Components of the Moons Acceleration

[neo32]

Hello, I need help on deriving equations for the moon's acceleration around the earth. I've found that the total acceleration is a=Gm/r^2 (G is gravitational constant, m is mass of moon and r is the Earth to moon distance) what i need to do is split this into its x and y components, and derive equations in terms of a, r, x, and/or y. I have ax=cos(x/r) and ay=sin(y/r) but it's not working. I also have to take into account the direction of the acceleration (towards the earth)

Any help would be really appreciated.

 

[vanesch]

I have ax=cos(x/r) and ay=sin(y/r) but it's not working.

Is x/r an angle ? Is y/r an angle ?
Should you take sin and cos of this quantity ?

 

[neo32]

well x and y are the coordinate positions from the Earth (which is at the origin) so this will make a triangle with the x-axis. Using trigonometry, x would be the adjacent side and y would be the opposite side, and r would be the hypotenuse (distance from Earth to moon). I'm supposed to use trig relationships so I would think that I need to take sin and cos of this quantity but I don't know why it won't work.

 

[HallsofIvy]

It won't work because it doesn't make sense! Saying you are "supposed to use trig relationships" doesn't mean you just apply sine and cosine to whatever numbers you want.
It is true that $sin(\theta)= \frac{y}{r}$ where $\theta$ is the angle the position vector makes with the x-axis. That is NOT
$cos(\frac{y}{r})$!

By the way- the "m" in your formula is mass of the earth, not the moon. The gravitational force formula involves both masses. Since that is equal to mass(of moon) times acceleration(of moon), it is the mass of the moon that cancels.

 

[neo32]

By the way i just noticed that those were supposed to be ASIN and ACOS. sorry. And thanks for clearing that up about the m value.

Anyways, what would i use then if I'm not supposed to apply sine and cosine? Should ASIN and ACOS work?

 

[neo32]

I fixed my equations: they're ax = a*cos(T) and ay = a*sin(T) where ax is the x component and ay is the y component of acceleration, r is the earth-moon distance, and T is the angle with the x axis). The problem is I need to derive T as well and plug that into the two equations. if those equations are wrong too then can someone put me on the right track? thanks

 

[jtbell]

Well, T is an angle, right? What's the angle corresponding to one complete orbit of the moon about the earth? And how much time does it take the moon to perform this orbit? Using those two bits of information, can you come up with an expression that gives you the angle for any given time?

Unless your assignment specifically states otherwise, it's probably reasonable to assume that the angle T=0 at time t=0.