Checking the dimensional consistency of equations | dimensional analysis kisembo academy

Checking the dimensional consistency of equations | dimensional analysis kisembo academy


in our previous video able to show how they derived quantities Relate with the fundamental quantities of
mass length and time and who are able to derive. How area of volume velocity? Acceleration and Force how they relate to
their physical quantities In our conclusion. We said that we use dimensions
of a physical quantity also To check for the consistency of equations,
and we also use it to derive equations in this video We shall concern ourselves with how we use
the dimensions of a physical quantity to check for the consistency of equations This is the same work adding me and thanks
for tuning in Now right in front of us. We have an equation
s is equal to ut Plus half it is where most of you know it
as the equation of motion Now we all know that s is equal to ut. Plus
half h squared is a correct equation but we want to use dimensions with its capacity
to check whether it is consistent now when we are checking for the consistency of equations we balance their units it means that we are
going to look at the Left-hand side of the equation and We do we see how that left-hand side is related
to the three fundamental quantities Then we also are going to look at the right-hand
side of the equation and we see how that right hand hand of the
equation is related to their physical quantities the basic ones Then if the to coincide if the two are the
same then we shall conclude and say that the equation is dimensionally consistent so Let’s look at this example. We have in on
our left hand side. We have s which is distance So that’s why we said on the left hand side
s is going to be equal to L it’s the length its s is distance, so The distance as far as the three basic quantities
are concerned distance is L. Then we shall look at our right hand side The right hand side is this other part? Our duty is to find the dimensions on the
right hand side to see if they are going to coincide with hell So all right hand side you have beauty plus
a half ay t squared So we find the dimensions of U times the dimensions
of t plus 1/2 times the dimensions of acceleration times
the dimension of time squared and Then from here. We shall end up with you.
We know that U is velocity so velocity The dimensions the velocity are Lt to the
point negative 1 which is that? So we look for the dimensions of displacement
and what are the dimensions of time? So now this placement is actually L divide
that by the time This is in front of me a basic quantity which
is t? So it means that the dimensions of velocity
are l t to the power negative 1 times time which is capital t plus 1/2 The dimensions of acceleration is l t to the
power negative 2 acceleration is velocity Over time by definition by definition we know
that acceleration is the rate of change of velocity So now what are the dimensions of velocity? The damage and the Velocity are L to the power
negative 1 we’ve got those ones before so for velocity it is L T to the Power negative 1 divided that by
the dimensions of time which is t? So our answer here is going to become l t to the Power Negative 2 multiply that by time square Which is t squared and of course lT to the
power negative 1 times t is t and that t? Cancel because this is to be the point negative
1 this is to the power 1 so They cancel and you remain with L and then
that is going to be plus Lt to the Power Negative 2 times t squared
this t squared and t to the power negative 2 will cancel out and You made with 1/2 l. L plus 1/2 L is going
to give us 3 over 2 times. L 3 over 2 times L this 3 over 2 is a constant
and as far as the emissions of a disco quantity are concern that constants are dimensionless, so because constants are dimensionless
it means that as far as the right hand is concerned you remain with L and So you realize that L at the L? They are coinciding the left hand side is
ill the right hand side is also. L. So you conclude by saying that since So we’re going to our next example. We have
V is equal to U cubed Plus 80 Now V is equal to U cube plus 80 look at we have the left hand side and the
right hand side so the left hand side is v V Is Velocity and the dimensions of velocity
are LG to the power negative 1 we derived those in our previous video? Then we have those of the right-hand side Which is this which is youtubed plus 80 so
now we work towards finding the dimensions on the right hand side The dimensions of the right hand side we have
u cubed The dimensions of U cubed plus the dimensions
of acceleration plus the dead times the directions of Time t the dimensions the velocity is going to be
allergic to the point negative 1 that’s for velocity but since it’s the path 3 so this is this
thing is to the power 3 plus the dimensions of Oscillation which is Lt to the power negative
2 this is acceleration times the dimensions of time which is capital t they’re then required and simplify this So this is the same as L to the power 3 Times Negative 3 times 3 is t to the Power
negative 3 that is plus L Energy to the point negative 1 cos t to the
point negative 2 times t to the Power 1 of course this t will cancel with one of those
t’s remain is the point negative 1 and When we add the new you can’t simplify this
father when you in our next episodes to have that So you realize that as far as our right hand
side is concerned. We have in just this I Do now left hand side. We are having lt to
the pole negative 1 now since the units of our right hand side I’m not coinciding with verse on our left-hand
side it means that this equation is not dimensionally consistent So in our conclusion. We shall say We’ll do one more example We have V squared is good 8 U squared plus
4 it is we are checking for the dimension of consistency of this equation So checking for the dimension of consistency
of this equation. We have the left-hand side which is v V Squared so the dimensions of V squared are
definitely l t to the Power Negative 1 that this is for
velocity Energy to the point negative 1 squared so
meaning that it’s going to be L Squared times t to the Power Negative 2 that is on the left hand side, so What about this on the right hand side so
let’s look for the right hand side On our right hand side. We have 8 U squared we need the dimensions of that Plus 4 times the dimensions of acceleration
times the dimensions of is the distance So now we’ll go ahead and say that this is
going to be 8 times Now U is velocity velocity is l t to the power
negative 1 This is squared plus 4 times the dimensions of acceleration
which is l t to the power negative 2 times Ss. Is distance so distance is L length? So this is going to become L to the Power 2 times t to the Power Negative 2 Plus that is going to be visit 8 this is 4
times L times L is L squared, so it’s L squared
times t to the power negative 2 So you have Via 8 L squared – a negative 2
and that? So this this plus that gives us 12 12 L squared t to the Power Negative 2 But as far as the emissions of physical quantities
concerned we know that constants are dimensionless, so this is the
same as L squared t to the point negative 2 so since The right hand side which is L squared t to
the power negative 2 is The same as L squared t to the power negative
2 which is the left hand side it means that this equation is dimensionally consistent now we only know that as far as the equations
of motion occurs and The third equation of motion is V squared
is equal to U square plus 2 a s but now this is a wrong equation as Velocity V squared is equal to 8 U squared
plus 4 s this is a wrong equation from our theory But then when we try to analyze it using the
dimensions of a physical quantity we find that the equation is dimensionally consistent So in other words, it’s the emission Dimensionally consistent but it is having
wrong values of 8 and for those other wrong constants now Why is this? So why is this thing consistent
it is having wrong values now. This is consistent here it’s having wrong values because 8 & 4 are
dimensionless Therefore this brings us to this conclusion but when we are doing Dimensional analysis, we can use it to eliminate
wrong equations? But we cannot use it to prove the correctness
of an equation, and why is this so? because you cannot use it to prove whether
the correctness of Factors or constants such as 1/2 8 4 pi et
cie are correct you can’t use it so That’s the conclusion here. We can use the
emissions of a physical quantity to eliminate wrong equations But then we cannot use it to prove the correctness
of an equation Please take note of that This brings us to the end of this video. Thanks
for watching Otherwise for more of these videos I encourage
you to subscribe and share this video It’s been a no-drama chromium for example
academy

