Answer:
a. F=GMm/r^2; a. M =
![M = \frac{Fr^{2} }{Gm}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7BFr%5E%7B2%7D%20%7D%7BGm%7D)
a. F=GMm/r^2; b. r =
![r = \sqrt{\frac{GMm}{F} \\}](https://tex.z-dn.net/?f=r%20%3D%20%5Csqrt%7B%5Cfrac%7BGMm%7D%7BF%7D%20%20%5C%5C%7D)
b. M=kxa^3/p^2; a. P =
![p = \sqrt{\frac{kxa^{3}}{M}}](https://tex.z-dn.net/?f=p%20%20%20%20%3D%20%20%20%5Csqrt%7B%5Cfrac%7Bkxa%5E%7B3%7D%7D%7BM%7D%7D)
b. M=kxa^3/p^2; b. a =
![a = \sqrt[3]{\frac{Mp^{2}}{kx} }](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7BMp%5E%7B2%7D%7D%7Bkx%7D%20%7D)
Step-by-step explanation:
For a. F=GMm/r^2; a. M =
To solve for M, we will rearrange the given equation F=GMm/r^2 such that M is the subject of the formula
From
F=GMm/r^2
![F = \frac{GMm}{r^{2} }](https://tex.z-dn.net/?f=F%20%3D%20%5Cfrac%7BGMm%7D%7Br%5E%7B2%7D%20%7D)
First, Cross multiplication, we then get
![Fr^{2} = GMm](https://tex.z-dn.net/?f=Fr%5E%7B2%7D%20%3D%20GMm)
Now, divide both sides by ![Gm](https://tex.z-dn.net/?f=Gm)
![\frac{Fr^{2} }{Gm} = \frac{GMm}{Gm} \\](https://tex.z-dn.net/?f=%5Cfrac%7BFr%5E%7B2%7D%20%7D%7BGm%7D%20%3D%20%5Cfrac%7BGMm%7D%7BGm%7D%20%20%5C%5C)
The equation becomes
![\frac{Fr^{2} }{Gm} = M](https://tex.z-dn.net/?f=%5Cfrac%7BFr%5E%7B2%7D%20%7D%7BGm%7D%20%3D%20M)
∴ ![M = \frac{Fr^{2} }{Gm}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7BFr%5E%7B2%7D%20%7D%7BGm%7D)
For a. F=GMm/r^2; b. r =
Also, to solve for r, we will rearrange the given equation F=GMm/r^2 such that r is the subject of the formula
From
F=GMm/r^2
![F = \frac{GMm}{r^{2} }](https://tex.z-dn.net/?f=F%20%3D%20%5Cfrac%7BGMm%7D%7Br%5E%7B2%7D%20%7D)
First, Cross multiplication, we then get
![Fr^{2} = GMm](https://tex.z-dn.net/?f=Fr%5E%7B2%7D%20%3D%20GMm)
Now, divide both sides by
, Such that we have
![\frac{Fr^{2} }{F} = \frac{GMm}{F} \\](https://tex.z-dn.net/?f=%5Cfrac%7BFr%5E%7B2%7D%20%7D%7BF%7D%20%3D%20%5Cfrac%7BGMm%7D%7BF%7D%20%20%5C%5C)
Then, ![r^{2} = \frac{GMm}{F} \\](https://tex.z-dn.net/?f=r%5E%7B2%7D%20%20%3D%20%5Cfrac%7BGMm%7D%7BF%7D%20%20%5C%5C)
∴ ![r = \sqrt{\frac{GMm}{F} \\}](https://tex.z-dn.net/?f=r%20%3D%20%5Csqrt%7B%5Cfrac%7BGMm%7D%7BF%7D%20%20%5C%5C%7D)
For b. M=kxa^3/p^2; a. P =
To solve for P, we will rearrange the given equation M=kxa^3/p^2 such that P becomes the subject of the formula
From
M=kxa^3/p^2
![M = \frac{kxa^{3}}{p^{2} } \\](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7Bkxa%5E%7B3%7D%7D%7Bp%5E%7B2%7D%20%7D%20%20%5C%5C)
First, Cross multiply, we then get
![Mp^{2} = kxa^{3}](https://tex.z-dn.net/?f=Mp%5E%7B2%7D%20%3D%20kxa%5E%7B3%7D)
Divide both sides by
, such that the equation becomes
![\frac{Mp^{2} }{M} = \frac{kxa^{3}}{M}](https://tex.z-dn.net/?f=%5Cfrac%7BMp%5E%7B2%7D%20%7D%7BM%7D%20%20%3D%20%5Cfrac%7Bkxa%5E%7B3%7D%7D%7BM%7D)
Then, ![p^{2} = \frac{kxa^{3}}{M}](https://tex.z-dn.net/?f=p%5E%7B2%7D%20%20%20%3D%20%5Cfrac%7Bkxa%5E%7B3%7D%7D%7BM%7D)
∴ ![p = \sqrt{\frac{kxa^{3}}{M}}](https://tex.z-dn.net/?f=p%20%20%20%20%3D%20%20%20%5Csqrt%7B%5Cfrac%7Bkxa%5E%7B3%7D%7D%7BM%7D%7D)
For b. M=kxa^3/p^2; b. a =
To solve for a, we will rearrange the given equation M=kxa^3/p^2 such that a becomes the subject of the formula
From
M=kxa^3/p^2
![M = \frac{kxa^{3}}{p^{2} } \\](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7Bkxa%5E%7B3%7D%7D%7Bp%5E%7B2%7D%20%7D%20%20%5C%5C)
First, Cross multiply, we then get
Now, Divide both sides by
, such that the equation gives
![\frac{Mp^{2}}{kx} = \frac{ kxa^{3}}{kx}](https://tex.z-dn.net/?f=%5Cfrac%7BMp%5E%7B2%7D%7D%7Bkx%7D%20%3D%20%5Cfrac%7B%20kxa%5E%7B3%7D%7D%7Bkx%7D)
Then, ![\frac{Mp^{2}}{kx} = a^{3}](https://tex.z-dn.net/?f=%5Cfrac%7BMp%5E%7B2%7D%7D%7Bkx%7D%20%3D%20a%5E%7B3%7D)
![a^{3} = \frac{Mp^{2}}{kx}](https://tex.z-dn.net/?f=a%5E%7B3%7D%20%3D%20%5Cfrac%7BMp%5E%7B2%7D%7D%7Bkx%7D)
∴ ![a = \sqrt[3]{\frac{Mp^{2}}{kx} }](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7BMp%5E%7B2%7D%7D%7Bkx%7D%20%7D)