Angular Momentum To Linear Momentum

Nosotros can answer this question in 2 ways: the short answer and the long answer. The short answer is No. Angular momentum cannot be transferred to linear momentum and neither can linear momentum exist transferred to angular momentum. Period.

The ii quantities are distinct and independent of each other with no link whatsoever them. Although it feels natural and appeals to our intuition that the two quantities are related somehow, nature just won't let that happen. But we shouldn't experience dismayed by this ugly truth, nature isn't in the habit of pleasing our whims. As biologist Thomas Henry Huxley famously put it, "the slap-up tragedy of scientific discipline is the slaying of a beautiful hypothesis by an ugly fact." The moral of the quote beingness, although common sense and intuition are important, the validity of a scientific theory is ultimately comes from observation and experimentation. A single ascertainment can pull the rug from a beautiful theory anytime.

For example, a long time ago, it felt natural for early astronomers to assume that the Earth was a perfect sphere and that it went around the Sun in a perfect circumvolve. Today nosotros know that both of those convictions are incorrect. Only 100 years agone, astronomers believed that the universe was static and unchanging – merely now we know it is expanding, and not slowing down. So bold that angular momentum and linear momentum are interchangeable based on a feeling is a risky motility. Only quoting a few examples here and in that location of when we call up 1 changed to the other isn't plenty. And we shall soon observe out near of those are a misconception of the ii quantities.

However, we wish to understand why angular and linear momentum aren't continued every bit we strongly suspect. Simply stating an ugly fact doesn't solve the conundrum so much as to sweep it under the carpet. For this, we need a long answer, one that takes u.s. dorsum to the nuts…

What is momentum in physics?

A few weeks agone, NASA launched a daring experiment aimed at assessing our readiness to gainsay a life-threatening and potentially extinction-level outcome resulting from an impact with an asteroid from infinite. In this experiment, a stray asteroid is assumed to be in a trajectory that will intersect with the orbit of World. The experiment, named Sprint – Double Asteroid Redirection Examination mission, involves NASA shooting a one,200 pound, fridge-sized spacecraft into an orbit doomed to intersect an asteroid named Dimorphos at 15,000 miles/ 60 minutes. The thought is to punch the asteroid hard enough that information technology changes course and avoids a theoretical rendezvous with Globe. This will be the beginning experiment of its kind to demonstrate humanity'southward ability to deflect a potentially dangerous extra-terrestrial object's rendezvous with Earth, fingers crossed for September 2022 when the "dial" is expected to occur.

The question, however, is what makes the Geeks at NASA and so sure that they will be able to sway the asteroid?

Just not bad a clamper of metal into it doesn't necessarily guarantee that an asteroid volition shift course. Think of the asteroid as a charging bull and Earth is the cowboy waving the red flag at it – only this time the poor cowboy is immobile, helpless, and armed with stones in a slingshot. How confident exercise you think the cowboy would be by hurling a stone at the charging bull in hopes of diverting him (save the David Vs Goliath story for Sunday School). To deflect the charging balderdash from its grade, the cowboy can promise for another bull-sized object (say a large boulder) to knock over the charging balderdash in such a way that he misses his target. You come across, more than of import than how fast the bull is charging at the cowboy, is how hard the balderdash charging at him. Momentum is a physical quantity that relates to this thought.

Momentum is a holding of a torso in movement that gives us a measure out of how stiff the body is moving. Information technology doesn't tell us how fast an object is moving (that would be velocity), but rather, how difficult it would exist to end (or to modify) the move of the body. Isaac Newton defined momentum as the quantity of movement of an object.

Angular momentum Vs linear momentum

Equally you lot are undoubtedly enlightened, there are two types of momentum:

Linear momentum – associated with the motion of a body in a straight line.

Athwart momentum – associated with the movement of a body in a rotational or circular motility.

Now, it is quite possible for a body to have both athwart momentum and linear momentum at the same time. For instance, an ice skater has both angular and linear momentum as they rotate and slide along the ice. A football game has both angular and linear momentum as it rotates along its length and travels down the pitch. There are also extended body systems such as a wheel, in which some parts take athwart momentum (the wheels) and it also has linear momentum from translational motion.

