Density

The asteroid Bennu is in an approximately circular orbit around the Sun. Scientists consider that Bennu may collide with Earth within the next century.

In 2060 Bennu will be 7.49×10⁸m from the Earth.

Show that the mass of Bennu is about 7.810¹⁰kg.

mean radius of Bennu = 245 m average density of Bennu = 1260 kg m^–3

Calculate the magnitude of the force F between Bennu and Earth in 2060.

mass of Earth = 5.9810²⁴kg

The orbital time of Bennu around the Sun is 1.17 years.

Calculate the mass of the Sun.

average distance of Bennu from the Sun = 1.7010¹¹m

Show that the mass of the ice sculpture is about 2.210⁵kg.

density of ice = 920 kg m^–3

Give two reasons why your analysis of the situation may be incorrect.

A student has 10 identical flat metal discs. One of the discs is shown. Each disc has a diameter d of about 5 cm and a thickness t of about 2 mm.

The student determined the density of the metal the discs are made from.

She used a half‑metre rule to measure d for one of the discs. She used vernier calipers to measure t for one of the discs.

The student then repeated this process for two more discs and calculated average values for d and t of the discs.

Describe improvements to the student’s method that would give more accurate values for d and t of the discs.

The student determined the average mass of a disc to be 42.5 g ± 0.5 g.

Calculate the density of the material the discs are made from and the percentage uncertainty in this value.

average diameter of disc = 5.2 cm ± 0.1 cm average thickness of disc = 1.9 mm ± 0.1 mm

The potential difference (p.d.) from a power pack alternates at a frequency of 50 Hz. The root-mean-square (r.m.s.) value of the output p.d. is measured to be 8.5 V using an a.c. voltmeter.

The output from the power pack is connected to a resistor of resistance 1.5 Ω.

Show that the rate of power dissipation in the resistor is about 48 W.

A lab technician used the resistor to heat 1.25 10^–4m³ of milk in a beaker.

Calculate the time taken for the milk to increase in temperature by 12 °C.

specific heat capacity of milk = 3930 J kg^–1K^–1

The lab technician used an oscilloscope to determine the supply frequency and check the r.m.s. output p.d. displayed by the voltmeter.

Describe the methods the technician should use.

A hot air balloon consists of a fabric envelope, heaters and a basket, as shown.

When the balloon is set up, the envelope is partly filled with air at 20°C. The air is then heated to 120°C and expands to fill the envelope and becomes less dense.

The air pressure inside the envelope is always equal to the air pressure outside the envelope because the envelope is open at the bottom.

The balloon takes off when the upthrust is more than the total weight of the balloon, the air in the envelope and the passengers.

Deduce whether the balloon can take off.

volume of air at 120°C in inflated envelope = 2800 m³ density of air at 20°C = 1.2 kg m^–3 mass of balloon = 380 kg mass of passengers = 340 kg upthrust when the envelope is full = 33 000 N

State one assumption of the kinetic theory of gases.

Derive an equation to show that, for a gas at temperature T, the mean kinetic energy of the molecules = kT

Calculate the root-mean-square speed of nitrogen molecules at a temperature of 120°C.

mass of nitrogen molecule = 28 u

A student made measurements to determine if some gold coins were made from pure gold. The coins that were available to the student are shown below.

The student used digital calipers to measure the thickness t and the diameter d of one of the coins.

Calculate the volume V of the coin, and the percentage uncertainty in V.

t = 1.54 mm d = 22.16 mm

The student measured the mass of the coin using an electronic balance. The balance had a resolution of 0.1 g.

Assess whether the coin could be made from pure gold.

density of pure gold = 1.93×10⁴kg m^–3 mass of coin = 11.2 g

The student’s experimental method could have been improved.

Explain two changes the student could have made to the experimental method.

A student investigated the behaviour of small, positively charged oil drops in an electric field.

He introduced an oil drop between two horizontal metal plates. A potential difference V was applied to bring the oil drop to rest as shown.

