Theorem of Parallel Axis Video Lecture - AP Physics C Mechanics - Grade

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particles and traditional motion part 26
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no move here from example now let us
look at the theorem of parallel axis so
this theorem is applicable so contrary
to the theorem of perpendicular axis
which is applicable only to cleaner
bodies this theorem is applicable to
bodies of any shape so it states that
the moment of inertia she out for body
about any axis is equal to the sum of
the moment of inertia of the body about
a parallel axis passing through its
center of mass and the product of its
mass and square of the distance between
the two parallel axis so that means let
us suppose if this is the axis this Z is
the axis which is passing through the
center of mass then we can say that the
moment of inertia about another axis
which is parallel to this axis is equal
to so we can say that is Z dash is equal
to the moment of inertia about the
parallel axis which is passing through
the center of mass plus mass in a square
where E is the distance between the two
axis so this is the distance between the
parallel axis that means these two axes
are parallel correct so that is why this
is known as theorem of parallel axis
that means the moment of inertia about
any axis is equal to the moment of
inertia about a parallel axis which is
passing through the center of mass plus
M into a square where E is the distance
between the two parallel axis right so
now let us look at the proof of the
theorem of pagli axis so it is also we
will try to prove in a similar way as it
we did for the theorem of perpendicular
axis so let us suppose we have any rigid
body so let us say this is the axis
which passes through the center of mass
of the object so let us call this axis
as the z axis there is another parallel
axis which is denoted by z
- okay the distance between the two axis
is given as D so what will be the moment
of inertia about this axis Z - now let
us consider any point inside any point
of the body let us say we consider any
point P so what would be the position
vector of the point P from Z if we take
it from this point similarly what would
be the position vector of the point P
from Z - if we join it to this point
right so now let us try to calculate the
moment of inertia about Z - axis now as
per definition this will be summation of
m-i into our eyes question what does Ri
Ri is nothing but distance come z - trip
so let us call this point as D so this
will be P DS where like so this will be
equal to summation of m-i and what is PD
square now this point P is has
coordinates x iy eyes and eye this point
B as coordinates X I will be equal to D
because this distance is small D so the
X becomes d what is y Y is equal to 0
and say this equal to say right so this
PD becomes equal to X I minus D whole
square plus Phi I minus 0 whole square
plus z i- z i whole square so this
becomes equal to summation of m-i x i
square plus d square minus 2 x ID plus y
i square so this becomes equal to
summation of m-i x i square plus y I
square plus summation of m-i d square
minus 2 D into summation of m-i X I now
summation of m-i
I want a summation of m-i X is nothing
but capital M because summation of M is
capital M and summation of m-i X is
nothing but M X center of mass which is
equal to zero now I said about season so
what is I said it is nothing but some
issue of M I P if let us call this as f
so this is P F square so what is this F
the coordinates of F is given as 0 0 0
so this will be equal to summation of
m-i into X I minus 0 whole square plus y
I minus 0 whole square plus z i- z i
whole square so this is equal to
summation of m-i x i square plus y I
square so this is my equation and this
was my equation 1 so this was equal to
AI z - so we see that the first term in
this equation is nothing but what
happened in the second equation
therefore for equation 1 & 2 we can see
that Isaac - so let us write it here so
from equation 1 and equation 2 we can
say that I said - is equal to I said
plus summation of m-i d square right
because a third term became equal to 0
from here so this term we can 0 this
term became equal to Isaac and this term
is nothing but summation of m-i into DS
4 so summation of m-i is nothing but
capital n so this is how the theorem of
parallel axis was also proved let us
look at an example which will make
things clear let us suppose
have to calculate the moment of inertia
of a solid cylinder about a line
parallel to the cylinder axis and on the
surface of the cylinder so that means it
would look something like this let us
suppose if you have a cylinder like this
so we have to calculate the axis of the
cylinder is this one so we have let us
call this axis as said so we have to
calculate the moment of inertia about a
line parallel to the cylinder axis and
on the surface of the cylinder that
means this is my axis so it is parallel
to this axis and also it is lying on the
surface of them selling them so now we
know that I said what is the moment of
inertia of a solid cylinder for about
its axis
I said is equal to M R square by 2 we
derived it in one of our previous slides
now as per this theorem we can say that
I said - is equal to AI z plus m into a
square that is the distance between
these two axis now what is the distance
between these two axes radius of the
cylinder that is M R square so this is
equal to M R square by 2 plus M R square
so this is equal to 3 by 2 M R square so
now you can see that how these theorems
of paddle axis and perpendicular axis
become very useful in calculating the
moment of inertia about different axis
of different bodies right ok thank you
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