HL Paper 2
A longitudinal wave travels in a medium with speed 340 m s−1. The graph shows the variation with time t of the displacement x of a particle P in the medium. Positive displacements on the graph correspond to displacements to the right for particle P.
Another wave travels in the medium. The graph shows the variation with time t of the displacement of each wave at the position of P.
A standing sound wave is established in a tube that is closed at one end and open at the other end. The period of the wave is . The diagram represents the standing wave at and at . The wavelength of the wave is 1.20 m. Positive displacements mean displacements to the right.
Calculate the wavelength of the wave.
Determine, for particle P, the magnitude and direction of the acceleration at t = 2.0 m s.
State the phase difference between the two waves.
Identify a time at which the displacement of P is zero.
Estimate the amplitude of the resultant wave.
Calculate the length of the tube.
A particle in the tube has its equilibrium position at the open end of the tube.
State and explain the direction of the velocity of this particle at time .
Draw on the diagram the standing wave at time .
A pipe is open at both ends. A first-harmonic standing wave is set up in the pipe. The diagram shows the variation of displacement of air molecules in the pipe with distance along the pipe at time t = 0. The frequency of the first harmonic is f.
A transmitter of electromagnetic waves is next to a long straight vertical wall that acts as a plane mirror to the waves. An observer on a boat detects the waves both directly and as an image from the other side of the wall. The diagram shows one ray from the transmitter reflected at the wall and the position of the image.
Sketch, on the diagram, the variation of displacement of the air molecules with distance along the pipe when t = .
An air molecule is situated at point X in the pipe at t = 0. Describe the motion of this air molecule during one complete cycle of the standing wave beginning from t = 0.
The speed of sound c for longitudinal waves in air is given by
where ρ is the density of the air and K is a constant.
A student measures f to be 120 Hz when the length of the pipe is 1.4 m. The density of the air in the pipe is 1.3 kg m–3. Determine the value of K for air. State your answer with the appropriate fundamental (SI) unit.
Demonstrate, using a second ray, that the image appears to come from the position indicated.
Outline why the observer detects a series of increases and decreases in the intensity of the received signal as the boat moves along the line XY.
A student is investigating a method to measure the mass of a wooden block by timing the period of its oscillations on a spring.
A 0.52 kg mass performs simple harmonic motion with a period of 0.86 s when attached to the spring. A wooden block attached to the same spring oscillates with a period of 0.74 s.
With the block stationary a longitudinal wave is made to travel through the original spring from left to right. The diagram shows the variation with distance x of the displacement y of the coils of the spring at an instant of time.
A point on the graph has been labelled that represents a point P on the spring.
Describe the conditions required for an object to perform simple harmonic motion (SHM).
Calculate the mass of the wooden block.
In carrying out the experiment the student displaced the block horizontally by 4.8 cm from the equilibrium position. Determine the total energy in the oscillation of the wooden block.
A second identical spring is placed in parallel and the experiment in (b) is repeated. Suggest how this change affects the fractional uncertainty in the mass of the block.
State the direction of motion of P on the spring.
Explain whether P is at the centre of a compression or the centre of a rarefaction.
Two loudspeakers, A and B, are driven in phase and with the same amplitude at a frequency of . Point P is located from A and from B. The speed of sound is .
In another experiment, loudspeaker A is stationary and emits sound with a frequency of . The microphone is moving directly away from the loudspeaker with a constant speed . The frequency of sound recorded by the microphone is .
Deduce that a minimum intensity of sound is heard at P.
A microphone moves along the line from P to Q. PQ is normal to the line midway between the loudspeakers.
The intensity of sound is detected by the microphone. Predict the variation of detected intensity as the microphone moves from P to Q.
When both loudspeakers are operating, the intensity of sound recorded at Q is . Loudspeaker B is now disconnected. Loudspeaker A continues to emit sound with unchanged amplitude and frequency. The intensity of sound recorded at Q changes to .
Estimate .
Explain why the frequency recorded by the microphone is lower than the frequency emitted by the loudspeaker.
Calculate .
There is a proposal to place a satellite in orbit around planet Mars.
The satellite is to have an orbital time T equal to the length of a day on Mars. It can be shown that
T2 = kR3
where R is the orbital radius of the satellite and k is a constant.
The ratio = 1.5.
Outline what is meant by gravitational field strength at a point.
Newton’s law of gravitation applies to point masses. Suggest why the law can be applied to a satellite orbiting Mars.
Mars has a mass of 6.4 × 1023 kg. Show that, for Mars, k is about 9 × 10–13 s2 m–3.
The time taken for Mars to revolve on its axis is 8.9 × 104 s. Calculate, in m s–1, the orbital speed of the satellite.
Show that the intensity of solar radiation at the orbit of Mars is about 600 W m–2.
