Centrifugal force on water in a glass

[srecko97]

Homework Statement

There is a cyllindrical glass (r=0.05 m) full of water. It is rotating around its vertical geometric axis with 3 turns per second. How many centimetres higher is water at the rim than in the centre of the glass . Water rotates together with the glass.

Homework Equations

p=ρah
ω=2πγ
a=ω²r

The Attempt at a Solution

All I know is that the shape of water is parabolic. In the centre there is no additional pressure from rotation, the pressure on the glass is p=ρah, a=ω²r
total pressure on any point is p_x + p_y (vectorial sumation)

 

[billy_joule]

All I know is that the shape of water is parabolic.

That's right. If we let the vertex be at (0,0) then you just need to find the Y value when X=0.05m.
Start by drawing a free body diagram for a piece of water on the surface.

 

[srecko97]

probably is y=Cx² ... but I do not know what is C, maybe a/g ... a=17.8 m/s^2, g=10 m/s^2 i do not know, honestly just guessing, as i do not know how to solve it

 

[billy_joule]

Start by drawing a free body diagram for a piece of water on the surface.

Post your FBD and we'll see where you've gone wrong.

 

[srecko97]

I do not know what free body diagram is. Sorry. Please explain to me and I will try

 

[haruspex]

I do not know what free body diagram is. Sorry. Please explain to me and I will try

Consider a small parcel of water at the surface, at some radius x. If the surface shape is described by y=f(x), what is the slope there? Draw a diagram of the parcel of water and the forces acting on it. Treat the rest of the water as a frictionless solid.

 

[srecko97]

 

[haruspex]

Ok. You left out a normal force, and you need to answer this question

If the surface shape is described by y=f(x), what is the slope there?

I.e., in terms of x and y.

 

[srecko97]

tanφ =ω²r / g

 

[srecko97]

and dy/dx = ω²r / g
so: y= ω²r x /g ... for r=x I get y= ω²x² /2g

 

[srecko97]

and I have y if i put r=x!

 

[haruspex]

and I have y if i put r=x!

Right. So what answer do you get?

 

[srecko97]

4,4cm

 

[haruspex]

4,4cm

Using g=10m/s², yes, but 4,5cm would be a bit closer.
Well done.