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We can answer this question in one of two ways: the short answer and the long answer.

The short answer is No. Angular momentum cannot be converted to linear momentum and neither can linear momentum be converted back to angular momentum. The two quantities are distinct and independent of each other. Period

For the long answer, let us go back to the basics…

What is momentum in physics?

Momentum is a property of a body in motion that gives us a measure of how “powerful” the body is moving. Momentum doesn’t tell us how fast an object is moving (that would be velocity), it tells us how hard it would be to stop (or to change) the motion of the body.

A truck and a supermarket trolley may both be moving downhill at 5mph, but the truck would be harder to stop because it has a bigger momentum.

Isaac Newton who conceived the idea defined momentum as the quantity of motion of an object.

Angular momentum and linear momentum

There are generally two types of momentum

Linear momentum – associated with the motion of a body in a straight line. A bullet fired from a gun is a good example of this.

Angular momentum – is associated with the motion of a body in a rotational or circular motion. A ceiling fan is a good example.

It is quite possible for a body to have both angular momentum and linear momentum at the same time. A football has both angular and linear momentum as it rotates along its length and travels down the pitch, the same can be said for a bullet or an arrow. There are also extended body systems such as a bicycle, which has linear momentum as it proceeds forward in general whilst some of its parts (the wheels) have angular momentum. In this article, I explain angular momentum in layman’s terms.

An interesting property of momentum is that it can be transferred from one body to another. But angular momentum will only be transferred as angular momentum and linear momentum will only be transferred as linear momentum. A cross-over isn’t possible and here is why.

1. They have different units

This is perhaps the first glaring difference between the two quantities. The units of linear momentum are kg.m/s whilst those of angular momentum are kg.m²/s. It immediately follows that it is impossible to perform basic operations of addition and subtraction with the two quantities. As one would expect from elementary mathematics, only quantities of the same units can be summed together. This implies that there is no such thing as the total momentum of a system in terms

Total momentum = linear momentum + angular momentum

As one would expect of energy, that is,

The total energy of a system = rotational kinetic energy + linear kinetic energy + potential energy + other forms of energy

This hints at a fundamental difference between the two quantities.

2. They are associated with different symmetries of nature

There is an interesting concept in physics that relates symmetry and conservation. Examples of symmetries include:

Translational symmetry – The laws of physics are the same everywhere. An experiment performed in your office could be replicated elsewhere with the same results. There is no preferred location.

Direction symmetry – The laws of physics do not depend on which direction you face. If a group of scientists was running experiments in a laboratory, they would get the same results if the lab was facing in any of the other directions (North, East, West, South, or elsewhere in between). There is no preferred direction

Time symmetry – an experiment performed in the morning can be replicated at a later time with the same results. There is no preferred time.

In 1918, a German mathematician Noether discovered an important connection between conservation laws and the symmetries of nature. She proved that “for every continuous symmetry, there is a corresponding conservation law”. A statement which bears her name, Noether’s theorem.

For example, time symmetry is linked with the conservation of energy. As a result, we know that when a ball is kicked upwards, its total energy (= potential + kinetic energy) will remain the same throughout the time of the travel.

In a similar way, translational symmetry is closely linked with the conservation of linear momentum. A pluck sliding on frictionless ice will retain its momentum in both magnitude and direction along the entire length (call it the x-axis) of the ice. This is also a nod to Newton’s first law in which the conservation of linear momentum is usually derived from classical physics texts.

In the same way, directional symmetry is linked with the conservation of angular momentum. Bodies in rotational motion retain their angular momentum in the absence of external torque.

The three symmetries are distinct and independent of each other. For example, time is independent of direction i.e. whichever direction you face, time symmetry holds true. The same can be said of the other symmetries. It also follows that conservation laws associated with them are distinct and independent of each other. In fact, momentum cannot even be transferred from one dimension to another; momentum conserved in the x-direction cannot be transferred to the y-direction.

