Pratica di Shadowing: Orbital Mechanics - Newton's Laws - Impara a parlare inglese con i video

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All right, so welcome back.

So today's lecture is going to be very short because all we need to cover to finish chapter one are Newton's laws.

And the reason why we need to review Newton's laws of motion as well as Newton's gravitational law is

because in the previous lecture we looked at kinematics where we saw techniques

and methods to study the motion in a force-free environment.

But when you think about a spacecraft orbiting a planet, there are obviously external forces being applied to our spacecraft and precisely gravitational forces.

So in order to be able to solve the motion, we need to invoke dynamics.

And again, the basis to start with dynamics is to review Newton's laws.

So first up, we have in 1.5.1, we have Newton's laws. of motion.

So there are three laws of motions derived by Newton.

The first one is known as the law of inertia.

That law says that unless being acted by an external force, a particle in motion will remain in uh in set motion

without the mean deviation will remain in a recti linear motion

okay uh let me just write it down for you for completeness

so this law states that unless acted upon by force

A particle will

remain in its

state of rest or will remain

in a straight line, motion with constant

inertial velocity.

Okay?

So all this to say that unless you perturb or push or pull on particle, well that particle will just continue its motion.

Okay, that's obvious.

And this is the law we observe in my laboratory where we have the granite table upon

which are floated spacecraft platform and because of

that they aren't experiencing any frictional effects and as such

if you were to push a platform in my lab the

platform will continue traveling with constant velocity in its initial direction

without any deviation thanks to the law of inertia as stated by Newton.

Okay so that was the first law of motion discovered by Newton.

The second one which is probably his most famous law is is f equal ma

okay so in a vector form you just say

that the net force being applied to the particle is equal

to the mass of the particle times its acceleration in a vector form

you could also reformulate that to solve for the acceleration

that the particle will experience given the net force being applied

to it all right pretty self-explanatory one thing to realize here is

that this acceleration you see on the left hand side of that equation acceleration is always the inertial acceleration.

Alright, so again, the acceleration in Newton's second law, F equal to mA, is always the inertial acceleration,

always, always, always, such that you could say

that this is the velocity vector inertial derivative or equivalently double inertial derivative of the position vector r

and this is why we studied previously how to relate the inertial acceleration to the non-inertial acceleration

because it was such a big deal to obtain the inertial acceleration right by virtue of newton's second line

that we're going to reuse ultimately in chapter two to come up with the spacecraft orbital dynamical equation okay

so second law of motion by newton finally we're down to the last one so this is the law of action

reaction, which says that for every applied force

is an equal and opposite reaction force.

Okay, so if you have a particle, say M1, and a particle M2 over there,

if M1 exerts a force on M2, say it's pulling on m2 like

that with f21 well then m2 will also exert the force on m1 of equal magnitude

but in opposite direction f12 all this to say that If F21 is the action,

you're going to get a reaction in the opposite direction,

like that, where F21 is equal to minus F12.

okay so same magnitude yet opposite direction what did I just say here that doesn't make any sense

f21 is unequal to minus f21 so if if you have f21 like that you get f12 like this

And they are in equal magnitude but opposite direction.

And that's all I wanted to say.

Apologies for that.

All right, so these are the three laws, as stated by Newton.

But perhaps the most important one in our context here is the law of gravitation.

Newton's law of gravitation. and 1.5.2.

So, that's considered a case again where we have two masses, M1 and M2.

And now we're looking specifically at gravitational forces.

So, we have M1 attracting M2 towards itself.

with a force being denoted F21 as previously.

So we're going to say that F21 vector

is the force of attraction on M2 by M1.

So M1 attracts M2 essentially.

And then you have the opposite direction, F12,

or F12 is the force of attraction applied on M1 exerted by M2.

and again by virtue of Newton's third law they are equal to magnitude yet in opposite direction

now Newton was able to quantify mathematically those gravitational forces

or attraction forces by virtue of the fact

that those point masses have finite mass of a kilogram

and the way they're quantified is F21 is equal to minus capital G where this is known as the universal

gravitational constant times m1

times m2 times r21 where r21 is the position vector of mass 2

with respect to mass 1 I'm just going to move my F12

and that makes a lot of sense because if R21 is to the right

then you get F, if R21 is to the right, F21 is to the left, like that, in opposite direction.

And all this is divided by R21 into the position of M2 with respect to M1.

The norm of that, cube.

So this is quantifying the gravitational forces of M1 attracting M2 towards it.

And similarly, you could solve for F21,

or sorry, F12, by saying that this is now a force in the same direction as R21.

So this time we don't have the minus in front of everything.

So G, M1, M2, R21,

position vector, over the distance between those two masses, Q.

By the way, this gravitational constant, you don't have to remember

that on top of your head in an exam I'm going to provide all constants for you

but for completeness this constant is given by 6.6732 times

10 to the negative 11 newton meter square over kilogram square.

And this concludes Chapter 1.

Alright?

So in Chapter 2, we're going to start right away by using those gravitational forces,

specifically in case of a planet attracting the spacecraft towards itself and thereby creating an orbit around itself.

Okay?

Hope you enjoyed Chapter 1.

Congratulations, you survived.

100

That wasn't an easy chapter to go through was pretty dry

101

a lot of new theory new mathematical operator mathematical objects the vector c's the hollow dot all those great things

102

but you made it thus far congrats and i'll see you again in chapter two soon see you

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