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Conservation laws: mass, energy and momentum

Filip Mroz

Conservation laws: mass, energy and momentum

Conservation laws are a fundamental characteristic of the universe and play an important part in physics. We see how mass, energy, momentum are conserved.

The whole universe is governed by a lot of laws about how things work. A subset of these laws, known as the laws of conservation, has to do with the idea of conserving observables, basically limiting what happens: preventing something from accelerating indefinitely without the help of an external agent or a mass suddenly disappearing. One would be thankful that such laws do exist, if one imagines what would have happened if they didn’t.

Mass

All principles of conservation say more or less the same thing, that the observable being conserved neither goes anywhere nor appears out of the blue. The law of conservation of mass says the same:

Mass is neither created nor destroyed. It can only be converted from one form to another.

Quite simple. It has a depth of meaning within it. All mass in this universe is conserved. No new mass can be created $meaning, no magic$, and, conversely, the mass already present can’t be destroyed. The meaning of ‘destruction’ here is to be understood properly. Sure, you can break a chalk piece, but $classically$ the number of chalk molecules before and after the breakage, on your hands, on the two pieces, and those that fell on the floor, remain the same.

The latter part of the law allows for the phase transitions of matter and for complex phenomena like radioactive decay. Also, from jewelery to furniture, anything that involves making items by melting and molding and such, is an example, and a proof of this law. To make an iron bar, one has to mine the ferrous ore, get iron from it, melt it into a liquid, pour it into a mold, and let it cool. No iron ore, no iron bar. That’s the iron rule, if you will excuse the pun.

A subtler, more restrictive condition also exists. Mass is localized, that is, it occupies a limited amount of space. It has to have a head and tail, so to speak, at some point. Mass retains it existence at all points of time. If a ball has to go from point A to point B, it has to go through some path between A and B. It has to travel the distance between. The ball will not suddenly disappear from the South Pole and reappear – with or without a time lag – on the North Pole.

Energy

The same Law that applies for mass applies for energy as well.

Energy is neither created nor destroyed. It can only be converted from one form to another.

Simple, again. The total amount of energy in this universe is conserved. What the universe had at the start of time, it has now, and it will have in the future. The second sentence is what clears the huge numbers of doubts that come to mind because of the first. Most natural displays of motion are instances of conversion of potential energy into kinetic energy. A spring jumping when the force pressing it down is released, anything falling from a higher place to a lower place, an arrow flying from a bow, etc.

Potential energy is an apt name. A body that possesses this energy has the potential to do something – jump or fall down or whatever the case may be. Let us go about it one by one:

1. When a spring is pressed, all the energy applied to it gets stored as potential energy, simply, , where is the co-efficient of stiffness of the spring and is the length by which it is compressed. When the pressure is released, there is nothing holding back the spring, and all the energy stored is converted into kinetic energy, and the spring jumps.
2. When something is lifted up, work is done against the force of gravity. All this work done is kept stored in the body as potential energy. When the object then has a chance of falling down, the potential energy slowly transforms itself into kinetic energy. This transformation goes on till either the object’s fall is broken or it reaches terminal velocity, by which time its potential energy is converted into kinetic energy, lost as friction etc. This increase in kinetic energy is basically increase in , and in usual cases, since the mass m is a constant, it is the velocity, or rather, that increases. This is another way of looking at why something accelerates while falling down, as opposed to the use of g.
3. As the arrow is pulled back, the bowstring is stretched taut, storing potential energy into it. When released, all of this potential energy is converted into kinetic energy as the string tries to get back to its original state. This violent jerk is passed on to the arrow, shooting it forward and off the bow.

In all these cases, one point is of importance. The kinetic energy that the body acquires is derived from the potential energy that it had, and nothing else. This implies that, in any case, the kinetic energy is equal to, or less than the potential energy, never more than it. In mathematical terms, the total energy of the body, , is equal to the sum of its kinetic and potential energies,

This implies the law of conservation – without any external source of energy, the LHS of the equation remains a constant, and, in turn, any change in the kinetic energy forces an equal negative change in the potential energy.