Add to the diagram to show the polarity of the power supply when the oil drop is at rest.

The oil drop was at rest when V = 4870 V.

The student expected the charge on the oil drop to be a whole number multiple of the charge on an electron.

Deduce whether this is confirmed by the experimental data.

distance between top plate and bottom plate = 1.55 cm

The student determined the terminal velocity of the oil drop in order to obtain a value for the weight of the oil drop.

He disconnected the power supply and the oil drop fell downwards. He viewed the oil drop through a microscope.

Explain why the student should wait a short while before starting to take measurements.

The view through the microscope is shown below. The scale is in millimetres.

The position of the oil drop at the start and end of a 120 s time interval is indicated by the black dots.

Determine the terminal velocity of the oil drop.

The student repeated the measurements on fifty oil drops. For each drop the student calculated the charge on the oil drop.

His results are shown in the bar chart.

The student predicted that the charge on each oil drop would be a whole number multiple of the charge on an electron.

Comment on the extent to which the bar chart supports the student’s prediction.

The photograph shows some metal washers. A student carried out an experiment to determine the density of the metal the washers are made from.

Each washer has a diameter d₁ of about 4.5 cm. The internal diameter d₂ of each washer is about 2.5 cm. Each washer has a thickness t of about 4 mm.

The student used a half metre rule to make measurements of a washer.

Comment on the student’s choice of measuring instrument.

The student measured t for each of the five washers and then calculated a mean value.

Explain how the student could modify her method to obtain a more accurate mean value for t.

Show that the percentage uncertainty in her value for the area of a washer is about 3%.

The student obtained the following values of t for each of the five washers.

The table shows the density of iron and steel.

Deduce whether the washers are made from iron or steel.

mean mass of a washer = 32.0 g

The uncertainty in the mass is negligible.

The harp is a musical instrument with many strings, as shown.

All the strings are under tension.

The strings on one type of harp are made from nylon of density 1070 kg m⁻³. One string has a diameter of 1.14 mm.

Show that the mass per unit length μ of the string is about 1.1 10⁻³kg m⁻¹.

When the middle of the string is plucked, a note of frequency 440 Hz is produced.

Calculate the tension in the string.

length of string = 41.0 cm

The graph shows how the Young modulus E of the nylon varies with temperature.

When the harp is played, the temperature of the string increases.

Explain how this temperature change would affect the frequency of the note produced when the string is plucked.

A student used a sonometer to investigate the properties of a stretched wire. The sonometer is a long hollow wooden box.

A steel wire is attached to one end of the box and rests on two wooden bridges. The wire is placed under tension T by hanging a mass from the end of the wire, as shown.

The student placed the base of a vibrating tuning fork in contact with the wire, at one of the bridges. This set the wire into oscillation. He adjusted the position of the other bridge until a single-loop standing wave was produced on the wire between the bridges.

Explain how an antinode is produced at the mid-point of the wire between the bridges.

The student repeated this for a series of tuning forks with different frequencies f. For each fork he measured the distance L between the bridges.

The steel wire, of mass per unit length μ, was placed under tension T by hanging a mass of 2.10 kg from the end of the wire.

State one safety precaution that should be taken when carrying out the investigation.

The student plotted a graph of L² against 1/f². Show that the gradient of this graph is equal to

The student then found a value of μ for a brass wire, using a different method.

He measured the diameter d of the wire using a micrometer.

Explain one technique the student should use when measuring d.

The student obtained the following data.

The stated value of μ for the brass wire used by the student was

Deduce whether the student’s data supports this value for μ.

density of brass = 8700 kg m⁻³ ± 200 kg m⁻³

Genuine crystal balls are made from clarified quartz rather than glass. A student was given a small crystal ball and wanted to know whether it was genuine.

The mean diameter of the crystal ball was measured to be 5.06 cm and the mass of the crystal ball was measured to be 175 g.

Show that the density of the material of the crystal ball is about 2600 kg m⁻³.

