Video instructions and help with filling out and completing Are Form 8453 S Contributing

Instructions and Help about Are Form 8453 S Contributing

Let's begin part two of rotations I thought I should summarize for you the main points because you learnt a whole lot of stuff but you don't have to carry all that in your head what I'm going to write down is the essential subject you should know so we were going to do rotation of rigid bodies which are forced to lie on the XY plane they have some shape and here are the main ideas we should know you first pick a point around which you want to rotate the body by driving a little skewer through that that's the point of rotation then here are the things we learned first of all the body has a mass M which is the simple thing you put it on a weighing machine that's the mass then it has a moment of inertia which if the body were made of a discrete set of points would be the mass of each point M sub I times the square of its distance from the point of rotation for a continuous distribution of matter like this one the Sun is replaced by an integral for example for a rod it was ml squared over 12 around one of the ends around the center of mass and for a disk it was M R squared over 2 so everybody has a moment of inertia here is the important caveat with respect to a particular point there is nothing called the moment of inertia because depending on the point these distances will be okay then if you want the rigid body to start doing interesting things you want it to move around it'll have an angular velocity Omega which is the rate at which it goes round and round it's the usual frequency times 2 pi because 2 pi radians are a full circle then it has an angular momentum which is the analog of the mass times the analog of the velocity which is universally denoted by the symbol L then if you want to change this angular momentum it's like asking how do you change the ordinary momentum of a particle you apply a force so now we apply what's called a torque the torque is the reason the angular momentum changes that the analog of forces the reason the usual momentum changes I remind you P is MB so it is really M DV DT equals MA so you can also write analogously here how equals I alpha because if you take the D by DT of this guy that's a constant and alpha is the rate of change of Omega last thing I have to tell you is what is the torque or the total torque on a body well if that is the force for example acting in that direction let me just take one force then it's separated by a distance R or a vector R from the point of rotation defined that angle theta as the angle between this point of separation distance of separation R and the applied force then the torque is F times R times sine theta if you took F to be perpendicular to this separation vector then sine theta is 1 if you took F to be parallel to the separation vector torque vanishes because it's no use no use applying a force in the line joining you to the center because you don't produce rotations that way so sine theta tells you how much of the force is good at producing rotations if you got many forces and many torques you got to add all the torques that's one thing to be careful about for example you can have a body where one force is trying to do that and here maybe a second force acting that way trying to do the opposite so we keep track of the torque by saying a there is clockwise or counterclockwise each force you can tell me intuitively by looking at it this guy is making it go counterclockwise this is rotating clockwise counter clockwise will we take and positive clockwise will be taken negative similarly angular acceleration will be positive if it's counterclockwise increasing so this is summary of what are everything we did yesterday on Monday you guys follow that so there's an analogy to F equal everything as a rotational counterpart but life is more difficult because in the case of F equals MA and was simply given to you here the moment of energy has to be calculated from scratch given the mass distribution that forms the body similarly you may have been simply given the forces in F equals MA but here even after I give you the forces you got to do some work calculating the torque but every force you got to find how far it is from the point of retention what's the angle between the force and the vector separating where the force is to where the rotation is taking place and calculating the sine theta okay that's all we did and what I did towards the end of the class was to calculate the moment of inertia for a couple of objects I showed you for a disc it's M R squared over 2 for a rod I did two calculations one was around this end and the moment of inertia around the end was ml squared over three and the moment of inertia around the center was ml squared over 12 and at the end of the lecture I said notice the following fact the moment of inertia through the end the moment of inertia through the center of mass are related in a very simple way which is the moment of inertia with respect to the new point this moment of inertia with respect to the center plus the mass of the entire rod times the square of the distance L over 2 squared separating the axis going through the center