Wouldn't it be nice if there was a way to make using conservation of energy easy? Well the good news is there is.
The other good news is: just doing this correctly will earn you a lot of partial credit.
As for the bad news...:
- what I am going to show you helps to apply the law of conservation of mechanical energy (kinetic energy as well, but you might not even need the trick in this case). However a problem that involves conservation of energy may involve other things as well. Or it may involve applying only this law, but more than once (i.e. perhaps applying it between two moments and then between another two, different ones).
- what it makes easy is to write down the law, the equation that represents conservation of energy for your particular problem. That does not mean that this equation is always going to look pretty and simple by your standards. Nor that what you'll have to do with that equation is going to be something you'll find easy if you don't get along with algebra.
THE SECRET
So you've decided that you need to use conservation of (mechanical) energy, but gosh your problem looks complicated (not an unlikely scenario if last test is anything to go by!). Here's the secret:
Don't waste your time staring at the problem in horror. Instead, use that time to DRAW A TABLE!
This is what the table should look like if you had time to make it pretty:
But we all know, you don't want to take the time to make it pretty in an exam. So just draw this in a corner somewhere, knowing what KL, etc... stand for (and if you draw it in pen and fill it up in pencil, you can even reuse it for several problems ;-):
where:
- KL: linear kinetic energy
- KR: rotational kinetic energy
- UG: gravitational potential energy
- US: spring potential energy
- KL: linear kinetic energy
- KR: rotational kinetic energy
- UG: gravitational potential energy
- US: spring potential energy
Notice the structure:
- you have 2 main columns, one for initial values of the energies, one for the final ones.
- Each of the main columns is split into more columns: one for each object in your system. Name them after the objects themselves, like, block, wheel, etc...
- There are 4 rows: 2 for the kinetic energies, 2 for the potential energies. These are the forms of energy that make up mechanical energy.
Once you've drawn your table, which should take you about one minute, fill it up.
For each cell, ask yourself whether the object has this kind of energy in this situation (i.e. initial or final).
If it doesn't, write 0.
If it does, insert a check mark.
Don't leave anything blank: you wouldn't be sure whether you considered all the cells or not. Yes, that will take you 2 long minutes. So all in all, you will have spent 3 minutes on this table. You have over 12 minutes per question, spending 3 of those 12 making sure that you don't waste them all by doing the problem wrong is well worth it ;-)
Also, as you know, for gravitational energy, you need to choose a reference height, where you take h = 0 m. You can choose a different one for each object.
Some rows you can fill up quick confidently: if there's not spring in the problem, the last row is 0 all the way. BUT, in general, be VERY careful in filling up the table. It's a lot like a free body diagram for Newton's second law: if you don't have all the correct forces on the diagram, you can forget about getting the problem right. Same here if you forget an energy.
Once you've got the table filled up, write down the equation: initial energies on the left of the equal sign and final ones on the right.
WARNING:
Use subscripts whenever needed: look carefully at the equation after you wrote it down, and if any two letters/symbols are the same, ask yourself whether they do represent the same thing physically. If they don't, add subscripts to both of them, so you can tell them apart. Together with forgetting an energy term, this is the most common source of mistakes.
Example 1:
For the following situation of a mass attached to a string that is wrapped around a wheel:
The table, and the resulting equation, would look like:
Only one symbol is repeated in this equation: m. In both cases, it represents the same thing (the mass of the block), so we don't need to add any subscript.
Example 2:
For the situation illustrated below, where the block now falls on an initially uncompressed spring. The block stops when the spring is completely compressed, but at that moment the wheel is still turning, just with a constant angular velocity now that the torque on it has disappeared - the string is about to go slack and is no longer pulling on the wheel. Pretty nasty-looking problem, right?!:
The table would look like the one below, and the equation that you would obtain for it would be the one under that:
See? No so bad after all!
Only one symbol is repeated in this equation: m. In both cases, it represents the same thing (the mass of the block), so we don't need to add any subscript.
Note that the spring does not have gravitational potential energy because it is assumed to have negligible mass.
