Properties of waves

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 transmitter produces microwaves with a frequency of 10.7 GHz. The microwaves are directed towards a metal plate. A microwave receiver R is placed between the transmitter and the metal plate, as shown.

When R is moved along the line AB, the intensity of microwaves received is observed to increase from minimum to maximum a number of times. The distance between successive minima is about 1.5 cm.

Explain these observations. Your answer should include a calculation.

A student wound a piece of string around the head of an electric toothbrush. The student attached a small mass to the other end of the string, as shown in Photograph 1.

The toothbrush was switched on and the head started to vibrate. The student rotated the toothbrush slowly to unwind the string. At a particular length, large vibrations were observed on the string. The string formed a loop, as shown in Photograph 2.

Explain how the vibrations of the toothbrush head caused the loop to form.

The student continued to unwind the string and the loop disappeared. When the length of the unwound string was twice that shown in Photograph 2 two loops were seen. Three loops were seen when the unwound length was three times that shown in Photograph 2 and so on.

Determine the frequency of vibration of the toothbrush head.

unwound length of string with 4 loops = 0.69 m mass on string = 0.010 kg mass per unit length of string = 9.1 × 10^–4kg m^–1

Rosslyn Chapel is a 15th century chapel in Scotland. Inside the chapel, small sandstone cubes protrude from a number of arches. It has been suggested that carvings on these cubes bear a resemblance to standing wave patterns that can be produced on a vibrating metal plate.

A metal plate is made to vibrate and sand is scattered evenly across its surface. At a certain frequency the sand moves to produce the standing wave pattern shown below.

(Source: https://skullsinthestars.com/2013/05/02/physics-demonstrations-chladni-patterns/#jp-carousel-7353)

Explain why the sand moves to different positions when a standing wave is formed on the plate.

The speed of sound in air can be determined by creating a standing wave in a column of air. The diagram shows a tuning fork just above the open end of a tube.

The tuning fork produces a sound wave of known frequency f. Several tuning forks are available, each with a different frequency.

A student adjusted the length L of the air column. A loud sound was heard when a standing wave was produced. A node was formed at the water surface, and an antinode was formed at the open end of the tube.

The student used values of L and f to determine a value for the speed of sound.

Describe a graphical method that the student could have used to determine a value for the speed of sound.

The photograph shows a stringed instrument called a cello being played with a bow.

(Source: © Vadim Ponomarenko/Alamy Stock Photo)

A standing wave forms on a cello string when the bow moves across the string.

Deduce whether a thicker string will produce a note of higher or lower frequency compared with a thinner string.

Assume each string is the same length and at the same tension.

A simple astronomical refracting telescope consists of two converging lenses. Light from a star is brought to a focus by the objective lens and then viewed through an eyepiece lens as shown.

In the arrangement shown, the final image is formed at infinity.

Explain why the separation of the objective and eyepiece lenses is equal to the sum of their focal lengths.

State why the final image is inverted.

Glass lenses used for optical instruments often have an anti-reflective coating. The coating is a thin layer of a transparent substance with refractive index n_c. Light is reflected from the coating surface and from the lens surface as shown. The reflected light interferes destructively.

When a single-layer coating is used, the coating thickness is chosen to eliminate reflections for green light, which is in the middle of the visible spectrum.

Calculate the minimum thickness d of the coating required for the reflection of green light to be eliminated.

frequency of green light = 6.00 10¹⁴Hz

State why white light reflected from coated lenses is seen as purple.

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.

In an experiment to determine the speed of sound in air a student connected two microphones to an oscilloscope, as shown.

The microphones detect sound from the loudspeaker, converting it to an electrical signal. The signal is displayed on the oscilloscope screen.

Both microphones were initially positioned the same distance from the loudspeaker. The two signals were in phase on the oscilloscope screen. The student slowly moved microphone 2 towards the loudspeaker, until the two signals on the oscilloscope were in phase again. He then measured the distance d between the microphones to determine the wavelength λ of the sound waves.

d = 20.5 cm

Comment on the student’s experimental technique to determine λ.

The oscilloscope trace for the signal from microphone 1 is shown below.

The time base of the oscilloscope was set to 0.20 ms div⁻¹.

Determine a value for the speed of sound in air.

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

The photograph shows a guitar.

When a guitar string is plucked, a standing wave is created.

Explain how a standing wave is created on the string.

The diagram shows a standing wave on a guitar string.

The oscillating length of the guitar string is 66 cm.

State the wavelength for this standing wave.

Calculate the frequency of vibration for this standing wave.

tension in guitar string = 88.6 N

mass per unit length of guitar string = 4.47 10⁻³kg m⁻¹

One end of the guitar string is wrapped around a cylindrical tuning peg. Turning the peg changes the total length of the string and hence changes the tension in the string. This changes the frequency of vibration of the string.

The length of one string is 68 cm.

Calculate the extension required to produce a tension of 93.4 N in the string.

Young modulus of string material = 1.8 × 10⁹N m⁻²

cross-sectional area of string = 6.6 10⁻⁷m²

The vibrating length of string is unchanged by turning the tuning peg.

Explain the effect that tightening the string has on the frequency of the sound produced.

The photograph shows a guitar.

When a guitar string is plucked, a standing wave is created.

Explain how a standing wave is created on the string.

The diagram shows a standing wave on a guitar string.

The oscillating length of the guitar string is 66 cm.

State the wavelength for this standing wave.

