Draining A Cone

Classical Mechanics Level 1

Each cone below has two holes: one at the apex and another in the base. They differ only in that the cone on the left is turned upside down.

If both cones initially contain the same amount of water and if they begin to drain simultaneously, which cone will take less time to empty out?

Both cones will take the same time The cone on the left The cone on the right

14 solutions

Robert Williams
Jan 7, 2018

Relevant wiki: Fluid Mechanics

The rate of flow is dependent on the pressure, which is proportional to the depth of water.

The cone on the left will maintain greater depth in relation to the remaining water, so will have greater pressure, and so empty more quickly.

Since the left side empties first, there will be a point where the depth is less than the right — which might provide inspiration for a more difficult question to find when that will occur.

is the depth a visual assumption

erica phillips - 3 years, 5 months ago

No, the picture makes it appear the depth is similar in both cones, but we are told that the volume is the same, so it would be deeper in the left cone — assuming they are not filled to the brim.

Robert Williams - 3 years, 5 months ago

sir why is it deeper in the left cone

erica phillips - 3 years, 5 months ago

@Erica Phillips – The right cone has bigger base so you have to put more water to it in order to have the same depth of water than to the one in the left (it will be equall eventually when you fill up the whole cone). Just imagine how it would look like in the real world... Or try to calculate what would be the amount of water in both cases to have the same (e.g. 1cm) water depth.

Andro Kamski - 3 years, 5 months ago

@Andro Kamski – thanks for the explanation

erica phillips - 3 years, 5 months ago

I would have thought the fluid flow would be hindered by slowed surface tension circumference ways and under acute slope and the greater volume of air in the hole vortex created by the flat bottomed plane. Youd also get water sticking to the plane unless it were friction free and repelled water. Thus it might take a verylong time to empty by evaporation in a cool place like your fridge.

Danny Robinson - 3 years, 5 months ago

That shows that the rate of the flow is momentarily fast. Is it true or false that the cone on the left would take the same time as that on the right to empty?

Agnishom Chattopadhyay - 3 years, 5 months ago

Pressure is more in the left come at the hole initially, but won't it change because height of the liquid column changes and therefore pressure changes. Also since initially the flow rate is faster in left cone, it's pressure also decreases faster than the right cone since the height decreases faster in left cone. Please correct me if I am wrong.

Troll guy - 3 years, 5 months ago

When the left cone caches the right depth both have the same pressure at the bottom so the flow should be equal, but since the left one has less volume at this point, it will empty faster. If for some reason the right one catches the left one again the same would happen, although I suspect the decrase in area at the top of the left's surface is greater than the decrease in the rate of flow, so the left one will surpass the right one and keep with the advantage.

Roger AB - 3 years, 5 months ago

I just thought that as the right cone empties towards the bottom the velocity slows given the flat surface whereas on the right the cone hole has the advantage of the angled slopes surrounding it and gravity. But now that I have read your answer many times over, it sounds like your technical answer is spot on.

Jerry Wong - 3 years, 4 months ago

My thinking was absolutely right thanks..

Onkar Gupta - 3 years, 4 months ago

I don't doubt the science of fluid mechanics but no one is mentioning the hole in the left cone. Usually fluids pour faster with a second hole in the container.

Titus-Nietzsche Anderson - 3 years, 4 months ago

Both have 2 holes!

Penna Nera - 3 years, 3 months ago

What people are missing is that the flat surface could cause the water to pool at the edges. This would cause a delay in the water leaving the country be with the flat base. Since this unknown variable is not included, the question can’t be answered.

Jody Mayfield - 3 years, 4 months ago

Agreeing to your first point. I was about to use fluid mechanics and just thought to myself, "not all the water will leave the cone on the right because of the flat base". It will stay beaded in certain parts and lose the race by a wide margin.

Carlos Hernandez - 3 years, 4 months ago

Bhimsen Padalkar
Jan 7, 2018

Flow rate increases with pressure.. Since both cones contain same amount of water, force(weight) exterted by water will be same in both cases.

But for left cone, area on which force is acted is less (just the hole), hence the more pressure.

P=F/A

how can u detect the area where it is not either given in the question or in the picture

erica phillips - 3 years, 5 months ago

On the left the area is the area of the hole, while on the right the area is the area of the base of the cone. We can safely assume the latter is bigger ;)

Penna Nera - 3 years, 3 months ago

No, the pressure in a static system is only dependent on depth, not on weight of fluid. eg, in a static system with 10m head of water, the pressure at the base is 1 bar, whether it is a 5mm upstand pipe or a 100m dia tank. If the depths were the same in this example and the holes were plugged then the same static pressure would apply over the whole flat surface of the base. It would of course apply greater force over the greater area.

A Former Brilliant Member - 3 years, 5 months ago

I was wondering if the volume of air over the water has a bearing.

Nandakumar Menon - 3 years, 5 months ago

I don't think the volume of air should have any effect. Both cones are open to the atmosphere (because of the hole in the top), so there is equal atmospheric pressure acting at the water surface for both cones.

Pranshu Gaba - 3 years, 5 months ago

I agree with Chris, the pressure depends on the height, which is greater for the cone on the left, so it drains out first. Pressure is independent of the volume of water.