Apart from this, there are other occurrences that are not simple changes from stored energy to moving energy. Riding a cycle, for instance. The rider uses his weight and muscular energy to rotate the pedal, which, using a system of gears, gives rotational energy to the wheel. The wheel rotates, and by friction between the tyres and the road, the cycle goes forward. In other words, a bicycle is a machine that allows you to convert muscular energy to rotational energy and that to kinetic energy. It wouldn’t work if the rider does not have enough strength in him.

Mass and energy

One of the most famous equations in physics, though many do not grasp its true meaning, is Einstein’s mass-energy equivalence, , where c is the speed of light. This is not to be taken at face value. Indeed, a coin or a button has nothing to do with the speed of light. What the equation means to say is that, if a coin of mass 2 grams was completely converted into energy, the result would be joule. Try to write down the number with all its zeroes and you realise how big it is.

Why is the equation so important ? That equation is what explains the energy released during a nuclear reaction, for example. During a nuclear reaction, it was observed that there is a slight difference between the sum of masses of the reactants and that of the products. This slight defect, , is converted to energy, and is numerically given by this equation, so, it is useful in reactors.

Consider another case when matter becomes energy: when matter and antimatter meet. Every particle has an antiparticle, like a positron for an electron and an antiproton for a proton, that is same in all respects except for the fact that it carries the opposite charge. Matter that is made of these antiparticles is aptly named antimatter. When a particle and its antiparticle meet, they annihilate each other and what is left is a lot of energy, numerically given by , where M is a particle’s mass.

But all of the above seemingly go against the law of conservation of mass. Solid mass seems to be destroyed! That’s where the equivalence of mass and energy plays its role. Since both independently have their own conservation laws, it is quite called for to have a law of conservation that takes both into account. Mass can become energy, and in some cases, vice versa. So it is not exactly destroyed, and we have

Mass and energy are neither created nor destroyed. They are simply interconverted, and the total mass and energy in the universe remains constant.

Now we have a general idea of what a conservation law says: ‘No magic allowed’. Everything that happens can be accounted for. Up next, we talk of momentum.

Momentum

Momentum is defined from Newton’s laws of motion, so it seems apt to refer back to them when we try to understand where the conservation laws come from. For example, Newton’s First law says,

Every object continues to remain in its state of rest or of uniform motion unless acted upon by an external unbalanced force.

And the Second law gives a hint of what the external unbalanced force does,

Force applied on a body gives rise to a proportional change in momentum.

Or, mathematically,

where is the momentum of the body. The First Law by itself is the Law of conservation of momentum. The Second is invoked just for a mathematical proof. In the case that the external force is non-existent, we have . That is, in the absence of external force, the momentum of the body does not change with time. is a constant with respect to time.

There are subsections of momentum conservation. Translational motion has rotational analogues. Angle for displacement, angular frequency for velocity, and so on. The analogue for force is torque, , and that for linear momentum, , is angular momentum, . Both linear and angular momenta are separately conserved in the absence of linear force and torque respectively. Momentum is conserved, no magic allowed.

Now to clear a doubt that might occur to you. When studying collisions, we are taught that in inelastic collisions, the momentum is not conserved, since the colliding bodies stick together. Whether this is right or not is a matter of perspective. But it does not violate the law of conservation. The reason for this is that in a collision, there is a non-zero external force involved. Our law is defined when there is no such thing. The two situations are mutually exclusive. So the next time you find yourself in a situation and seem to be missing mass, energy or momentum, know that they are always conserved. They are still out there and you will find them if you look closely.

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V.H. Belvadi is an Assistant Professor of Physics. He teaches postgraduate courses in advanced classical mechanics, astrophysics and general relativity. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin. He currently serves as the Editor-in-Chief of Physics Capsule.

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