3. The question of direction

Angular and linear momentum are conserved in both quantity and direction. If we convert angular momentum to linear momentum, how does direction fit into the equation?

The angular momentum of an arbitrary particle about a point in space is the cross product of the position vector and linear momentum of the particle in question. This implies (from the right-hand rule of vector cross product) that the direction of angular momentum will be at right angles to the direction of linear momentum. Thus if direction is to be conserved in the transfer of linear momentum to angular momentum (or vice-versa), the body in question ought to rotate itself in space somehow. But physical bodies cannot rotate themselves in space without the presence of an external torque to facilitate the motion as required by the laws of motion.

4. Sir Isaac Newton would be disappointed

Isaac Newton went through a deal of work to establish the laws of motion. But if angular momentum and linear momentum are transferrable, his whole system comes down crashing. Let me explain.

Consider an isolated bicycle wheel in deep space spinning about its center. Assume that from your point of reference, this wheel is stationary, i.e. even though it is spinning, it does not have translational motion (linear momentum = 0). Then somehow, this wheel converts its angular momentum (from spinning) to linear momentum and starts moving away from you!

That would defy Newton’s first law of motion which requires all stationary bodies to remain at rest unless an external force acts on them. Newton wouldn’t be too happy about that!

This logic would allow us to ride a bicycle in space since all we would need is to convert the angular momentum of the wheels to linear momentum and we’re off!

5. There wouldn’t be conservation laws

Angular momentum and linear momentum are separately conserved quantities.

If linear momentum can be converted to angular momentum, then the total linear momentum of a system wouldn’t stay the same before and after a physical interaction. Any apparent loss in linear momentum would be attributed to any gain in angular momentum. Thus the total momentum of the system would be the sum of the linear and angular momentum. But as we have seen, one cannot add linear momentum and angular momentum together.

Angular momentum and linear momentum are always conserved separately; this is true for collisions involving galaxies to those involving sub-atomic particles. There’s no physical evidence whatsoever of the cross-over from linear momentum to angular momentum or vice-versa.

Why do people say angular momentum converts to linear momentum?

When people say, “it is possible to convert angular momentum to linear momentum.” They usually come up with very convincing examples. But upon close scrutiny, it is usually just a misconception. Here are some examples.

Riding a bicycle

The idea seems pretty straightforward: a cyclist pedals a bicycle; the wheels acquire angular momentum and as they rotate the system acquires linear momentum and moves forward. When the cyclist stops pedaling, the wheels stop rotating and the bicycle loses its linear momentum eventually coming to a halt.

All right, let’s poke a few holes in this argument.

First, the bicycle is not an isolated system, which is a necessary condition when dealing with quantities of conservative nature, such as momentum. For the bicycle to move forward at all, you need the Earth; an external agent which interacts with the bicycle and helps it move forward. In essence, the wheel continuously pushes the Earth, and in turn, the Earth continuously pushes on the wheel causing it to move forward. It’s a classic example of Newton’s third law of motion.

So where does the angular momentum of wheels go?

Answer: the Earth

The wheels of the bicycle acquire angular momentum due to their rotation. In the Earth-Bicycle system, the angular momentum of the bicycle is transferred to the angular momentum of the Earth. Believe it or not, you are altering the angular momentum of the planet when riding your bicycle. Put in another way, the wheel dumps its angular momentum to Earth. As they say, nature always balances her books.

A water turbine

When the water in a pipe passes through a turbine in a pipe (water wheel) it slows down, apparently converting its linear momentum to angular momentum of the wheel.

For this, I will link out to an answer by “Mike W” from the department of physics Illinois – Urbana. 

A Yo-yo

Again, seems quite straightforward: as it drops straight down (linear momentum along the string becomes rotational). Snap the string and the Yo-Yo climbs straight up (angular momentum becomes linear).

All right, here we go again.

Phew! I suppose you can only read as much.

Conclusion

Angular momentum cannot be transferred to linear momentum and neither can linear momentum be transferred to angular momentum. Thank you for reading and see you on the next one!