Proving t^2/r^3: What Went Wrong?

[jackrc11]

Homework Statement

In our unit on centripetal force, a large portion of the unit has been focusing on gravity and how the orbit is related to centripetal force. Our teacher told us to prove t^2/r^3 using F=(mv^2)/r and F=G(mM/r^2) as a small side part of our homework

Homework Equations

t^2/r^3
F=(mv^2)/r
F=G(mM/r^2)

where G = 6.67*10^-11
M and m are the objects' masses

The Attempt at a Solution

(mv^2)/r = G(mM/r^2)
mv^2 = G(mM/r)
v^2 = GM/r

giving

4(pi^2)(r^2)/t^2 = GM/r
and
GM = v^2 / r = 4(pi^2)r

GM/r = (4pi^2)r / r = 4pi^2

4(pi^2)(r^2)/t^2 = (4pi^2)

leaving the ratio
r^2 / t^2

When it should be r^2 / t^3. What did I do wrong? Was there something I forgot?

 

[SammyS]

Homework Statement

In our unit on centripetal force, a large portion of the unit has been focusing on gravity and how the orbit is related to centripetal force. Our teacher told us to prove t^2/r^3 using F=(mv^2)/r and F=G(mM/r^2) as a small side part of our homework

Homework Equations

t^2/r^3
F=(mv^2)/r
F=G(mM/r^2)

where G = 6.67*10^-11
M and m are the objects' masses

The Attempt at a Solution

(mv^2)/r = G(mM/r^2)
mv^2 = G(mM/r)
v^2 = GM/r

giving

4(pi^2)(r^2)/t^2 = GM/r $\ \$ How do you go from this line
and
GM = v^2 / r = 4(pi^2)r $\ \$ to this line ?

GM/r = (4pi^2)r / r = 4pi^2

4(pi^2)(r^2)/t^2 = (4pi^2)

leaving the ratio
r^2 / t^2

When it should be r^2 / t^3. What did I do wrong? Was there something I forgot?

It's just bad algebra.