‫Welcome back.
‫In this video, we will drive a dynamic's equation for the translational motion of the body, meaning
‫that we will not touch the orientation of a body for now.
‫If you remember then we had a vector.
‫Gummo in the inertia frame equals X, Y and Z transposed.
‫That was our position vector in the inertial frame or in the earth frame.
‫Now you have probably heard the Newton's second law, right?
‫Force equals mass times acceleration or you can read it like this.
‫Mass times the time derivative of the velocity vector.
‫That would be the Newton's second law because DVD is acceleration, a more general case is force vector
‫equals the time derivative of the linear momentum.
‫This is a more general case.
‫Linear momentum is calculated by multiplying mass times the velocity vector.
‫In other words, force vector equals d m times the vector divided by the T.
‫If you take its time derivative, then it looks like this.
‫The force vector equals M that times V vector and M that is the time derivative of mass.
‫It's how much mass changes with respect to time plus mass times, the velocity that vector.
‫And this is a simple product rule from calculus.
‫Right.
‫If you have a product of two variables and you want to take their derivative, then you first take the
‫derivative of the first variable times, the second variable, and then you add the first variable times
‫the derivative of the second variable.
‫So that's just calculus.
‫And just to make clear, this DVD is the dot vector and M Dot is D.M. Over DETI.
‫And by the way, keep in mind that the velocity here is a vector quantity and also the acceleration
‫is a vector quantity.
‫However, mass and the change of mass with respect to time, this is a scalar quantity.
‫It doesn't have a direction.
‫It's just a number.
‫It's a scalar.
‫Now, since the mass of outdrawn remains constant in time, then that means that M dot equals zero kg
‫per second.
‫That means that this term here becomes zero Newtons and you're only left with force equals mass times
‫the V dot vector or the acceleration.
‫So this classical second law of Newton, it's a specific case when your mass is constant.
‫Now let's rewrite our position vector in a different form.
‫Gurmai equals X plus Y, J plus Z, K.
‫It's the same thing simply written in a different form.
‫Remember the I, J and K where unit vectors whose values were one but who pointed in the X, Y and Z
‫direction respectively.
‫And so let's think the derivative of the position vector gulman that E equals and now we're going to
‫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
‫dot.
‫Now pay attention that according to the product rule from calculus, you also have to take the derivatives
‫of I, J and K with respect to time.
‫And so the idot Jadot and K Dot give you information on whether the direction of your X, Y, Z axis
‫is constant or whether it changes, since right now we are in the inertial frame, the axis of the inertia
‫frame are fixed.
‫Their direction doesn't change in time, so if you look at this inertia from here, then these axes,
‫they don't turn they don't move anywhere.
‫So they are just fixed to the ground and they don't change.
‫That means that I dot equals zero, g dot equals zero and K dot equals zero.
‫And so this term, this term and this term, they all become zeros.
‫And you're only left with this term, this term in this term.
‫So I have written it again here.
‫But now that's just the change of position with respect to time.
‫That's our velocity vector.
‫But I need the V dot.
‫I need the acceleration vector.
‫I need the change of velocity with respect to time.
‫And so I'm going to take another derivative.
‫I'm going to have Gunma Double Dot and that will be the second time derivative of the position vector
‫and that's how you will get your acceleration.
‫And if you take the second derivative of it, then this is what you get.
‫So you take this guy here and you apply another time derivative to it and then you have this rule and
‫I have simply rearranged the terms here.
‫So whenever I had I Dot Jadot and K Dot, I put them here as the fourth, fifth and sixth term.
‫And again, since the axis of the inertia frame are constant, the change of the unit vectors with respect
‫to time are zero units per second.
‫So that's what I don't think that are there zero units per second.
‫So this term, this term, in this term they will be zero and you will be left with these three terms
‫here.
‫And now we can see that the force vector in the inertial frame equals mass times, Gunma, double dot
‫vector in the inertial frame and this force vector here is the sum of the forces, right.
‫It's the net force.
‫So you can also write it down like this.
‫The sum of the forces equals mass times acceleration, which is this Gummo double dot in the inertia
‫frame.
‫You can also write it down like this.
‫If X, F, Y and Z equals mass times, X, double dot, Y, double dot and Z double dot.
‫Because this vector here is equivalent to this vector here, right.
‫I mean, if you want, you can also write this equation down like this.
‫Let me just squeeze the final piece of information into the screen.
‫You can have F, X, I, plus F, Y, J, plus F, Z, K equals mass times X, double that I plus Y,
‫double dot J plus Z, double dot K.
‫Like this.
‫It's the same thing.
‫It's just a different way to represent the same thing.