I should know this, but (solving equation for M)

[ljwilder]

I got two different answers, so maybe someone can show me what I've done wrong.

F = (G)(M/r^2)(m/r^2) Solve for M

1st try:
Fr^2 = [(G)(M/r^2)(m/r^2)]/r^2
Fr^2 = (Gr^2)(M)(m)
(Fr^2)/(Gr^2m) = [(Gr^2)(M)(m)]/(Gr^2m)
F/Gm = M

I multiplied each side by r^2 (r squared) to get rid of the denominator, then I divided each side by Gr^2m to get the M alone.

2nd try:
F = (G)(M/r^2)(m/r^2) Solve for M
F/G = [(G)(M/r^2)(m/r^2)] / G
F/G = (M/r^2)(m/r^2)
(F/G)(r^2/m) = (M/r^2)(m/r^2)(r^2/m)
(Fr^2 )/(Gm) = M/r^2
[(Fr^2)/(Gm)]r^2 = (M/r^2)(r^2)
(Fr^4)/Gm = M

I divided each side by G, then I multiplied each side by r^2/m, then I multiplied each side by r^2.

Can anyone tell me if either one of these is right and if not give me an idea where I made my mistakes?

Thanks

 

[Mrspi]

Re: I should know this, but

I got two different answers, so maybe someone can show me what I've done wrong.

F = (G)(M/r^2)(m/r^2) Solve for M

1st try:
Fr^2 = [(G)(M/r^2)(m/r^2)]/r^2
Fr^2 = (Gr^2)(M)(m)
(Fr^2)/(Gr^2m) = [(Gr^2)(M)(m)]/(Gr^2m)
F/Gm = M

I multiplied each side by r^2 (r squared) to get rid of the denominator, then I divided each side by Gr^2m to get the M alone.

2nd try:
F = (G)(M/r^2)(m/r^2) Solve for M
F/G = [(G)(M/r^2)(m/r^2)] / G
F/G = (M/r^2)(m/r^2)
(F/G)(r^2/m) = (M/r^2)(m/r^2)(r^2/m)
(Fr^2 )/(Gm) = M/r^2
[(Fr^2)/(Gm)]r^2 = (M/r^2)(r^2)
(Fr^4)/Gm = M

I divided each side by G, then I multiplied each side by r^2/m, then I multiplied each side by r^2.

Can anyone tell me if either one of these is right and if not give me an idea where I made my mistakes?

Thanks

It looks to me like the common denominator for the fractions in this problem should be r⁴.

Multiply both sides of the equation by r⁴>:

r⁴*F = r⁴ * (G)(M/r²)*(m//r²)

r⁴ F = G(M)(m)

Now, it should be relatively simple to solve for M....if you have further questions, please show us all of the steps you have taken.

 

[ljwilder]

Thanks for your help, but it seems to me that multiplying by r^4 would not work because each term needs to be multiplied by it and that would cancel out the r^2s in the denominators but leave extra r^2s on each side of the equation. To get rid of the denominators of r^2 shouldn't I multiply each side of the equation by r^2 and use the distributive property on the right hand side to multiply each term?

I just saw that I wrote my first try wrong, I actually did this:

F = (G)(M/r^2)(m/r^2) Solve for M

1st try:
(F)r^2 = [(G)(M/r^2)(m/r^2)]r^2
(F)r^2 = (G)(r^2)(M)(m)
(F)r^2)/(G)(r^2)(m) = [(G)(r^2)(M)(m)]/(G)(r^2)(m))
F/Gm = M

Thanks

I'm sorry if this is confusing since I don't know how to make the (r squared) look right.

 

[tkhunny]

If r^4 does not work, you have written it incorrectly. Notice how you correctly stated that it would be inappropriate for "each term". You have only one term! Those items you are talking about are factors, not terms. Different set of rules. Listen to Mrspi, excepting references to a "common" denominator. There is only one denominator.

 

[ljwilder]

2nd try was right?

You're right, I stated it incorrectly. They are factors not terms, but the only denominators are already common, I just need to eliminate them so in my first try if I multiply each side of the equation by that one denominator which means I would multiply each "factor" on the both sides of the equal sign. I don't see what was wrong with the way I did it.

But if you and Mrspi are correct then my second try was right, because even though I didn't start out by multiplying by r^4 I did end up with the same answer as Mrspi if he had finished the equation. I just used different steps to get there. But I think I just saw where I made a mistake in line 4 below, I didn't distribute the m/r^2 to both factors on the right hand side. ARGHHHH!

2nd try:
F = (G)(M/r^2)(m/r^2) Solve for M
F/G = [(G)(M/r^2)(m/r^2)] / G
F/G = (M/r^2)(m/r^2)
(F/G)(r^2/m) = (M/r^2)(m/r^2)(r^2/m)
(Fr^2 )/(Gm) = M/r^2
[(Fr^2)/(Gm)]r^2 = (M/r^2)(r^2)
(Fr^4)/Gm = M

I divided each side by G, then I multiplied each side by r^2/m, then I multiplied each side by r^2.

Only problem is the answer in the back of the book was Fr^2/Gm which didn't match either of my tries and is why I got on this board to begin with.

Can anyone tell me for sure. Perhaps my way of typing it out is too confusing. Maybe if I spell it out it will be easier to understand.

F equals (G) times (M over r squared) times (m over r^2)
Solve for M

Thanks for all any and all input

 

[ljwilder]

been thinking

I've been thinking and see where I was making mistakes, distribution is used when you are adding or subtracting and this is a multiplication problem, hence I needed to multiply by r^4 to be able to reduce both fractions with denominators.

I've also realized my other problem was that the way it was written in the textbook was very confusing because it showed the M with a line under it and then the m with a line under it (these lines did not connect with each other) and the r^2 was underneath them both in the middle. I took this to mean that they each had the same denominator. They should have made it more clear by having the M and the m next to each other with one line under them and then the r^2 as the denominator. Here is my last attempt:

F = G * Mm/r^2 (whole different problem, isn't it?)
r^2 * F = G * Mm/r^2 * r^2
r^2 * F = GMm
(r^2*F)/Gm = (GMm)/Gm
(r^2*F)/Gm = M

which matches the book's answers.

Thanks tkhunny and Mrspi for helping me see my mistakes and working through this problem. I appreciate the time you put into this board.