A 'hole' new dimension

Classical Mechanics Level 3

A small hole is made in the side of a full oil drum (barrel). A jet of oil shoots from the drum and lands on the ground some distance from the drum. At what height (in metres) above the ground should the hole be made to maximise the horizontal distance between the initial splash down point and the drum? Give your answer to three decimal places.

Note . It is not too hard to find the answer using standard formulas from mechanics and hydrodynamics. My challenge is to find the answer from first principles using dimensional analysis without appealing to a formula sheet (not even a memorised one!).

Assumptions

The drum is 1 metre tall and open to the atmosphere at the top. acceleration due to gravity g = 10 m s − 2 density of oil ρ = 900 k g m − 3 \text{The drum is 1 metre tall and open to the atmosphere at the top.}\\\text{acceleration due to gravity }g=10ms^{-2} \\ \text{density of oil }\rho=900 kgm^{-3}

Air resistance to the jet of oil may be ignored.


The answer is 0.500.

Peter Macgregor by Peter Macgregor , 64, United Kingdom

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1 solution

Peter Macgregor
Mar 20, 2017

The answer is 0.500.

We will use M,L and T to stand for the dimensions 'mass', 'length' and 'time' respectively. Let h t h_t be the distance of the hole from the top of the drum, and h b = 1 − h t h_b=1-h_t be the distance of the hole from the bottom of the drum.

Instead of borrowing Torricelli's theorem ready made we will work out the heart of the formula for the exit speed of the oil by dimensional analysis. Let us make the reasonable assumption that this speed depends on gravity, the density of the oil, and the distance from the top of the drum. Then

v ∝ ρ α g β h t γ … ( 1 ) v\propto \rho^\alpha g^\beta h_t^\gamma\dots(1)

or in terms of dimensions

[ L T − 1 ] = [ M L − 3 ] α [ L T − 2 ] β [ L ] γ [LT^{-1}]=[ML^{-3}]^\alpha[LT^{-2}]^\beta [L]^\gamma

Comparing the powers of M,L and T on each side of the equation gives us three simultaneous equations

α = 0 1 = − 3 α + β + γ − 1 = − 2 β \alpha=0\\1=-3\alpha + \beta + \gamma\\-1=-2\beta

yielding

α = 0 β = γ = 1 2 \alpha=0\\\beta=\gamma =\frac{1}{2}

Substituting these results into (1) gives

v ∝ g h t v\propto \sqrt{gh_t}

and so

v = k 1 g h t v=k_1\sqrt{g h_t} for some dimensionless constant k 1 … ( 2 ) k_1\dots(2)

Now we will use dimensional analysis again to find the heart of the formula telling us how long it takes the jet to drop from the height h b h_b to the ground. We start with the reasonable assumption that

t ∝ g α ρ β h b γ t\propto g^\alpha \rho^\beta h_b ^\gamma so that

[ T ] = [ L T − 2 ] α [ M L − 3 ] β [ L ] γ [T]=[LT^{-2}]^\alpha[ML^{-3}]^\beta [L]^\gamma

This time equating the powers of the dimensions and solving the simultaneous equations leads to

β = 0 α = − 1 2 γ = 1 2 \beta =0 \\ \alpha =-\frac{1}{2} \\ \gamma =\frac{1}{2}

and so

t = k 2 h b g … ( 3 ) t=k_2 \sqrt{\frac{h_b}{g}} \dots(3)

where k 2 k_2 is a dimensionless constant of proportionality.

Combining the horizontal exit velocity (2) with the time taken for the jet to fall to the ground (3) shows that the distance travelled is

k 1 k 2 h t h b = k h b ( 1 − h b ) k_1 k_2 \sqrt{h_t h_b}=k \sqrt{h_b (1-h_b)} where k = k 1 k 2 k=k_1 k_2

By applying the AM-GM inequality to the terms h b h_b and 1 − h b 1-h_b this distance is less than or equal to k 2 \frac{k}{2} with the maximum being attained when

h b = 1 − h b h_b=1-h_b

in other words when h b = 1 2 \boxed{h_b=\frac{1}{2}}

Notice that neither the density of the oil nor the acceleration due to gravity affect the answer.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

That's very cool. You could probably program computers to come up with formulae automatically using this approach. Just give them access to huge quantities of empirical data and let them go, maybe with a little up front guidance about what you want.

Steven Chase - 4 years, 2 months ago

Thanks for your comment Steven. As you say upfront guidance would be needed to automate this process because some physical intuition is needed to solve a problem using dimensional methods. For instance in this problem you need to know that the depth of the hole influences the speed of the jet, while the height of the hole influences the time to fall. You also need to intuit what variables are relevant - why not, for example, include the viscosity of the oil or the atmospheric pressure or the diameter of the small hole in the analysis? Usually solving a problem by dimensional analysis leaves you with an undetermined constant of proportionality which can be found either by a more detailed analysis of the problem, or by experimental work.However these constants are so often about one, i.e in the range 0.1 < k < 10 0.1<k<10 that the result of a dimensional analysis usually gives a good ball-park feeling for what is going on.

Peter Macgregor - 4 years, 2 months ago

When I opened up "Discuss Solutions" That huge thing came up and I was like, Seriously, that much math!

Rida Khan - 2 years, 6 months ago

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Hi Rida,

Yes there is a lot of maths, because the answer is worked out 'from first principles'.

I like brief solutions as well, but sometimes it's good to try a different approach.

Peter Macgregor - 2 years, 6 months ago

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