The student measured the diameter of the crystal ball using vernier calipers with a resolution of 0.01 cm. She measured the mass of the crystal ball using a balance with a resolution of 1 g.

The table gives the densities of clarified quartz and glass.

Determine whether the crystal ball was genuine.

A spring is made from loops of thick steel wire as shown.

There are two extra loops, one on each end of the spring.

A student determined the length of steel used to make the spring by using vernier calipers to measure the width w of the spring. The length of wire l on each loop is given by l = πw

The student obtained the following values for w.

Calculate l.

Estimate the percentage uncertainty in your value for l.

Calculate the total length L of wire used to make the spring.

The student measured the diameter d of the steel wire and obtained a value of 2.52 mm.

Explain which instrument he used to measure the diameter.

Estimate the percentage uncertainty in the student’s value for d.

The student used a balance to measure the mass m of the spring. He obtained a value of 32.0 ± 0.5 g.

Estimate the percentage uncertainty in the mass of the spring.

Determine whether the data collected leads to a value for the density of steel in agreement with the standard value.

density of steel = 7 800 kg m⁻³

A spring is made from loops of thick steel wire as shown.

There are two extra loops, one on each end of the spring.

A student determined the length of steel used to make the spring by using vernier calipers to measure the width w of the spring. The length of wire l on each loop is given by l = πw

The student obtained the following values for w.

Calculate l.

Estimate the percentage uncertainty in your value for l.

Calculate the total length L of wire used to make the spring.

The student measured the diameter d of the steel wire and obtained a value of 2.52 mm.

Explain which instrument he used to measure the diameter.

Estimate the percentage uncertainty in the student’s value for d.

The student used a balance to measure the mass m of the spring. He obtained a value of 32.0 ± 0.5 g.

Estimate the percentage uncertainty in the mass of the spring.

Determine whether the data collected leads to a value for the density of steel in agreement with the standard value.

density of steel = 7 800 kg m⁻³

Impact craters are formed when meteorites strike the surface of a planet. A student investigated some factors that might influence the formation of impact craters. He did this by dropping spheres of modelling clay into a tray of sand.

The diameter of the crater produced by each sphere was measured using vernier calipers as shown.

This process was repeated for spheres of different diameters.

In one test, the spheres were dropped from the same height.

Determine the factor by which the kinetic energy of the sphere just before impact increases when the sphere diameter is increased from 2.0 cm to 4.0 cm.

State a suitable measuring instrument that could be used with a single disc to measure t.

A balance which can measure mass with a resolution of 0.2 g is available.

Determine the minimum number of discs that should be placed on the balance together if the percentage uncertainty in the measurement of the mass is to be less than 0.5%.

density of steel = 7900 kg m^–3

The measured uncertainty in d is ±0.1 mm and the measured uncertainty for t is ±0.05 mm.

Determine the percentage uncertainty in the calculated volume of the disc.

A student is investigating a ‘Cartesian diver’.

The diver is made from a plastic pipette. When placed in a plastic bottle full of water the diver rises to the top and touches the lid.

Show that the downward force of the lid on the diver is about 0.02 N.

volume of diver = 8.010^–6m³

mass of diver = 0.0059 kg

density of water = 1.010³kg m^–3

When the pressure is increased by squeezing the bottle, water is forced into the diver increasing its weight. The diver sinks, then remains at rest in the position shown.

The volume of air in the empty pipette in part (a) was 8.010^–6m³.

Show that the volume now occupied by the air is about 610^–6m³.

The pressure of the air in the empty pipette in part (a) was 1.0110⁵Pa.

Calculate the pressure of the air in part (b).

A student carries out measurements to determine the density of glass. The student has 20 glass microscope slides available.

The approximate dimensions of one slide are shown.

The density is calculated using the equation

Describe how the student can determine an accurate value for the density of the glass. Your answer should include the measuring instruments required.

State one precaution that the student should take to ensure the measurements are accurate.