Solving for the Depth of Release: Pressure on a Ball

[BoostAdiction]

[SOLVED] Pressure on a ball

Homework Statement

Suppose you release a small ball from rest at a depth h below the surface of water. The ball shoots upwards above the surface by 1m. If the density of the ball is .400 that of water and drag force on the ball from water is negligible, at what depth was the ball released?

Homework Equations

I'm assuming p(2)=p(1) + dg(y(1)-y(2))

The Attempt at a Solution

I think I have to find the pressure on the ball in order to start solving this. I also know that the force due to gravity on the ball is less than that of the buoyancy force because the ball shoots upwards. I also know that the pressure increases as h below the water increases

 

[tiny-tim]

Welcome to PF!

Hi BoostAdiction! Welcome to PF!

Hint: Try this as a potential energy problem rather than a pressure problem.

What is the potential energy difference between the ball being distance h below the surface and being at the surface?

 

[BoostAdiction]

If you set the ball below the surface to be 0, then the potential energy is 0. Thus, the ball at the surface has a potential energy equal to mgh. Both m and h are unknown. And thanks for the welcome!

 

[tiny-tim]

Thus, the ball at the surface has a potential energy equal to mgh.

Whoa!

It isn't g, is it?

What is the effective force of gravity (in other words, the actual acceleration) under the water?

For a start, it's up and not down, isn't it?

Try again!

 

[BoostAdiction]

So it has an acceleration in the positive direction. I understand that much, then its all a lil fuzzy

 

[tiny-tim]

So it has an acceleration in the positive direction. I understand that much, then its all a lil fuzzy

Hi BoostAdiction!

Normally, I'd recommend looking at wikipedia … but I've just checked the page on Buoyancy, and it looks terribly long and confusing!

So let's ignore wikipedia … what are the forces on the ball, in terms of volume, density of ball, and density of water?

So what is the overall force?

And then what is the acceleration?

 

[BoostAdiction]

haha, good ole' wiki. Theres a force due to gravity pushing down on it, and one pushin up (buoyancy force) and I think the net upward force is that buoyancy force...

 

[tiny-tim]

haha, good ole' wiki. Theres a force due to gravity pushing down on it, and one pushin up (buoyancy force) and I think the net upward force is that buoyancy force...

Yeeees … but how much is it?!

 

[BoostAdiction]

Well the acceleration has to be greater than 9.8 to have an upwards effect on the ball...So would i take .4 x 9.8 to get the force due to gravity?

 

[tiny-tim]

Well the acceleration has to be greater than 9.8 to have an upwards effect on the ball...So would i take .4 x 9.8 to get the force due to gravity?

erm … would you take 1 x 9.8 if it had the same density as water?

Or 5 x 9.8 if it was 5 times as dense?

Either do the equations …

… what are the forces on the ball, in terms of volume, density of ball, and density of water?

So what is the overall force?

And then what is the acceleration?

… or make a good guess!