folds his hands slowly, the moment of inertia about the axis decreases as the distribution of mass is closer to the axis of rotation. In order to conserve angular momentum, the turn table and person starts rotating at greater angular velocity.

ω f > ω i ω f > ω i

3: Ice skater

An ice skater rotates with outstretched hands on one leg. When he/she folds the hands and legs closer to the axis of rotation, the moment of inertia of the body decreases. As a consequence, ice skater begins to spin at much greater angular velocity.

4: Spring board jumper

The spring board jumper follows a parabolic path of a projectile. In this case, the motion is about an accelerated axis of rotation, which moves along the parabolic path. In the beginning, the jumper keeps hands and legs stretched. During the flight, he/she curls the body to decrease moment of inertia. This results in increased angular velocity i.e. more turns before the jumper hits the water.

Figure 5: The jumper follows a parabolic path under gravity.

Spring board jumping

When the spring board jumper nears the water surface, he/she stretches hands and legs so that he/she she has straight line posture ensuring minimum splash while hitting water surface.