Pressure in fluids

Part of a machine to generate electrical energy using waves is a cylindrical drum whose cross-section is shown below. The height of the drum is 2.0 m and its cross-sectional area is 0.79 m².

The mass of the drum is 1200 kg.

The centre of mass of the drum is 0.50 m from one of the circular faces, on a line joining the centres of the circular faces.

The density of water is 1000 kg m^–3.

The drum floats in flat water with its long axis vertical. The base of the drum is a distance d below the surface, where d is measured in m.

Explain why the weight of water displaced must be equal to the weight of the drum.

Calculate d.

Upthrust forces act at the centre of mass of the fluid displaced.

Explain why the drum cannot float in equilibrium on its side with the long axis horizontal.

When the vertical drum is displaced a distance x deeper into the water from the equilibrium position, the magnitude of the resultant force on the drum, F, is given by

F = –kx

where k is a constant and d is measured in m.

By considering the magnitude of the additional upthrust on the drum, show that k is about 8000 Nm^–1.

When the drum is released, it oscillates vertically. Explain why the oscillations are simple harmonic motion.

Calculate the natural frequency f of the oscillations.

* Two designs, A and B, of the machine are tested in a large ripple tank.

The ripple tank can produce waves whose amplitude and frequency can be varied over wide ranges.

The natural frequencies of the two machines are identical. The graphs show the results of the tests but are incomplete.

• Describe how the amplitude of the machine could be measured at each frequency tested. You do not need to describe how to vary the amplitude or frequency of the waves.

• Complete the graphs with suitable labels and annotations.

• Discuss which of the designs is the most effective at generating electrical energy.

Additional space if required

A sealed container contains n moles of an ideal gas. The gas has pressure p, absolute temperature T and occupies volume V.

The mass of one mole of the gas is M.

Use an ideal gas equation to show that the density of the gas is given by the expression

An airship has a cabin suspended underneath a gasbag inflated with helium.

The airship is floating above the ground and is stationary.

The volume of the gasbag is 12 000 m³. The temperature of the helium and the surrounding air is 20 °C. Atmospheric pressure is 1.0 × 10⁵ Pa. The molar mass of air is 0.029 kg mol^–1. The volume of the cabin is negligible compared to the volume of the gasbag.

Show that the density of air under the conditions described is about 1.2 kg m^–3. (i)

Calculate the weight of air displaced by the airship.

weight of air........ N

Explain why the weight of air displaced by the airship has the same magnitude as the weight of the airship and its contents.

The pressure of the helium in the gasbag is maintained at a value only slightly greater than atmospheric pressure. Suggest why a larger pressure is not used.

The airship engine drives a fan which moves 7.8 kg of air per second at a relative speed of 45 m s^–1, so the airship starts to move.

All other conditions given in (b) remain the same.

Calculate the thrust that the engine produces.

The airship has a higher maximum speed at high altitudes, but also produces less thrust from the engine.

Explain these observations.

The table shows some data on the planet Venus.

Calculate the magnitude of the gravitational field strength g at the surface of Venus.

Give your answer to 3 significant figures.

=........ N kg^–1 g

Two identical space probes, A and B, land on a flat surface on Venus.

Probe A lands at the north pole. Probe B lands on the equator.

Each probe has mass 760 kg and volume 1.7 m³.

Calculate the centripetal acceleration a of probe B at the equator due to the rotation of Venus about its axis.

=........ m s^–2 a

The atmosphere exerts the same upthrust on each probe.

Using your answer to (a), calculate the upthrust acting on each probe.

Explain which probe will experience the greater normal contact force from the surface of Venus.

A long wooden cylinder is placed into a liquid and it floats as shown.

The length of the cylinder below the liquid level is 15 cm.

State Archimedes’ principle.

The pressure exerted by the liquid alone on the bottom of the cylinder is 1.9 × 10³ Pa. (ii)

Calculate the density ρ of the liquid.

=........ kg m^–3

The cylinder is pushed down into the liquid and then allowed to oscillate freely. The graph of displacement x against time t is shown below.

The cylinder oscillates with simple harmonic motion with frequency of 1.4 Hz.

