The Young and The Restless 7/10/17
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let's do a quick recap of what we have discussed in this chapter so far so what
we have talked about are the kinematic properties of physical objects Model as
miniscule particles where all their masses are concentrated and some of the
kinematic quantity is that we have been interested in are the position vector
the differential displacement velocity and acceleration and we studied
these quantities in three different coordinate system one was Cartesian
coordinate system and the other two were curvilinear coordinate system namely
tangent normal and polar also known as r-theta system and we looked at the
velocity expression and acceleration expression in each of these three
coordinate systems now we're in a good position to actually write down a
general method for solving all the kinematics problems for the particles
and general method is actually fairly simple so let me start by writing method
for solving particle kinematics problems and as long as you follow this method
consistently to solve the problems you see that you will have no trouble
pretty much solving any problem in this particular chapter so the step one of
this method is to look at the problem look at the input information given to
you and choose either one or many coordinate systems so I will write choose
coordinate system and in parenthesis I will write s because in some problems you may have
to choose two different point systems and be able to relate the various
kinetic properties in the two coordinate system to be able to solve the problem
and we'll see an example of that so choose correct system
okay so this could be we have three choices you have cartesian
you have tangent normal I'll write it as TN and you have r-theta okay step two is
show your chosen coordinate system as well as position variables so this could
amount to you know just showing your coordinate system like this this is oxy
and here is my particle and then that's my position vector of the
particle P okay or if it was a tangent-normal system or r-theta system then you
might show a path you know if the particle you say this is my tangent
vector and this is my normal vector okay and if you have r-theta system you
do something similar so just showing the coordinate system and position variable
is going to be very important because if I am looking at you know your solution to
the problems I need to know where your coordinate system is located and how your
position variables are indicated step 3 of this process is to write down
velocity and acceleration expression in chosen coordinate systems
so if you have chosen only Cartesian coordinate system they need to write its
velocity you will start with position vector and then velocity and acceleration
if you choose tangent normal then you will write the velocity expression in
tangent normal system and acceleration and same thing with the polar
coordinate system and finally you will solve for the unknowns from the velocity
and acceleration expressions and that's all there is to solve problems in this
chapter so let's do a simple example ok what I will try to do is I'll try
to bring in all the three coordinate system and you can see how they would
all relate to one another ok and I'll choose a simple example of a particle
moving along the circular track along a circular path so this could be a car
that's moving along a circular track this could be an aeroplane you know
traveling along a circular path it could be a vertical plane it could be a horizontal
plane or it could be just an electron actually orbiting around the nucleus ok
so it doesn't matter but they're all going to be modeled as a particle P
which is located at a certain distance away from the center of this path all right
so let's start first with the Cartesian coordinate system so i'm going to locate
the origin of coordinate system right at the center of this circle and this
will be my x this is my y which means that this is my i hat direction and
that's my j hat direction ok so at any particular instant let's say this angle is
theta and it is moving along counterclockwise direction the position
of the particle is given as x i hat plus y j hat ok now since I know the distance from
the origin i can also write this as r cosine theta i hat plus R sine theta j
hat right and there is no problem with that I want to get the velocity I
differentiate it and i get minus r sine theta theta dot i hat plus R cosine
theta theta j hat ok so that gives me the velocity I want to get the
acceleration I differentiate it once more and I'll take my r outside then
I'll write this as minus cosine theta theta dot square plus sine theta theta
double dot i hat Plus r this will be minus sine theta
theta dot squared plus cosine theta theta double dot and that's j hat and that's
your acceleration in Cartesian coordinate system so if you know theta dot you know
theta double dot you know r you can compute any of these quantities right
okay now i'm going to pick a different color let's say i want to indicate its
velocity and acceleration in tangent normal system what would that be so at
this particular instant where the particle P is what is the tangential
direction? that is same as the direction of velocity so that's et hat right right
over here and the en hat would be pointing this way okay so if i want to
write velocity in tangent normal system that would be whatever its magnitude is
the speed times et hat right and if I want to write acceleration that would
be v dot et hat so tangential component plus v square over radius of
curvature which is r en hat now if somebody says can you go from
this to this you can easily go that way because you know at this point your i
hat is pointing this way and j hat is pointing that way and you can relate to
the i hat j hat with et hat and en hat in fact they're quite a few problems in
this chapter would be just relating two coordinate system so if i want to
write let's say i had in terms of et hat and en hat it would be a simple matter
of just showing the two coordinate system so i'm going to draw this separately so
this is my i hat this is my j hat right and et hat is pointing this way and en
hat is perpendicular to et hat of course right this is perpendicular this
is perpendicular and angle that en hat is making from horizontal is actually
theta so if I said can you write let's say et hat in terms of i hat and j hat
or i hat in terms of et hat and en hat you could do that it's not a
problem so if this angle is theta this angle is Theta as well so i can write et
hat equal to cosine theta j hat minus sine theta i hat right so it's
just a simple matter of writing a particular vector and in a chosen coordinate
system so in this case I'm choosing to write e sub t hat in terms of i hat an j
hat what about en hat en hat would be equal to minus cosine theta I hat minus
sine theta j hat ok and if you are having trouble seeing how we obtained
this you will need to review the review material that i have posted on blackboard and
go to the section where we talked about the vectors and i have a couple of
examples of how you go from one coordinate system to another so now if you take
this ok and you substitute in over here in the expression for velocity as well
as in the acceleration and you do the same thing with the en hat you will
actually get the velocity and acceleration in terms of i hat j hat
and they would be same as what you have over here in one and two they wont be any
different ok all right so what about r theta well at this point P and I'll
choose a different kind of this time the what is the direction of er hat remember your
hat points radially outward so that would be your hat and e theta hat points
in the increasing theta direction so that actually would be same as et hat
direction ok so you could also write velocity and acceleration in r theta
system so what would that be so let's do that so velocity of p is r dot er hat
plus r theta dot e theta hat and acceleration is r double dot minus r
theta dot square er hat plus 2 r dot theta dot plus R theta double dot e theta
hat so now again if you want to relate the velocity and acceleration in r theta
system with tangent normal or cartesian system you can clearly see how they
would relate in fact er hat at this instant is equal to minus en hat right
because they're pointing in opposite direction so you could write er hat is
nothing but minus en hat ok what about e theta hat e theta hat is same as tangent
direction so e theta hat is equal to tangent direction right and that's it so if you
know how to go from one coordinate system to another you can easily relate any of
these quantities without a problem
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