What is non possible, still, is for one to be transferred to the other – and for expert reason. As we will soon find out, if angular and linear momentum are interchangeable, that detail system would disobey the laws of physics.

A spinning skater in translational motion is an example of a body having both angular and linear momentum

To be clear, momentum tin can be transferred between different bodies through common interaction but can't spring boat from one blazon of momentum to the other. Angular momentum volition remain every bit angular momentum and linear momentum will be linear momentum.

Related: What is athwart momentum in literal layman'southward terms?

Why angular momentum isn't transferrable to athwart momentum (and vice-versa)

1. The question of units

This is mayhap the commencement glaring difference between the ii quantities. The units of linear momentum are kg.m/s whilst those of angular momentum are kg.m²/southward. It immediately follows that the two quantities cannot be combined together with the basic operations of addition and subtraction. As 1 would expect from simple mathematics, merely quantities of the same units tin can be summed together. This implies that in that location is no such affair as the total momentum of a arrangement in terms of say,

Total momentum = linear momentum + angular momentum

As i would expect of energy, that is,

Total energy of a organization = rotational kinetic energy + translational kinetic energy + potential energy + any other form of energy

This hints at a fundamental difference between the two quantities. Equally we dive deeper, this incompatibility volition become even more credible.

2. The issue of symmetry

Nosotros have a very powerful relation in physics by the name of Noether's theorem – named after a German mathematician, Emmy Noether, who discovered it. She showed that conservation laws (such every bit conservation of free energy or momentum) originate from different symmetries of nature. Symmetry of nature is a condition in which a concrete arrangement behaves the aforementioned way as before despite undergoing a transformation of some kind. For instance:

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

Management symmetry – if a group of scientists was running experiments in a laboratory, they would get the same results if the lab was facing in whatsoever of the other directions. The laws of physics do not depend on which direction you face. There is no preferred direction.

Fourth dimension symmetry – an experiment performed in the morning tin can be replicated at a afterwards time with the same results. There is no preferred time.

In 1918, Noether showed through painstaking mathematics that "for every continuous symmetry, there is a respective conservation law associated with it".

Noether was able to mathematically deduce that fourth dimension symmetry is linked with the conservation of energy. Equally a result, we at present know that when a ball is kicked upwards, its full free energy (= potential + kinetic energy) will remain the same throughout the time of the travel.

In a similar way, translational symmetry corresponds with the conservation of linear momentum. Unlike time, which is non-directional, translational symmetry relates to the spatial directions of space, and the corresponding conserved quantity is a vector. For example, a pluck sliding on frictionless water ice will retain its momentum in both magnitude and direction along the entire length (call it x-centrality) of the water ice. This is also a nod to Newton'south first law in which the conservation of linear momentum is usually derived from in classical mechanics.

In the same way, directional symmetry corresponds with the conservation of angular momentum. Bodies in rotational motion retain their angular momentum in magnitude and direction in the absence of external torque.

These three symmetries are distinct and independent of each other. For example, time symmetry is independent of management i.e. whichever direction you face, the laws of physics will nonetheless be contained of fourth dimension. In the same mode, directional symmetry is contained of translational symmetry .i.eastward. whichever management you face, the laws of physics will still exist independent of translational motion in that direction. Noether theorem further implies that the respective conserved quantities deduced from each symmetry are as well contained and distinct from each other.

So energy cannot cross over and exist conserved as momentum and neither tin momentum exist conserved as free energy. The same applies to athwart momentum and linear momentum. In fact, momentum cannot fifty-fifty be transferred from one dimension to another without violating the laws of physics; momentum conserved in the x-direction cannot be transferred to the y-direction or whatsoever other direction for that thing.

3. The question of direction

Angular momentum and linear momentum of a system are conserved vector quantities: they remain the same in magnitude and direction in the absenteeism of external forces.

Related: Understanding the parallelogram constabulary of vectors with real-life application

Then when nosotros transfer athwart momentum to linear momentum. How does direction fit in?

Remember the angular momentum of an arbitrary particle well-nigh a signal in space is the cross-product of the position vector and linear momentum of the particle in question. This implies that angular momentum and linear momentum don't face in the same management. In fact, according to the right-mitt rule of the vector product, the direction of athwart momentum will be at right angles to the management of linear momentum. Thus if the management is to exist conserved in the transfer of linear momentum to angular momentum (or vice-versa), the trunk in question ought to realign itself in space. Just physical bodies cannot rotate themselves in infinite without the presence of an external torque to facilitate the rotation as required past the laws of motion.