Pranshu Gaba - 3 years, 5 months ago

I find that the concept of pressure is a bit confusing. If we imagined that we had a tall cylinder with holes on both ends and water suspended (temporarily) inside, wouldn't it just fall out of the cylinder due to gravity, and the pressure would still be zero? edit: I imagine the cylinder positioned vertically

Morten Silcowitz - 3 years, 5 months ago

Muhammad Rasel Parvej
Jan 7, 2018

Relevant wiki: Fluid Mechanics

The flow rate is directly proportional to pressure, which is, in turn, proportional to the depth, not to volume.

Could you explain why flow rate is directly proportional to pressure? Thanks.

Pranshu Gaba - 3 years, 5 months ago

When sizes of the holes are equal.

Correct me if I am wrong, please!

Muhammad Rasel Parvej - 3 years, 4 months ago

Hi Pranshu We don’t have to assume that flow is directly proportional to pressure, only that it continuously increases with pressure. We know that pressure is directly proportional to depth, and, for the same volume of water the depth will be greater in the narrow part of the cone, on the left, than in the wide part on the right.

Robert Williams - 3 years, 4 months ago

Hey. But that's mean after some time left cone will have the same depth as right one... so... there is a point of time after start, where depth is the same so pressure is the same.. it's not make sense to me :(((

Maciej Kozłowski - 3 years, 4 months ago

Note that when the height of the water is same in both cones, there is less water in the left cone because area of cross section is less. So although the rate of loss of water is the same at that time, there is less water to be drained in the left cone and it empties first.

Pranshu Gaba - 3 years, 4 months ago

Ander Ander
Jan 8, 2018

In the case of an incompressible fluid, the product of the velocity of a fluid and the section area should remain constant.

In the left cone, since the section is decreasing, the velocity of the fluid should increase. In the right cone, it happens the opposit, the velocity of the fluid is decreasing.

Therefore the answer is the left cone.

The product of velocity and the section area gives the amount (volume) of liquid flowing out. If you say that the product is constant, then it means that water is flowing out at a constant rate. This is not true, since the amount of water coming out is maximum initially when the height of the water is maximum and decreases as the height of the water column decreases.

Pranshu Gaba - 3 years, 5 months ago

Shankar Pariasamy
Jan 11, 2018

Hydrostatic pressure, P = ρ (density) x g (gravitational acceleration) x h (height of column of water).

ρ and g are equal in both cases. However, h is higher in the left cone due to its geometry and orientation, despite having the same volume of water - therefore the rate of flow is higher in the left cone.

Eliud Alejandro Maldonado Sanchez
Jan 8, 2018

The cone on the right dissipates more energy from the vortex created while the level of water is going down, to the surrounding water that is farer from the vortex, so it is slower.

Laurent Shorts
Jan 11, 2018

If I did my math correctly, the right one will take $\dfrac{8}{3}$ more time.

Could you post your working for how you got that number? Thanks!

Pranshu Gaba - 3 years, 4 months ago

Young Brigham Bautista
Jan 8, 2018

The right one will take more time because the pressure is equally surrounds the base. Unlike the on the left cone the pressure is focused on the center.

Shuto Nakai
Jan 8, 2018

This question is all about gravity and pressure, : The flow rate is directly proportional to pressure, which is, in turn, proportional to the depth, not to volume.

You copied this off someone else called Muhammad Rasel Parvej Shuto

Aiden Cooper - 3 years, 5 months ago

No i didnt!

Shuto Nakai - 3 years, 5 months ago

Karl-Otto Markussen
Jan 13, 2018

Just a remark. Why is it the cone on the left that is turned upside down? - it might as well be the one on the right. Traffic cones or ice cream cones - that is the question.

It shows that the time it takes for the cone to completely drain out can be different depending on which end it is emptied from.

Pranshu Gaba - 3 years, 4 months ago

Yes, I got that point.

My point concernes the way you label the orientation of the cones: "They differ only in that the cone on the left is turned upside down." This implies that the one on the right is upside up. You might as well have labelled the one on the right upside down. Or in my mind more correctly said "If you turn one of the cones upside down they will be the same."

A cone does not have an upside or a downside per se. It has an orientation, but none of them are per definition upside.

In context there might be an upside - for instance can both an icecream cone and a traffic cone be turned upside down - but cone-vice then they don't turn the same way.

Karl-Otto Markussen - 3 years, 4 months ago

Harish N
Jan 11, 2018

According to Torricelli's Law
v=sqrt(2 * density * height)
Since the density of liquid does not change.
So, velocity of liquid coming out depend on height upto which liquid is filled.

Wasiff Afzal
Jan 11, 2018

Doesn't the cone on the left just have less water so it'll empty out faster? Why is no one noting that?

The question says "... both cones initially contain the same amount of water ...". The interesting point is that the left cone drains out faster in spite of both cones starting with the same amount of water initially.

Pranshu Gaba - 3 years, 5 months ago

Au Alvin
Jan 8, 2018

Pressure = Force/Area. The one on the left has less surface area as it drains compared to the one on the right, so it drains slower.

Bro.... Pressure=Force/Area

Matin Ahmed - 3 years, 5 months ago

Aiden Cooper
Jan 8, 2018

The flow rate is proportional to depth, not volume.

Draining A Cone

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