Gravity Basics: Negative Signs & Direction

[Hunt4Higgs]

Another stupid thing I've come across and thought; what?

Mainly down to not having many example questions to look at, I'm sure that'd help a bunch.

I am self studying this topic so apologies for what may be silly questions...

Looking at the formulae for Gravity, Gravitational Field Strength, Gravitational Potential, and Gravitational Potential Energy...

Some of those should end up with negative answers, and some dont.
I understand that obviously the direction is towards the mass (example earth), but some of the minus signs in the formulae say they are just to show direction.
Which are for direction and which are in there to provide you with a negative answer at the end? And why, if possible to explain?

I've briefly looked at gravity wells and I have seen that work done in taking a mass to infinity is positive work...and work done in bringing a mass from infinity is negative work.

As an example of where I'm stuck.
Gravitational Field Strength is a Vector Quantity and when the formula is in front of you, it is given with a minus sign, and the answer comes out with no minus sign.

Gravitational Potential is a scalar quantity and the formula is again given with a minus sign, yet the answer that comes out is supposed to be negative?

Why are some supposed to be negative and some arent but are still given with minus signs in their individual formulae?

 

[tiny-tim]

Hi Hunt4Higgs!

Gravitational Field Strength is a Vector Quantity and when the formula is in front of you, it is given with a minus sign, and the answer comes out with no minus sign.

when field strength comes with a minus sign,

that's because field strength is a vector, so it'll be minus whenever it happens to be opposite to whatever convenient coordinate system you're using

eg we usually like up to be positive, so gravity is negative

Gravitational Potential is a scalar quantity and the formula is again given with a minus sign, yet the answer that comes out is supposed to be negative?

the value (and sign) of the potential depends on where you choose your zero

if you choose it at ground level, then the potential (above ground) is always positive, so higher points have greater potential than lower ones

if you choose it at infinity, then the potential is always negative, but higher points still have greater potential than lower ones, since they're less negative!

 

[technician]

As tiny tim points out the ZERO of potential energy is taken to be at infinity.
This makes sense because infinity is a reference point we can all agree on and supplying energy raises you towards infinity.
By analogy heights on Earth are taken with reference to sea level so heights are positive, depths are negative.

 

[Naty1]

Those signs do take some thinking about...it's not necessarily 'obvious'...but fortunately those who have come before are smart and have thought about it a lot!

Try Wikiepdia for some additional information:

http://en.wikipedia.org/wiki/Gravitational_potential

[and note the diagrams with summaries under each]

In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that is done by the force of gravity as an object moves to that location from a reference location. It is analogous to the electric potential with mass playing the role of charge. By convention, the gravitational potential is defined as zero infinitely far away from any mass. As a result it is negative elsewhere.

Can you figure out why the sign is negative here:

http://en.wikipedia.org/wiki/Gravitational_field


And another insight:

as posted already:

if you choose it [your reference] at infinity, then the potential is always negative, but higher points still have greater potential than lower ones, since they're less negative!

The amount of positive energy in the form of matter is exactly canceled out by the negative energy in the form of gravity.

http://en.wikipedia.org/wiki/Zero-energy_universe

I've briefly looked at gravity wells and I have seen that work done in taking a mass to infinity is positive work

Earth is a 'gravity well'...As an example, when we lift a space probe off Earth we need a rocket...energy to do work against gravity...so no matter what sign convention you decide to use, the only 'free ride' is on the way back WITH gravity.

NOT basic:
[Regarding gravitational potential: in relativity you'll eventually find that there are other 'strange' effects...gravitational potnetial slows time [causes time dilation] while the rate of change of gravitational potential [how fast it changes] changes spacetime curvature! Took an 'Einstein" to first figure those out]

 

[Hunt4Higgs]

Tiny Tim,

Just looking at what you said about the formula for field strength coming with a minus sign because it is a vector.

So the minus sign is put in with the original formula just to show it is a vector quantity and seeing as gravity is normally taken to be in the negative (down) direction, it is left with a minus in the formula.

Back to anyone and everyone that can help,

I have a question in my textbook, a worked example that is calculating the gravitational field strength at the surface of Venus. Obviously dodging the numbers on this occasion - the formula is g= -GM_v / R_v² (as I'm sure you will all know already)
Then the answer given is a positive number, specifically 9.1 N kg^-1

However when placing that calculation into a calculator the answer is negative.

Is this purely because the question is only asking for the strength of the gravitational field, therefore you can say it is this positive number? Would it matter if you left it as a negative number?

I hope this isn't too bold, or too blatently obvious, but is this just a general rule for vector quantities? And if so is there a rule for scalar quantities whereby if they have a negative sign in the formula, they will always be a negative number?

(Apologies if this is obvious information I should already know, I don't have the best tutor and the majority of this I am self studying)

Thanks for the help so far guys/girls.

 

[tiny-tim]

Is this purely because the question is only asking for the strength of the gravitational field, therefore you can say it is this positive number? Would it matter if you left it as a negative number?

Yes, although the field is a vector, everyone knows what its direction is (!), so really we're only interested in its magnitude, which is a (positive) scalar.

This really only applies to gravity.

(there are other anomalies of language with gravity … eg, g-force is an acceleration, not a force! )

 

[Hunt4Higgs]

Just a quick check with you guys before I move on.

Gravitational potential is defined at a point as being numerically equal to the work done in bringing a unit mass to that point.

However the formulae for GPE and for work done are the same right?

But is this connection between gravitational potential and work done being equal because a "unit mass" has a mass that is negligible? Hence making the formulae for gravitational potential the same as gpe and work done?

 

[tiny-tim]

Hi Hunt4Higgs!

… Hence making the formulae for gravitational potential the same as gpe

Potential is not the same as potential energy​

Gravitational potential = gravitational potential energy per mass

(Similarly, if you've done electric charge, you'll know:

electric potential = electric potential energy per charge)

 

[Hunt4Higgs]

Hi Tiny Tim! :)

So how can the gravitational potential be equal to the work done if the formulae are different?

Are the formulae for work done and gpe not the same?

Was the last part of what I said correct other than the GPE part? Is it numerically equal because a unit mass is negligible in the calculation?

(sorry, I keep getting stuck with silly mundane things).

 

[tiny-tim]

Gravitational potential = gravitational potential energy per mass

Gravitational potential = gravitational work done per mass

… Is it numerically equal because a unit mass is negligible in the calculation?

it isn't negligible, it's very large , it's 1 !