Uses of Bernoulli's principle

Torricelli's law

A tank is open to the air, and a small hole is made $h$ below the surface. At the surface and at the hole the pressure is the air's, and the water at the top hardly moves. Bernoulli's equation then gives

the speed of a stone dropped through $h$. The jet does not depend on the liquid's density.

You should find that the jet goes farthest from a hole halfway down: a deep hole is fast but low, a high hole is slow.

The Venturi meter

A pipe narrows to a throat and widens again. The water speeds up in the throat, and its pressure drops. A manometer measures that drop, and Bernoulli's equation with continuity gives the speed, and so the flow rate.

Fast air, low pressure

• A spray or a carburettor: fast air across the top of a tube lowers the pressure there, and the liquid below is pushed up into the stream.
• A wing: its shape and tilt make the air above move faster, so the pressure below is higher, and the difference lifts the plane.
• A spinning ball: one side drags the air along faster, and the ball swerves toward that side. This is the Magnus effect.
• A roof in a storm: fast wind above, still air below, and the roof is pushed up.

Watch out. A fast stream has low pressure inside it. The slow air beside it does the pushing.

A worked example

A Venturi meter in a water pipe $12\ \text{cm}$ across has a throat $6.0\ \text{cm}$ across. The pressure in the throat is $15\ \text{kPa}$ lower. Find the flow rate.

1. Continuity. Half the diameter is a quarter of the area, so $v_2 = 4v_1$.
2. Bernoulli. $15\,000\ \text{Pa} = \tfrac12 \times 1000\ \text{kg/m}^3 \times (16 - 1)v_1^2$, so $v_1^2 = 2.0\ \text{m}^2/\text{s}^2$ and $v_1 = 1.41\ \text{m/s}$.
3. Flow rate. $\pi(0.060\ \text{m})^2 \times 1.41\ \text{m/s} = 0.016\ \text{m}^3/\text{s}$, $16$ litres each second.

Practise with new numbers

Try it in Python

The program finds the speed and the landing point of jets from several holes.

Remember these

• Torricelli: $v = \sqrt{2gh}$, whatever the liquid.
• Venturi meter: the pressure drop in the throat gives the speed.
• Fast air has low pressure: sprays, wings, swerving balls, roofs in storms.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How fast does water leave a hole $3.2\ \text{m}$ below the surface? (Take $g = 10\ \text{m/s}^2$.)

2. Would mercury come out of the same hole faster or slower than water?

3. Why does a strip of paper rise when you blow across its top?

4. A spinning cricket ball swerves. Toward which side?