Determine, in K, the mean surface temperature of Mars. Assume that Mars acts as a black body.
The atmosphere of Mars is composed mainly of carbon dioxide and has a pressure less than 1 % of that on the Earth. Outline why the mean temperature of Earth is strongly affected by gases in its atmosphere but that of Mars is not.
A beam of microwaves is incident normally on a pair of identical narrow slits S1 and S2.
When a microwave receiver is initially placed at W which is equidistant from the slits, a maximum in intensity is observed. The receiver is then moved towards Z along a line parallel to the slits. Intensity maxima are observed at X and Y with one minimum between them. W, X and Y are consecutive maxima.
Explain why intensity maxima are observed at X and Y.
The distance from S1 to Y is 1.243 m and the distance from S2 to Y is 1.181 m.
Determine the frequency of the microwaves.
Outline one reason why the maxima observed at W, X and Y will have different intensities from each other.
The microwaves emitted by the transmitter are horizontally polarized. The microwave receiver contains a polarizing filter. When the receiver is at position W it detects a maximum intensity.
The receiver is then rotated through 180° about the horizontal dotted line passing through the microwave transmitter. Sketch a graph on the axes provided to show the variation of received intensity with rotation angle.
The ball is now displaced through a small distance x from the bottom of the bowl and is then released from rest.
The magnitude of the force on the ball towards the equilibrium position is given by
where R is the radius of the bowl.
A small ball of mass m is moving in a horizontal circle on the inside surface of a frictionless hemispherical bowl.
The normal reaction force N makes an angle θ to the horizontal.
State the direction of the resultant force on the ball.
On the diagram, construct an arrow of the correct length to represent the weight of the ball.
Show that the magnitude of the net force F on the ball is given by the following equation.
The radius of the bowl is 8.0 m and θ = 22°. Determine the speed of the ball.
Outline whether this ball can move on a horizontal circular path of radius equal to the radius of the bowl.
Outline why the ball will perform simple harmonic oscillations about the equilibrium position.
Show that the period of oscillation of the ball is about 6 s.
The amplitude of oscillation is 0.12 m. On the axes, draw a graph to show the variation with time t of the velocity v of the ball during one period.
A second identical ball is placed at the bottom of the bowl and the first ball is displaced so that its height from the horizontal is equal to 8.0 m.
The first ball is released and eventually strikes the second ball. The two balls remain in contact. Determine, in m, the maximum height reached by the two balls.
A student investigates how light can be used to measure the speed of a toy train.
Light from a laser is incident on a double slit. The light from the slits is detected by a light sensor attached to the train.
The graph shows the variation with time of the output voltage from the light sensor as the train moves parallel to the slits. The output voltage is proportional to the intensity of light incident on the sensor.
As the train continues to move, the first diffraction minimum is observed when the light sensor is at a distance of 0.13 m from the centre of the fringe pattern.
A student investigates how light can be used to measure the speed of a toy train.
Light from a laser is incident on a double slit. The light from the slits is detected by a light sensor attached to the train.
The graph shows the variation with time of the output voltage from the light sensor as the train moves parallel to the slits. The output voltage is proportional to the intensity of light incident on the sensor.
Explain, with reference to the light passing through the slits, why a series of voltage peaks occurs.
The slits are separated by 1.5 mm and the laser light has a wavelength of 6.3 x 10–7 m. The slits are 5.0 m from the train track. Calculate the separation between two adjacent positions of the train when the output voltage is at a maximum.
Estimate the speed of the train.
Determine the width of one of the slits.
Suggest the variation in the output voltage from the light sensor that will be observed as the train moves beyond the first diffraction minimum.
In another experiment the student replaces the light sensor with a sound sensor. The train travels away from a loudspeaker that is emitting sound waves of constant amplitude and frequency towards a reflecting barrier.
The graph shows the variation with time of the output voltage from the sounds sensor.
Explain how this effect arises.
On a guitar, the strings played vibrate between two fixed points. The frequency of vibration is modified by changing the string length using a finger. The different strings have different wave speeds. When a string is plucked, a standing wave forms between the bridge and the finger.
The string is displaced 0.4 cm at point P to sound the guitar. Point P on the string vibrates with simple harmonic motion (shm) in its first harmonic with a frequency of 195 Hz. The sounding length of the string is 62 cm.
Outline how a standing wave is produced on the string.
Show that the speed of the wave on the string is about 240 m s−1.
Sketch a graph to show how the acceleration of point P varies with its displacement from the rest position.
Calculate, in m s−1, the maximum velocity of vibration of point P when it is vibrating with a frequency of 195 Hz.
Calculate, in terms of g, the maximum acceleration of P.
Estimate the displacement needed to double the energy of the string.
