# How do I calculate the acceleration of gravity underwater?

1. Jul 15, 2015

### Charlie Kay

Does anyone know a simple formula to calculate the acceleration of gravity underwater?

2. Jul 15, 2015

### Charlie Kay

I have this, "m(d2x/dt2)=(rhoB-rhoA)(g*V)-0.5(Cd)(rhoA)(A)(dx/dt)^2"
When m=body's mass, rhoA=fluid density, rhoB=body density, g=gravity, x=displacement, t=time, Cd=drag coefficient, V=body volume and A=body area

Help??!

3. Jul 15, 2015

### SteamKing

Staff Emeritus
What makes you think that the acceleration due to gravity is different underwater?

4. Jul 15, 2015

### SteamKing

Staff Emeritus
And where did you get this formula?

5. Jul 15, 2015

### Charlie Kay

a) there is more drag underwater and b)why, have you seen it before?

6. Jul 15, 2015

Staff Emeritus
This. Ever see a movie set on a submarine? That's underwater, and the gravity looks normal.

7. Jul 15, 2015

### CWatters

True but that's nothing to do with gravity.

I think your original question is wrong. I think what you are asking is.. What is the net vertical force (eg the sum of weight and buoyancy) ?

8. Jul 15, 2015

### my2cts

Eureka !

9. Jul 15, 2015

### Staff: Mentor

The acceleration due to gravity is g, by definition. That equation you have is the net force due to gravity + buoyancy + drag.

10. Jul 15, 2015

### XZ923

Not to throw fuel on the fire, but I'm just curious. Isn't the gravitational constant g defined as being ~9.81 m/s^2 at sea level? If so, couldn't "below sea level" factor into the equation? My inclination is that the answer is basically "yes, but by such a small amount it's insignificant". Is this correct?

11. Jul 15, 2015

### nasu

Yes, but this has nothing to do with the presence of water.
The sea level is just a convenient reference level. You can be under the sea level but not underwater. Or way above the sea level and underwater.
It would be misleading to attribute the change in g due to altitude change to being (or not) "underwater".

12. Jul 15, 2015

### Noctisdark

g doesn't change a lot when you go up/down few meters, in fact the change is so small that it can be noticed, I will assume that you mean the net acceleration under water, you can use newton's second law, F = ma, so ma = ρgV - mg (+drag force that I will ignore, considering a point particle), you get a = ρgV/m -g, if our particle has constant density then a = g*(ρmediumparticle-1), this is a very particular case :)

13. Jul 15, 2015

### XZ923

Thanks that's pretty much what I figured. I knew that the presence of water wouldn't matter, I was just curious as to the sea level reference. I understand that it's simply the reference point that was chosen when they calculated the gravitational constant. I guess my question was more along the lines of "is the difference in gravitational force at the bottom of the Mariana's trench and the peak of Mt. Everest (because point A is closer to the center of the earth's mass than point B; the water of course is irrelevant) significant enough to factor into calculations?

14. Jul 15, 2015

### nasu

That will depend on what is the calculation and what is the purpose of it.
And also, on what is the accuracy of the other parameters entering the calculation.

15. Jul 15, 2015

### SteamKing

Staff Emeritus
This article has a discussion on the variation of g due to various factors, including altitude and latitude:

https://en.wikipedia.org/wiki/Gravity_of_Earth

At the top of Mt. Everest for instance (approx. 9000 m above sea level), the value of g is about 0.29% lower than g measured at sea level. At the altitude the Space Shuttle formerly orbited, about 400 km above the surface, g was still about 90% of its value at sea level. There is a relatively simple formula to calculate g at altitude.

Going in the other direction, the variation of g with depth is a little trickier to calculate, because you must account for the mass of whatever solid (or liquid) material lies between the measurement depth and sea level. See more about this in the discussion of the Shell Theorem.

16. Jul 15, 2015

### davenn

careful with your choice of words there

Gravitational Constant = G = approximately = 6.674×10−11 N⋅m2/kg2 a whole different thing

acceleration due to gravity ( eg at Earth's surface) = g = ~9.81 m/s2
which is the one being discussed

Dave

17. Jul 16, 2015

### XZ923

I stand corrected. Thank you sir. You're right, poor choice of words on my part. And SteamKing thanks for satisfying my curiosity!

18. Aug 21, 2015

### Charlie Kay

When I said underwater I mean deep, as in 11,000 m below sea level!

19. Aug 21, 2015

### PeroK

With $M$ as the mass of the Earth, $R$ the radius and assuming uniform density, the force of gravity a distance $r$ from the centre of the Earth, with $r< R$, is given by:

$\frac{GMr}{R^3}$

It would be a good exercise to derive this equation for yourself.

20. Aug 21, 2015

### nasu

This is pretty deep for diving or submarines.
But for change in gravity is not much. You compare 11 km with the radius of the Earth which is around 6,400 km. About 0.2%.
You should not expect a significant change. As g goes like r^2, the change in g may be some 0.4% or about 0.04 m/s^2.