Calculate the displacement, in cm, at time t = 0.60 s.

Calculate the maximum speed of the oscillating cylinder.

maximum speed =........ m s^–1

The cylinder is now pushed down further into the liquid before being released. As before, the cylinder oscillates with simple harmonic motion.

State the effect this has on

1 the amplitude

2 the period.

A toy rocket is made from a 1.5 litre plastic bottle with fins attached for stability.

The bottle initially contains 0.30 litres of water, leaving 1.2 litres of trapped air at a temperature of 17 °C. A pump is used to increase the pressure of the air within the plastic bottle to 2.4 × 10⁵ Pa at the start of lift-off.

Fig. 1.1 shows the rocket at the start of lift-off.

1 litre = 10^–3 m³

Calculate, in moles, the amount of trapped air in the bottle at the start of lift-off.

The trapped air pushes the water downwards out of the hole, causing the rocket to rise. The temperature of this air remains constant.

Calculate the final pressure of the trapped air just before all the water has been released.

Use the data above to show that the upwards force on the rocket at the start of lift-off is about 15 N.

Hence calculate the initial vertical acceleration of the rocket.

initial acceleration =........ m s^–2

Discuss whether adding more water initially would enable the rocket to reach a greater height.

A group of students are conducting an experiment in the laboratory to determine the value of absolute zero by heating a fixed mass of gas. The volume of the gas is kept constant. Fig. 17.1 shows the arrangement used by the students.

The gas is heated using a water bath. The temperature of the water is increased from 5 °C to 70 °C. The temperature of the water bath is assumed to be the same as the temperature of the gas. The pressure p of the gas is measured using a pressure gauge.

The results from the students are shown in a table.

/ °C	p / kPa
5 ± 1	224 ± 3
13 ± 1	231 ± 3
22 ± 1	238 ± 3
35 ± 1	248 ± 3
44 ± 1
53 ± 1	262 ± 3
62 ± 1	269 ± 3
70 ± 1	276 ± 3

Describe and explain how the students may have made accurate measurements of the temperature .

Fig. 17.2 shows the pressure gauge. Measurements of p can be made using the kPa scale or the psi (pounds per square inch) scale. The students used the psi scale to measure pressure and then converted the reading to pressure in kPa.

Suggest why it was sensible to use the psi scale to measure p.

Plot the missing data point and the error bars on Fig. 17.3.

* Explain what is meant by absolute zero. Describe how Fig. 17.3 can be used to determine the value of absolute zero. Determine the value of absolute zero. You may assume that the gas behaves as an ideal gas.

Describe, without doing any calculations, how you could use Fig. 17.3 to determine the actual uncertainty in the value of absolute zero in (c)(ii).

The experiment is repeated as the water bath quickly cools from 70 °C to 5 °C. Absolute zero was found to be −390 °C.

Compare this value with your value from (c)(ii) and explain why the values may differ. Describe an experimental approach that could be taken to avoid systematic error in the determination of absolute zero.

A swimming pool designer investigates the depth d below a water surface reached by a diver when diving from a height h above the water surface. The designer models the diver as a uniform wooden cylinder. The experimental arrangement is shown in Fig. 18.1.

Calculate the density of the wood.

density =........ kg m⁻³

Suggest why wood is an appropriate material to model the depth reached by a diver.

The cylinder is released from rest from a trapdoor. The base of the cylinder is at a height h = 0.30 m above the water surface. Calculate the speed of the cylinder just before the base hits the water. Ignore air resistance.

speed =........ m s⁻¹

Fig. 18.2 shows the cylinder fully submerged under the water surface before it has come to rest. The cylinder is moving vertically down.

Add arrows to Fig. 18.2 to show the three forces acting on the wooden cylinder. Label

the arrows.

Describe and explain how the resultant force on the wooden cylinder varies from the moment the cylinder is fully submerged until it reaches its deepest point.

The graph of Fig. 18.3 shows the depth d reached for different initial drop height h.

The designer is required to double the height of a diving board for an existing swimming pool. He suggests that the depth of the pool also needs to be doubled. Use Fig. 18.3 to explain whether you agree with this suggestion.