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‫Welcome back.

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‫In this video, we will drive a dynamic's equation for the translational motion of the body, meaning

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‫that we will not touch the orientation of a body for now.

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‫If you remember then we had a vector.

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‫Gummo in the inertia frame equals X, Y and Z transposed.

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‫That was our position vector in the inertial frame or in the earth frame.

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‫Now you have probably heard the Newton's second law, right?

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‫Force equals mass times acceleration or you can read it like this.

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‫Mass times the time derivative of the velocity vector.

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‫That would be the Newton's second law because DVD is acceleration, a more general case is force vector

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‫equals the time derivative of the linear momentum.

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‫This is a more general case.

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‫Linear momentum is calculated by multiplying mass times the velocity vector.

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‫In other words, force vector equals d m times the vector divided by the T.

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‫If you take its time derivative, then it looks like this.

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‫The force vector equals M that times V vector and M that is the time derivative of mass.

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‫It's how much mass changes with respect to time plus mass times, the velocity that vector.

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‫And this is a simple product rule from calculus.

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‫Right.

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‫If you have a product of two variables and you want to take their derivative, then you first take the

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‫derivative of the first variable times, the second variable, and then you add the first variable times

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‫the derivative of the second variable.

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‫So that's just calculus.

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‫And just to make clear, this DVD is the dot vector and M Dot is D.M. Over DETI.

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‫And by the way, keep in mind that the velocity here is a vector quantity and also the acceleration

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‫is a vector quantity.

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‫However, mass and the change of mass with respect to time, this is a scalar quantity.

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‫It doesn't have a direction.

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‫It's just a number.

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‫It's a scalar.

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‫Now, since the mass of outdrawn remains constant in time, then that means that M dot equals zero kg

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‫per second.

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‫That means that this term here becomes zero Newtons and you're only left with force equals mass times

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‫the V dot vector or the acceleration.

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‫So this classical second law of Newton, it's a specific case when your mass is constant.

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‫Now let's rewrite our position vector in a different form.

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‫Gurmai equals X plus Y, J plus Z, K.

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‫It's the same thing simply written in a different form.

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‫Remember the I, J and K where unit vectors whose values were one but who pointed in the X, Y and Z

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‫direction respectively.

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‫And so let's think the derivative of the position vector gulman that E equals and now we're going to

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‫use the product rule X that I plus X idot plus Y dot J plus Y Jadot plus Z dot k and plus Z times K

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‫dot.

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‫Now pay attention that according to the product rule from calculus, you also have to take the derivatives

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‫of I, J and K with respect to time.

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‫And so the idot Jadot and K Dot give you information on whether the direction of your X, Y, Z axis

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‫is constant or whether it changes, since right now we are in the inertial frame, the axis of the inertia

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‫frame are fixed.

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‫Their direction doesn't change in time, so if you look at this inertia from here, then these axes,

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‫they don't turn they don't move anywhere.

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‫So they are just fixed to the ground and they don't change.

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‫That means that I dot equals zero, g dot equals zero and K dot equals zero.

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‫And so this term, this term and this term, they all become zeros.

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‫And you're only left with this term, this term in this term.

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‫So I have written it again here.

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‫But now that's just the change of position with respect to time.

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‫That's our velocity vector.

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‫But I need the V dot.

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‫I need the acceleration vector.

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‫I need the change of velocity with respect to time.

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‫And so I'm going to take another derivative.

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‫I'm going to have Gunma Double Dot and that will be the second time derivative of the position vector

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‫and that's how you will get your acceleration.

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‫And if you take the second derivative of it, then this is what you get.

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‫So you take this guy here and you apply another time derivative to it and then you have this rule and

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‫I have simply rearranged the terms here.

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‫So whenever I had I Dot Jadot and K Dot, I put them here as the fourth, fifth and sixth term.

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‫And again, since the axis of the inertia frame are constant, the change of the unit vectors with respect

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‫to time are zero units per second.

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‫So that's what I don't think that are there zero units per second.

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‫So this term, this term, in this term they will be zero and you will be left with these three terms

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‫here.

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‫And now we can see that the force vector in the inertial frame equals mass times, Gunma, double dot

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‫vector in the inertial frame and this force vector here is the sum of the forces, right.

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‫It's the net force.

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‫So you can also write it down like this.

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‫The sum of the forces equals mass times acceleration, which is this Gummo double dot in the inertia

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‫frame.

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‫You can also write it down like this.

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‫If X, F, Y and Z equals mass times, X, double dot, Y, double dot and Z double dot.

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‫Because this vector here is equivalent to this vector here, right.

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‫I mean, if you want, you can also write this equation down like this.

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‫Let me just squeeze the final piece of information into the screen.

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‫You can have F, X, I, plus F, Y, J, plus F, Z, K equals mass times X, double that I plus Y,

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‫double dot J plus Z, double dot K.

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‫Like this.

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‫It's the same thing.

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‫It's just a different way to represent the same thing.