This poses a contradiction in the laws of nature

4. Newton would be dissapointed

It is possible for an isolated system or an isolated torso to accept only angular momentum and no linear momentum. Consider a wheel in deep space spinning about its center. Assume that from your frame of reference this wheel is stationary .i.e. even though it is spinning, it does not have translational motility (linear momentum = 0). Then suddenly, this bike converts its angular momentum (from spinning) to linear momentum and starts moving abroad from you!

That would defy the offset police of motion which requires all stationary objects to remain stationary unless an external force acts on them. Newton wouldn't exist 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!

It would seem that for angular momentum and linear momentum to interchange, an external agent is necessary. But that would defy the principle of conservation upon which the two quantities are founded. More on this below.

Related: Crazy Isaac Newton story of how he discovered the notion of gravity

v. There would exist no such matter every bit conservation of linear momentum or angular momentum

The 2 quantities: athwart momentum and linear momentum share a very important holding in physics, conservation. If linear momentum tin can be converted to angular momentum, then the total linear momentum of an isolated system wouldn't stay the aforementioned earlier and later on a physical interaction. That is,

Total linear momentum before ≠ Total linear momentum after.

The apparent loss in linear momentum would then be attributed to the proceeds in angular momentum. That is,

Total linear momentum earlier = Total linear momentum after + modify in angular momentum

But as nosotros accept seen, one cannot add linear momentum and angular momentum together. Angular momentum and linear momentum are always conserved separately; this is truthful for collisions involving galaxies to those involving sub-diminutive particles. There's no physical show whatsoever of the cantankerous-over from linear momentum to angular momentum or vice-versa.

Mutual quoted examples of when angular momentum and linear momentum interchange

Hither are some examples from everyday life where folks would quote equally "angular momentum to linear momentum" or vice-versa situations.

Riding of a cycle

The thought seems pretty straightforward. A cyclist pedals a bicycle; the wheels acquire angular momentum and as they rotate, the bicycle acquires linear momentum and moves forrard. When the cyclist stops pedaling, the wheels stop rotating and the bicycle loses its linear momentum somewhen coming to a halt.

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

Outset of all, angular momentum is never lost in an isolated system. True, the rotation of the wheels causes the bicycle to move forward. But the bicycle itself is not an isolated system. For the wheel to move forward at all, you need the Earth; an external agent, to push the bicycle forward. In essence, the wheel continuously pushes the Earth, and in turn, the World continuously pushes on the cycle causing information technology to movement forward. Information technology's a classic case of Newton's 3rd police force of move. Angular momentum has nothing to do with the bike's forward move.

And then where does the athwart momentum of wheels go?

Answer: the Earth

The moving wheel exerts a torque on Earth at the point of contact, thanks to friction, the wheels transfer their angular momentum to Earth. You come across, the bike-World arrangement is as close to an isolated system as it tin get. Then, a loss in angular momentum of the wheels means an equal gain in angular momentum of the Earth equally required past the police force of conservation. Ideally speaking, you modify the rotation of the Earth every time y'all ride a cycle or drive a motorcar. Geologically speaking though, you're insignificant, and that change is minuscule. Put in another way, the wheel dumps its angular momentum to World.

Momentum earlier = momentum after

Then, athwart momentum isn't converted to linear momentum, information technology simply goes from the wheels to the World.

As they say, nature always balances her books.

A water turbine

When the water in a pipe passes through a turbine in a pipe (h2o wheel) information technology slows downwards, evidently converting its linear momentum to angular momentum of the wheel.

For this, I will link out to an answer by "Mike Due west" from the department of physics Illinois – Urbana.

A Yo-yo

Once again, seems quite straightforward: every bit it drops direct downwardly (linear momentum along the string becomes rotational). Snap the cord and the Yo-Yo climbs straight up (angular momentum becomes linear).

All right, here we go again.

Phew! I suppose you lot tin just read every bit much.

Conclusion

Angular momentum cannot exist transferred to linear momentum and neither can linear momentum exist transferred to angular momentum. Cheers for reading and come across you on the next one!