The string is made to vibrate in its third harmonic. State the distance between consecutive nodes.
A loudspeaker emits sound towards the open end of a pipe. The other end is closed. A standing wave is formed in the pipe. The diagram represents the displacement of molecules of air in the pipe at an instant of time.
X and Y represent the equilibrium positions of two air molecules in the pipe. The arrow represents the velocity of the molecule at Y.
The loudspeaker in (a) now emits sound towards an air–water boundary. A, B and C are parallel wavefronts emitted by the loudspeaker. The parts of wavefronts A and B in water are not shown. Wavefront C has not yet entered the water.
Outline how the standing wave is formed.
Draw an arrow on the diagram to represent the direction of motion of the molecule at X.
Label a position N that is a node of the standing wave.
The speed of sound is 340 m s–1 and the length of the pipe is 0.30 m. Calculate, in Hz, the frequency of the sound.
The speed of sound in air is 340 m s–1 and in water it is 1500 m s–1.
The wavefronts make an angle θ with the surface of the water. Determine the maximum angle, θmax, at which the sound can enter water. Give your answer to the correct number of significant figures.
Draw lines on the diagram to complete wavefronts A and B in water for θ < θmax.
A beam of coherent monochromatic light from a distant galaxy is used in an optics experiment on Earth.
The beam is incident normally on a double slit. The distance between the slits is 0.300 mm. A screen is at a distance D from the slits. The diffraction angle θ is labelled.
The graph of variation of intensity with diffraction angle for this experiment is shown.
A beam of coherent monochromatic light from a distant galaxy is used in an optics experiment on Earth.
The beam is incident normally on a double slit. The distance between the slits is 0.300 mm. A screen is at a distance D from the slits. The diffraction angle θ is labelled.
A series of dark and bright fringes appears on the screen. Explain how a dark fringe is formed.
Outline why the beam has to be coherent in order for the fringes to be visible.
The wavelength of the beam as observed on Earth is 633.0 nm. The separation between a dark and a bright fringe on the screen is 4.50 mm. Calculate D.
Calculate the angular separation between the central peak and the missing peak in the double-slit interference intensity pattern. State your answer to an appropriate number of significant figures.
Deduce, in mm, the width of one slit.
The wavelength of the light in the beam when emitted by the galaxy was 621.4 nm.
Explain, without further calculation, what can be deduced about the relative motion of the galaxy and the Earth.
The diagram shows the direction of a sound wave travelling in a metal sheet.
The sound wave in air in (c) enters a pipe that is open at both ends. The diagram shows the displacement, at a particular time T, of the standing wave that is set up in the pipe.
A particular air molecule has its equilibrium position at the point labelled M.
Sound of frequency f = 2500 Hz is emitted from an aircraft that moves with speed v = 280 m s–1 away from a stationary observer. The speed of sound in still air is c = 340 m s–1.
Particle P in the metal sheet performs simple harmonic oscillations. When the displacement of P is 3.2 μm the magnitude of its acceleration is 7.9 m s-2. Calculate the magnitude of the acceleration of P when its displacement is 2.3 μm.
The wave is incident at point Q on the metal–air boundary. The wave makes an angle of 54° with the normal at Q. The speed of sound in the metal is 6010 m s–1 and the speed of sound in air is 340 m s–1. Calculate the angle between the normal at Q and the direction of the wave in air.
The frequency of the sound wave in the metal is 250 Hz. Determine the wavelength of the wave in air.
On the diagram, at time T, draw an arrow to indicate the acceleration of this molecule.
On the diagram, at time T, label with the letter C a point in the pipe that is at the centre of a compression.
Calculate the frequency heard by the observer.
Calculate the wavelength measured by the observer.
Titan is a moon of Saturn. The Titan-Sun distance is 9.3 times greater than the Earth-Sun distance.
The molar mass of nitrogen is 28 g mol−1.
Show that the intensity of the solar radiation at the location of Titan is 16 W m−2.
Titan has an atmosphere of nitrogen. The albedo of the atmosphere is 0.22. The surface of Titan may be assumed to be a black body. Explain why the average intensity of solar radiation absorbed by the whole surface of Titan is 3.1 W m−2.
Show that the equilibrium surface temperature of Titan is about 90 K.
The mass of Titan is 0.025 times the mass of the Earth and its radius is 0.404 times the radius of the Earth. The escape speed from Earth is 11.2 km s−1. Show that the escape speed from Titan is 2.8 km s−1.
The orbital radius of Titan around Saturn is and the period of revolution is .
Show that where is the mass of Saturn.
The orbital radius of Titan around Saturn is 1.2 × 109 m and the orbital period is 15.9 days. Estimate the mass of Saturn.
Show that the mass of a nitrogen molecule is 4.7 × 10−26 kg.
