CONSERVATION LAWS

CONSERVATION LAWS AND WHEN THEY APPLY

A conservation law is a law that tells you that some quantity (momentum, some energy, etc...) remains the same for a certain system (remember that "system" in physics is just a jargon way of saying an object or a group of objects).

When expressed in equation form, a conservation law always has the structure:

initial quantity = final quantity

Initial and final just refer to the two instants of time that you've decided to care about (the quantity in question is also conserved at other moments in between, but for the purpose of solving a problem you need to pick the two that you are given information about).

CONSERVATION OF MOMENTUM:

CONSERVATION OF LINEAR MOMENTUM:

p_i = p_f

It is always true for a system that is "isolated", that is on which there is no net force at any time between the moments you consider.

Example: think about the carts that you were colliding during the lab. If you consider just one cart, and you consider only what happens before the collision, the linear momentum of this cart does not change, it is conserved. The mass remains the same, and the cart moved at the same speed in the same direction. Until it hit the second cart...

Once it hits the other cart, that other cart exerts a force on it and changes its momentum (that's what you were calculating in one of the labs too: you were checking that the impulse due to the second cart was equal to the change in momentum of the first). So if you compare the momentum of the cart before the collision, to its momentum after, you will find that its momentum has changed. It is not conserved. HOWEVER, the momentum of the two carts considered together as one system is not changed by the collision. That "total" momentum was conserved at times before the collision, and it is also conserved at times before and after the collision.

This is why when it comes to momentum, whether or not it is conserved depends on which system you are talking about. In order to be able to apply this law, you've got to pick a system that is "isolated". In the context of collisions, that means that you need to apply the law to a system made up of all the objects that interact with one another between the times you care about.

CONSERVATION OF ANGULAR MOMENTUM:

L_i = L_f

Same idea and rules as above, except that instead of an external net force, what would change the momentum is an external net torque.

Conservation of angular momentum and linear momentum are separate laws: each form of momentum is conserved independently of each other, so it has its own conservation law, so that we have two different laws, expressed by two equations:

p_i = p_f

L_i = L_f

but NOT p_i +L_i = p_f + L_f, unlike you do when dealing with energy: in the case of energy, linear and angular kinetic energies all get to sit in one equation.

CONSERVATION OF ENERGY:

If by "energy" we mean the total energy, talking into account all possible forms it can take, then this law would be always true. But that's not very useful in practice: in the situations described in the problems given to you, you either don't need to take into account all these forms of energy, or you just cannot. 

The other conservation of energy laws that you are given are special cases of this one. The idea is this: 

- energy, like momentum, can be transferred between objects. So here as well, you need to apply the law to a system made up of all the objects that interact with one another between the times you care about.

Often there is only one object you need to care about, but sometimes there's more: for example in the collisions we did in the lab, you had to consider the energy of both carts: your system was the two carts.

- energy can change form. 

example: one of many! Kinetic energy can be transformed into potential gravitational energy. Linear kinetic energy can be transformed into angular kinetic energy. This is why linear and angular kinetic energies get to sit in one equation.

This is what gives rise to the different energy laws: each one takes into account only some forms of energy. Consequently, they are only true when these forms of energy are not transformed into yet different ones (see below).

CONSERVATION OF KINETIC ENERGY:

K_i = K_f

This one is true when the kinetic energy is not transformed into any other kind of energy. It is useful when kinetic energy is merely transferred from one object to another. 

This happens for example in what we call elastic collisions. So if a problem tells you that a collision is elastic, then this law applies. 

Note that as with momentum, there are TWO kinds of kinetic energies: linear, and angular.

CONSERVATION OF MECHANICAL ENERGY:

E_Mi = E_Mf 

Many problems involve this one. 

"Mechanical energy" is the energy that takes into account, that contains, the following forms of energy:

- kinetic energy K, which can be linear or angular.

- mechanical potential energy U, which can be gravitational or spring potential energy.

So there are FOUR kinds.

So whenever you have a situation when kinetic energy is transformed into potential energy, or vice versa, that's the one to pick. 

HOWEVER, it does NOT apply when kinetic and/or potential energy is transformed into yet other forms of energy, like heat. This usually happens when friction is involved, or an object is permanently deformed (not elastically). When it does not apply, you may be able to use the work energy theorem instead (it all depends on the problem of course, the information given in it, the question...)

There is a trick to apply conservation of mechanical energy easily (and correctly!) to complicated problems. To find out which, go to this tutorial.

WORK ENERGY THEOREM

This rule does not have the form of a conservation law, but you need it in your tool kit too, it is related to the issue of conservation of energy. 

It state that the change in kinetic energy is equal to the work done by the net force. 

Another way to say this is that the change in kinetic energy is equal to the sum of the works done by all the forces.

This rule is very similar to the conservation of total energy (not just mechanical), and indeed you'll find that calculating the work done by the weight gives you the same expression as the gravitational potential energy. Like energy, work is expressed in joules. So the two laws are closely related, they're almost different versions of the same thing. Concretely what this implies for you is two things:

- it always applies (whether there is friction involved or not).

- it is most useful when friction takes energy out of the system. 

This is because you have no way to handle this situation with a conservation of energy law, which in turn is because we don't have a neat expression for that form of energy. And that, in turn, is due to the fact that how much energy is transformed by friction depends not just on the situation of the object at the two moments you want to consider, but also on what has happened to the object in between. That's what we mean by friction is a non-conservative force: the work it does can't be expressed as a change of energy that depends only on the situation at two different moments, and not what happens in between, as it can be done for weight and spring force.