S1lent Echoes said:
After a lot of research and much studying, I have come up with two possible understandings.
The gravitational constant can be understood more simply as 6.67*10^-11 N. The dimensional analysis lead me to believe that the other letters were just a way of expressing what was going on in relation to what the letters were representing.
No, the gravitational constant cannot be interpreted more simply as just having units of Newtons. It has precisely the units that are listed there: (Nm
2)/kg
2
S1lent Echoes said:
For instance -
70 kg * 70 kg / (1 m)^2 is another way of saying 4900 kg traveling at a meter per second per second and since a Newton is basically 1 kg traveling at a meter per second per second then you could say that the 4900 kg is 4900 N.
Everything you've said above is completely nonsensical. The 4900 is a quantity that has dimensions of (mass
2)/(length
2). There is no need to try to get an "intuitive" sense of what that means. Newton's universal law of gravitation says that the gravitational force between two masses is proportional to the product of the two masses, and inversely proportional to the square of the distance between them. When you compute the product of the two masses and divide by the square of the distance between them, you get this number which has those units. (kg
2/m
2). That's all there is to it.
S1lent Echoes said:
When I multiply the 4900 N by the 6.67 * 10^-11 N, I get 32683 * 10^-11 N. Add in the decimals to simplify and I get 3.2683 * 10^-7 N or 3.3 * 10^-7 N rounded up, which is the answer given.
OR
Do the letters cancel each other out in the equation?
For instance -
In N * m^2 / kg^2 * 70 kg * 70 kg / (1 m)^2, do the m^2's cancel each other out and kg^2's cancel each other out leaving only N * 4900? Which then multiplies by the 6.67 * 10^-11, resulting in the numbers I gave above.
It seems to me that it has to be one or the other explanations as they give the correct answer, I am just not sure which one it is and why. And again, I want to understand what I am doing, not just come to the right conclusion.
Look, I don't know what you're trying to do here, but to work out the units of the final answer,
just follow the rules of algebra. Let's start with the equation$$F = G\frac{m_1m_2}{r^2}$$ $$= \frac{(6.67 \times 10^{-11}~\textrm{Nm}^2/\textrm{kg}^2)(70~\textrm{kg})(70~\textrm{kg})}{1~\textrm{m}^2}$$ $$ = \frac{(6.67 \times 10^{-11}~\textrm{Nm}^2/\textrm{kg}^2)(4900~\textrm{kg}^2)}{1~\textrm{m}^2}$$
Now let's collect all of the units together and simplify:$$ = (6.67 \times 10^{-11})(4.9 \times 10^3) \frac{\textrm{Nm}^2}{\textrm{kg}^2} \frac{\textrm{kg}^2}{\textrm{m}^2}$$ $$ = 32.683 \times 10^{-8}~\textrm{N} \frac{\textrm{kg}^2}{\textrm{kg}^2} \frac{\textrm{m}^2}{\textrm{m}^2}$$
After simplifying the units by cancelling out things that appear in both numerator and denominator (remember the statement in bold above), we are left with:$$ = 3.2683 \times 10^{-7}~\textrm{N}$$
So, as you can see, the units work out just fine! We're left with Newtons, which is good, because this equation is supposed to be for computing the gravitational force. The reason why G has the units that it does is precisely in order to make the units work out. Because on the left hand side of the equation, you have something with dimensions of force, which means that the right hand side also has to have dimensions of force in order for the equation to be dimensionally consistent. But mass*mass/length*length doesn't have dimensions of force. The force is *proportional* to this, with the constant of proportionality G having the dimensions to make it work out. IF you look at the right hand side without G, it has units of "kg^2/m^2", which I'll call "blah" for short. Therefore, G has units of "N/blah" so that when you multiply by G, you just end up with units of N.