Estimate the root mean square speed of nitrogen molecules in the Titan atmosphere. Assume an atmosphere temperature of 90 K.
Discuss, by reference to the answer in (b), whether it is likely that Titan will lose its atmosphere of nitrogen.
A vertical solid cylinder of uniform cross-sectional area floats in water. The cylinder is partially submerged. When the cylinder floats at rest, a mark is aligned with the water surface. The cylinder is pushed vertically downwards so that the mark is a distance below the water surface.
At time the cylinder is released. The resultant vertical force on the cylinder is related to the displacement of the mark by
where is the density of water.
The cylinder was initially pushed down a distance .
Outline why the cylinder performs simple harmonic motion when released.
The mass of the cylinder is and the cross-sectional area of the cylinder is . The density of water is . Show that the angular frequency of oscillation of the cylinder is about .
Determine the maximum kinetic energy of the cylinder.
Draw, on the axes, the graph to show how the kinetic energy of the cylinder varies with time during one period of oscillation .
Two loudspeakers A and B are initially equidistant from a microphone M. The frequency and intensity emitted by A and B are the same. A and B emit sound in phase. A is fixed in position.
B is moved slowly away from M along the line MP. The graph shows the variation with distance travelled by B of the received intensity at M.
Explain why the received intensity varies between maximum and minimum values.
State and explain the wavelength of the sound measured at M.
B is placed at the first minimum. The frequency is then changed until the received intensity is again at a maximum.
Show that the lowest frequency at which the intensity maximum can occur is about 3 kHz.
Speed of sound = 340 m s−1
Loudspeaker A is switched off. Loudspeaker B moves away from M at a speed of 1.5 m s−1 while emitting a frequency of 3.0 kHz.
Determine the difference between the frequency detected at M and that emitted by B.
Monochromatic coherent light is incident on two parallel slits of negligible width a distance d apart. A screen is placed a distance D from the slits. Point M is directly opposite the midpoint of the slits.
Initially the lower slit is covered and the intensity of light at M due to the upper slit alone is 22 W m-2. The lower slit is now uncovered.
The width of each slit is increased to 0.030 mm. D, d and λ remain the same.
Deduce, in W m-2, the intensity at M.
P is the first maximum of intensity on one side of M. The following data are available.
d = 0.12 mm
D = 1.5 m
Distance MP = 7.0 mm
Calculate, in nm, the wavelength λ of the light.
Suggest why, after this change, the intensity at P will be less than that at M.
Show that, due to single slit diffraction, the intensity at a point on the screen a distance of 28 mm from M is zero.
A buoy, floating in a vertical tube, generates energy from the movement of water waves on the surface of the sea. When the buoy moves up, a cable turns a generator on the sea bed producing power. When the buoy moves down, the cable is wound in by a mechanism in the generator and no power is produced.
The motion of the buoy can be assumed to be simple harmonic.
Water can be used in other ways to generate energy.
Outline the conditions necessary for simple harmonic motion (SHM) to occur.
A wave of amplitude 4.3 m and wavelength 35 m, moves with a speed of 3.4 m s–1. Calculate the maximum vertical speed of the buoy.
Sketch a graph to show the variation with time of the generator output power. Label the time axis with a suitable scale.
Outline, with reference to energy changes, the operation of a pumped storage hydroelectric system.
The water in a particular pumped storage hydroelectric system falls a vertical distance of 270 m to the turbines. Calculate the speed at which water arrives at the turbines. Assume that there is no energy loss in the system.
The hydroelectric system has four 250 MW generators. Determine the maximum time for which the hydroelectric system can maintain full output when a mass of 1.5 x 1010 kg of water passes through the turbines.
Not all the stored energy can be retrieved because of energy losses in the system. Explain two such losses.
An experiment to investigate simple harmonic motion consists of a mass oscillating at the end of a vertical spring.
The mass oscillates vertically above a motion sensor that measures the speed of the mass. Test 1 is carried out with a 1.0 kg mass and spring of spring constant . Test 2 is a repeat of the experiment with a 4.0 kg mass and spring of spring constant .
The variation with time of the vertical speed of the masses, for one cycle of the oscillation, is shown for each test.
Calculate the frequency of the oscillation for both tests.
Deduce .
Determine the amplitude of oscillation for test 1.
In test 2, the maximum elastic potential energy Ep stored in the spring is 44 J.
When t = 0 the value of Ep for test 2 is zero.
Sketch, on the axes, the variation with time of Ep for test 2.
The motion sensor operates by detecting the sound waves reflected from the base of the mass. The sensor compares the frequency detected with the frequency emitted when the signal returns.
The sound frequency emitted by the sensor is 35 kHz. The speed of sound is 340 m s−1.
Determine the maximum frequency change detected by the sensor for test 2.