0 - 2020-04-14

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Knight

08:09

@AnnaV Hello ma’am! How are you?

2 hours later…

Knight

10:04

@JohnRennie Hello! Sir how are you?

Sir if we take a unit volume of water under gravitational field such that the surface of water is at $z=0$ and we are taking positive $z-$ axis as vertically downwards. Then, $$ \mathbf F = F_z \hat k \\ \textrm{where} ~~~~ F_z = \rho g = \gamma$$

What I’m unable to understand is when my book writes (just after the things which I have stated above) $$ U = -\rho g z $$. Sir how the potential energy of whole liquid is $\rho g z$ ? (Or whatever he wanted to denote by that expression)

John Rennie

@Knight hi :-) Are you there?

Knight

10:26

@JohnRennie Yes, :-)

John Rennie

Consider some small volume element of the water $dV$. The mass of this element is $dm = \rho dV$.

Knight

Yes

John Rennie

Now move this element a height $z$, and the potential energy changes by $dU = g z dm = g z \rho dV$

Knight

Yes

John Rennie

The final step is that the potential evergy is an extrinsic quantity i.e. it is the potential per unit volume of the liquid. So just divide through by $dV$ and yout get $U = \rho g z$

Knight

10:29

Okay. Got it thanks

:-)

He meant “potential energy per unit volume”, ha?

John Rennie

Well $\rho g z$ is the pressure at a depth $z$, isn't it?

Knight

Yes

John Rennie

$dW = PdV$ is the work, and the work is just the change in PE $dU$

Knight

Yes

Sir please excuse me, I got attend Nature’s call :-) . I really apologise, I’m coming as early as possible

John Rennie

OK :-)

Knight

10:42

@JohnRennie I’m back

If we take a unit volume of water, then the potential energy of a layer of water $dz$ units thick and situated at $z$ is $$ dU = g dm z = g \rho 1\cdot dz ~z$$ now potential energy of whole liquid is $$\int _{0}^{1} g \rho ~z ~dz \\ g \rho |\frac{z^2}{2} |_{0}^{1} = g \rho /2$$

Why I’m not getting $\rho g z$ ?

John Rennie

Yes. The $U$ that you're given isn't the potential energy of the whole volume of the water. It's the potential energy per unit volume of a small element at depth $z$.

Knight

“It’s the potential energy per unit volume of a small element at depth $z$” didn’t get you.

John Rennie

The function $U(z)$ is defined such that if we take some small volume element $dV$ at a depth $z$ then the potential energy of that element is $U(z)dV$.

Knight

Okay. Got you!

Thanks

Yuvraj Singh...

11:02

@JohnRennie hi

John Rennie

@YuvrajSingh... hi :-)

Yuvraj Singh...

u ignored my ping ?

@JohnRennie

John Rennie

Sorry :-( I get a lot of pings and sometimes I miss pings.

Yuvraj Singh...

no issue .

i can understand ,that there were many of the such users

who need ,s you for their help!

Knight

Sir 76 cm Hg means the pressure that I would feel if I were to stand at the bottom of 76 cm of mercury column? (Zero degrees Celsius is ensured)

John Rennie

11:06

@Knight suppose you have a column of mercury with a cross sectional area of 1 m^2 and a height of 76cm.

Knight

Okay

John Rennie

Then the force at the base of the column is the weight of the mercury i.e. $F = 0.76\rho g$

And the pressure is the force divided by the area. We chose the area to be 1m^2 so the pressure is also $P = F/1 = 0.76\rho g$ Newtons.

Knight

Sir have you missed $g$ ?

John Rennie

@Knight oops, yes :-)

Knight

Yes.

John Rennie

11:08

@YuvrajSingh... you're always welcome to ping me and I'll try to reply quickly.

Knight

So, what does 76 cm Hg actually mean?

John Rennie

It's the pressure at a depth of 76cm in mercury.

Knight

Oh kaye!

A very weird unit ! Beacuse I wouldn’t think of going under the sea of mercury (no matter how small ) :)

John Rennie

@Knight it's a historical artefact. In the old days barometers were built using mercury columns, and it was convenient to take the pressure of a 1mm height of mercury as a unit. This unit is called the torr.

So 1 atm = 760 torr.

Torr

The torr (symbol: Torr) is a unit of pressure based on an absolute scale, defined as exactly 1/760 of a standard atmosphere (101325 Pa). Thus one torr is exactly 101325/760 pascals (≈ 133.32 Pa). Historically, one torr was intended to be the same as one "millimeter of mercury", but subsequent redefinitions of the two units made them slightly different (by less than 0.000015%). The torr is not part of the International System of Units (SI), but it is often combined with the metric prefix milli to name one millitorr (mTorr) or 0.001 Torr. The unit was named after Evangelista Torricelli, an Italian...

Knight

@JohnRennie When we measure blood pressure the mercury rises up in column, what actually happens there?

John Rennie

11:16

The mercury rises up to a height $z$ such that the pressure $P = \rho g z$ is equal to your blood pressure.

Knight

Yes, I think we have derived it just few minutes before. :-)

John Rennie

I need to go now but I'll be around later or tomorrow morning as usual.

Knight

11:51

Okay sir