all right what's up everybody it's been a long time since I did some dynamics so
by special request I've had some requests to do some at least talked a
little bit about pulley problems so here you know I'm going to talk about
absolute dependent motion which for all intents and purposes our pulley problems
here and the thing that we're going to do here is we're just define what
dependent motion is and describe an approach to solve these pulley problems
I'll give you a general overview on how to do that and then probably another
video or something I'll do some example problems but really you know dependent
motion as you could imagine is motion of one particle that depends on another so
basically one particle connected to another and you can imagine all kinds of
you know like pulley systems maybe where you have mass here and then another
pulley on the ground over here and a mass over here BAM there's a pulley
system right here this has like a mass a and a mass and B or or you could think
of more complicated things whatever they are right but that's a pulley system
cranes are probably the you know they have a dependent motion as well
elevators depending on how they were built and even gosh what else just I
guess just anything that's connected right with the chord and and you can you
can model it if you will have ads as to particles so let's talk now I guess I
give you enough overview let's talk a little bit about what a pulley problem
or how to solve these things right here so let's see my approach and that I you
know that you can read in a lot of hip textbooks I like Hibbler but you know
there was like beer and Johnston and and Miriam and Craig and stuff for dynamics
those are part of the three most popular but anyway you know the general approach
like in any textbook is is you have a pulley system here this drawing was
inspired by Hibbler so so I'm not I'm not trying to I'm not saying its mine but
you know it's just I used Hibbeler to inspire myself but here I've got this
mass at a mass of B here connected by this cable and really what you want to
is is for each particle the first thing you want to do is select a reference or
datum for each particle and that's important that each particle and so here
so for particle a a characteristic of a reference or datum location is it has to
be fixed and you have to be able to at least what's given you have to be able
to find the distances so a fixed location for me for a you know this
might be a pop so ninety degrees to the direction of motion so let's see the
datum should be fixed and the the lawn it should be perpendicular to the motion
of the particle motion of particles so this is kind of like our our
characteristics of the of the datum here so for a I would choose here this would
be my reference or datum datum for a and then for B I would choose ninety degrees
to this and that would be right about like that right here so that a datum for
a for particle a and datum for particle B and I can't tell you enough that this
is far the most important thing is defining this datum and then saying what
is positive with respect to the reference and so I'm going to say motion
this way is considered positive or direction this way is my positive
orientation or my positive distance that's very important what is positive
that way when a is moving towards the data or if it moves towards the data
then it's that's considered negative motion okay versus be moving towards the
datum so if I'm moving away from the datum in this case that's considered
positive motion and now now I have this the next thing I want to do is to find
the distances from the datum to the to the particle or to the first part of the
particle so here in this case right here a particle a is really this this little
fixed pulley here to accommodate the cable and and and then the mass of a
here so really the distance for a here is I know I can choose probably maybe I
would choose right here and I would say the distance from the datum to here
right here to define this distance is I'll call this plus s
ehh ok so that's the distance from here to there right here or I could you know
I could you could argue that that that distance is really to that first
location right here where that little polling dot is there because you know
when a moves that much or did that speed that point right here on the pulley
wheel also moves at that same speed because they're all connected that's one
big particle if you will all right so this is si here and then for B the
distance from B right here ma'am to right here the distance the position of
B from the particle from the reference right here is that distance there okay
and these distances are going to be really important for us when we develop
an equation right here for the length of the cable right and and so here in this
case right here when we want to develop a length for this cape for this cable an
equation for length of cable it's just saying I broke this cable up into parts
and I want to sum them up using how I define these parts okay so so to do that
the blue line is the cable in this case I would have for the cable so let me use
blue cable length cable length is equal to let's see I have one of these SB
lengths so the distance from the datum to here this portion right here to here
is a length sb+ I have a distance from here to here and
I have two of these distances so that we plus two si and if I want to be nitpicky
you know I've got really this little portion right here so I'll call that
shoot what should I call that I'll call that l1 and then I've got even this
portion around the pulley and I'll call that l2 which our end those plus l1 plus
l2 and those links right here L 1 and L 2 are constant
they're also constant and so this cable length assuming it doesn't stretch which
is true right we're going to assume a assume cable does not stretch does not
stretch right here and so here if I take a time derivative which is next right
here if I take the time derivatives right here bye-bye
if I take the time derivatives sorry for the red line if I take these time
derivatives here of this equation right here if I take a time derivative blue
this cable length equation let's see the cable length is constant so the time
derivative of that would be zero that would be d SB DT plus two da DT plus D
well the l1 length here no matter where I am is going to be zero or I'm sorry
the derivative will be time to be zero because L 1 is always remains constant
as does l2 so essentially plus zero this is the same as saying zero is equal to
the velocity of B plus the velocity of a okay which means that VB is equal to
minus VA and that makes sense it means that if VA is moving away right here in
a positive direction away from the data then B is going to be negative right the
velocity will be negative or it's going to be moving towards the data which
which is we define as negative and what's more if I take a second
derivative second time derivative so if I take another derivative of this
velocity you know that it becomes acceleration and so that would tell me a
B plus 2 oh snap I almost messed this up there should be 2 VA right here and this
should be minus 2 VA like that right here to a a is equal to zero a B plus 2
a and so my acceleration relationship between these two particles is that a B
is equal to minus 2 AAA right and that just says about the acceleration okay so
let's see here that's I think that's good and then once you have these
usually these end up being either the velocity or the acceleration
relationship it ends up being the the one other equation you need to solve a
kinematic problem and and so we'll do example of that in second all right see
you soon hope you enjoyed it bye