18 thoughts on “Checking the dimensional consistency of equations | dimensional analysis kisembo academy

  • May 23, 2017 at 3:25 pm
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    Helpful anx

    Reply
  • August 25, 2017 at 11:39 am
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    thanx

    Reply
  • September 23, 2017 at 4:33 am
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    thank you

    Reply
  • January 31, 2018 at 8:54 pm
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    Wonderful method of teaching! Helpful!

    Reply
  • April 18, 2018 at 2:27 pm
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    From which country you are?

    Reply
  • May 25, 2018 at 12:05 pm
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    I liked the way of teaching. Very simple and understandable.
    Thanks a lot.

    Reply
  • July 14, 2018 at 8:07 am
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    Thanks a lot

    Reply
  • October 5, 2018 at 10:08 pm
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    How come 1+1÷2 =3÷2

    Reply
  • November 10, 2018 at 6:41 pm
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    thanks

    Reply
  • December 25, 2018 at 9:07 pm
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    Nice teaching… please explain to me about the cube on the initial velocity… because am only familiar with v=u+at and not v=u³+at..thank u

    Reply
  • January 15, 2019 at 3:21 am
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    very helpful on a topic I was not able to find many videos on, thanks!

    Reply
  • February 4, 2019 at 10:09 pm
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    Mate, I’m currently under a Aeronautical systems degree and you’ve summed my whole 2 week learning package in 11 mins. Thank you hugely!

    Reply
  • September 2, 2019 at 3:32 am
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    How is 3/2L = L? I'm confused, please help

    Reply
  • September 7, 2019 at 12:19 pm
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    You are a great teacher thank you so much

    Reply
  • September 12, 2019 at 1:30 pm
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    Good job

    Reply
  • September 12, 2019 at 1:30 pm
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    Good teaching carry on..

    Reply
  • January 7, 2020 at 7:28 am
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    Great job

    Reply
  • January 7, 2020 at 1:18 pm
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    Thanks, it helped me a bunch!

    Reply

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