Rotations
It's been quite a while since I've done anything on this site- too long! I really want to make an excuse that I've been too busy with Chemistry work (or some other not-quite-truth) but I think I've just been feeling a bit out of it since Christmas.
A quick update firstly: I'm going to change the layout of the study notes pages pretty soon, as soon as I get some free time. Also, I really need to update the list of recommended books and put in some reviews for books I hadn't read back when I wrote it. Including the FEYNMAN LECTURES!! Best. Christmas Present. Ever. I still haven't finished them though (or even finished vol. I) but I do want to mention them. Oh, and for those with a less serious bent, this series is both enlightening and entertaining. That's the only fanfic I've ever read but it's amazing to be honest.
Okay, so about a week ago I went to UCL to see a public lecture. The topic was suprisingly hardcore- it was on rotations and the coriolis effect. The lecture relied on a small amount of maths, but luckily it also had plenty of demonstrations and I'm told by those in the know that the lectures there normally rely much less on mathematics and are more accessible. Nevertheless, I enjoyed it and I believe I understood most of it as well!
First topic was the concept of 'moment of inertia'. Qualitatively, your moment of inertia is to rotation as mass is to linear movement- it resists changes in angular velocity. You moment of inertia depends partly on your mass, but also on the distribution of mass within a system. For example, if you have a metre long rod and hold it at the end, the moment of inertia is larger than if you pivot it around the centre instead.
Qualitatively, what does a moment of inertia mean? Well, we can return to the rod pivoting about one end. If we want to find the total energy, we go from one end to the other and add the energy of small 'chunks' of the rod. Of course, we should let the size of the chunks go to zero and the number go to infinity, but that would take too long! We go from one end, and find the total energy E=0.5mv^2, but since we are working with 'chunks' we must use a little mass, denoted (dm). The 'd' is short for delta, which means a change if you're a mathematician (if you're a geographer, then delta means something to do with rivers, and if you're both then you're simply confused).
The v also needs to be rewritten, because different parts of the rod move at different speeds. This is not good, because as aspiring physicists we need to look for constants. We need to find something related to speed, but which is constant throughout a rotating rod. We might try angular velocity- a funny concept which means how fast you are turning but not how fast you are actually moving. Anything given in rpm is in fact an angular velocity as revolutions are a fixed amount of angle (2 pi to be precise)- so a ball spinning at 100rpm is spinning with angular velocity w=100(2pi)/60=10.5 rad per sec. You can either trust me or prove to yourself that the angular velocity w is related to tangential velocity v by the following reliationship:
v=wr, where r is the radius of the circle something is rotating in.
If you want to prove that result, you either need to know the formula for arc-length in radians (l=rt where t is angle) or consider the time is takes to complete a full circle.
Ok, so we now know how to find the energy of a rotating rod: we "sum" a lot of little bits of energy where dE=0.5dm(wr)^2=0.5w^2(dmr^2)
It's a little hard to manipulate this in text only, so pleas bear with me. The above formula consists of two main factors, a factor equal to 0.5w^2 and another equal to dm(r^2). The first is constant everywhere, so we can factorise it out of the sum, and we are left with:
Total Energy=0.5w^2 * sum_(dmr^2)
We call this sum, the moment of inertia. As you can see, if you have the same angular velocity but double the moment of inertia then you need twice as much rotational kinetic energy. Moment of inertia is given the symbol I (usually italicised). We can summarise this by stating the rotational and translational formulas for kinetic energy:
Firstly, translational energy is given by E=mv^2/2
Secondly, rotational energy is given by E=Iw^2/2
Hopefully you can see the similarities, and why I said earlier that moment of inertia was to rotation as mass is to translation. The analogy to translational equivalents is also true for angular velocity w and if you want to, you can derive a set of equations that perfectly parallel the SUVAT ones for translational kinematics, but using rotations instead. That would require too sophisticated typesetting for here, but it's worth a go if you want some confidence deriving formulas like that. With that in mind, here are some simple things you can try to work out. If you get stuck just look them up online, they're not that important:
What are the units for moments of inertia?
What is the moment of inertia of a rod of length L pivoted around the end?
What is the moment of inertia of a rod of length L pivoted around the centre?
What are the kinematics equations for constant angular acceleration, and explain why these have the same form as the equations for linear motion. (If you have time, derive the linear equations but for constant jerk, i.e. rate of change of acceleration. What do you notice?)
I'll post soon on the next part of the lecture- angular momentum and gyroscopes!