Calculate the frequency of vibration for this standing wave.

tension in guitar string = 88.6 N

mass per unit length of guitar string = 4.47 10⁻³kg m⁻¹

One end of the guitar string is wrapped around a cylindrical tuning peg. Turning the peg changes the total length of the string and hence changes the tension in the string. This changes the frequency of vibration of the string.

The length of one string is 68 cm.

Calculate the extension required to produce a tension of 93.4 N in the string.

Young modulus of string material = 1.8 × 10⁹N m⁻²

cross-sectional area of string = 6.6 10⁻⁷m²

The vibrating length of string is unchanged by turning the tuning peg.

Explain the effect that tightening the string has on the frequency of the sound produced.

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

Solar panels consisting of combinations of photovoltaic cells use energy in the radiation received from the Sun to generate electricity.

An advertisement for solar panels claims that the intensity of radiation from the Sun incident at the top of the Earth’s atmosphere is more than 2 kW m⁻².

Assess the validity of this claim.

radius of Sun = 6.96 × 10⁸m

surface temperature of Sun = 5790 K

distance from Sun to Earth = 1.50 × 10¹¹m

The average intensity of radiation from the Sun incident at the Earth’s surface over a 24-hour period has been determined to be 164 W m⁻².

The average intensity of radiation from the Sun at the Earth’s surface is much less than the intensity incident at the top of the Earth’s atmosphere.

Explain why.

It is claimed that the area of solar panels needed to generate 100 GW of power is about 0.5% of the surface area of the Earth.

Assess the validity of this claim.

radius of Earth = 6.4 × 10⁶m

typical efficiency of solar panels = 25%

Scientists are developing a space station equipped with large solar panels. The space station would be located in a geostationary orbit. The space station would transfer energy to Earth as microwaves.

A space station in a geostationary orbit is above the equator and has a period of 24 hours.

Explain one advantage of locating the space station in a geostationary orbit.

Calculate the height h of the space station above the equator when it is in a geostationary orbit.

mass of Earth = 6.00 × 10²⁴kg

24 hours = 8.64 10⁴s

Solar panels consisting of combinations of photovoltaic cells use energy in the radiation received from the Sun to generate electricity.

An advertisement for solar panels claims that the intensity of radiation from the Sun incident at the top of the Earth’s atmosphere is more than 2 kW m⁻².

Assess the validity of this claim.

radius of Sun = 6.96 × 10⁸m

surface temperature of Sun = 5790 K

distance from Sun to Earth = 1.50 × 10¹¹m

The average intensity of radiation from the Sun incident at the Earth’s surface over a 24-hour period has been determined to be 164 W m⁻².

The average intensity of radiation from the Sun at the Earth’s surface is much less than the intensity incident at the top of the Earth’s atmosphere.

Explain why.

It is claimed that the area of solar panels needed to generate 100 GW of power is about 0.5% of the surface area of the Earth.

Assess the validity of this claim.

radius of Earth = 6.4 × 10⁶m

typical efficiency of solar panels = 25%

Scientists are developing a space station equipped with large solar panels. The space station would be located in a geostationary orbit. The space station would transfer energy to Earth as microwaves.

A space station in a geostationary orbit is above the equator and has a period of 24 hours.

Explain one advantage of locating the space station in a geostationary orbit.

Calculate the height h of the space station above the equator when it is in a geostationary orbit.

mass of Earth = 6.00 × 10²⁴kg

24 hours = 8.64 10⁴s

The mouse repeller produces ultrasound of frequency 26.0 kHz.

A converging lens can be used to produce a real image on a screen.

A converging lens of focal length 15.0 cm is used to project an image of an illuminated object onto a screen. The object is a circle of diameter 4.0 mm and the image must be as large as possible on a screen of size 0.75 m by 1.25 m.

Calculate the distance between the lens and the screen for this image to be displayed.

A magazine article includes the statement:

If the distance from the lens to the screen is doubled, the brightness of the image is halved.

Assess the validity of this statement.

In 2016 scientists at the Laser Interferometer Gravitational‑Wave Observatory (LIGO) announced that gravitational waves had been detected.

The signal they detected is shown on the graph.

Gravitational waves travel at the speed of light.

Determine the mean wavelength of the waves detected between 0.30 s and 0.35 s on the graph.

Gravitational waves alternately compress and stretch matter by very small amounts as they pass through.

The LIGO detector has two arms, at 90° to each other, each 4 km long. As a gravitational wave passes the detector, the arms change length. The detector continuously compares the lengths of the two arms.

An article states that ‘the maximum change in the 4 km length of the arm is about 0.001 times the diameter of a proton’.

Determine whether this statement applies to the gravitational wave shown in the graph.

diameter of proton = 8.810^–16m

In the LIGO detector, any change in the lengths of the arms is detected using a laser beam and photodetector.

The laser beam is split into two at B, one beam travelling to one mirror and the other beam travelling to the other mirror. After reflection at the mirrors, the beams are recombined at B and reach the photodetector. The photodetector measures the intensity of the incident light.

The system is arranged so that when no gravitational waves are present, the beams have a path difference of half a wavelength at the photodetector.

Explain how the photodetector detects very small changes in the length of one arm, when the other arm stays the same length.

The system could be arranged so that when no gravitational waves are present, the beams have zero path difference at the photodetector.

Explain whether using an initial path difference of half a wavelength is a more sensitive way of detecting changes in length than having an initial path difference of zero.