Density

State why thermal energy passes into the tank.

Explain, in terms of particle energies, why the temperature of the oxygen remains constant despite the transfer of thermal energy into the tank.

Show that in one hour the volume of liquid oxygen that evaporates is about 0.02 m³. (c)

After one hour the number of moles of gas above the remaining liquid oxygen is 710.

Calculate the pressure of the gas.

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.

A pulsar is a rapidly rotating neutron star that emits radio waves.

Describe the formation of a neutron star.

State one characteristic of a neutron star.

An astronomer uses a radio telescope to observe a pulsar.

The graph below shows the power that the telescope receives due to the radio waves from one full rotation of a pulsar.

By calculating the area between the curve and the horizontal axis, estimate the total energy received by the telescope in one full rotation of the pulsar.

The surface area of the telescope is about 3000 m². (ii)

The distance to the pulsar is about 300 pc.

By assuming that the radiation from the pulsar is emitted equally in all directions, estimate the total energy emitted in one full rotation.

Describe the origin of the microwave background radiation.

The figure below shows how the intensity of the microwave background radiation varies with its wavelength .

The peak intensity is at a wavelength of 1.1 mm.

This spectrum of microwave background radiation changes with temperature according to Wien’s displacement law.

Suggest and explain how the spectrum might have looked in the distant past. You may draw on the figure to support your answer.

Calculate the energy of a photon which has a wavelength of 1.1 mm.

Estimate the number of photons of microwave background radiation incident per second on the back of your hand.

Assume that all emitted photons have the energy calculated in (ii), and that the back of your hand has a surface area of 150 cm².

number of photons per second =........ s^–1

Use your answer in (i) to calculate the distance d of Proxima Centauri from the Earth in light-years (ly).

1 pc = 3.26 ly

Estimate the gravitational force between two neighbouring galaxies.

Show that the mean density of the Universe is about 7 × 10^–29 kg m^–3. (ii)

Suggest why the actual mean density of the Universe is different from the value calculated in (ii).

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 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.