HL Paper 2
Two identical positive point charges X and Y are placed 0.30 m apart on a horizontal line. O is the point midway between X and Y. The charge on X and the charge on Y is +4.0 µC.
A positive charge Z is released from rest 0.010 m from O on the line between X and Y. Z then begins to oscillate about point O.
Calculate the electric potential at O.
Sketch, on the axes, the variation of the electric potential V with distance between X and Y.
Identify the direction of the resultant force acting on Z as it oscillates.
Deduce whether the motion of Z is simple harmonic.
The graph shows the variation with diffraction angle of the intensity of light after it has passed through four parallel slits.
The number of slits is increased but their separation and width stay the same. All slits are illuminated.
State what is meant by the Doppler effect.
A plate performs simple harmonic oscillations with a frequency of 39 Hz and an amplitude of 8.0 cm.
Show that the maximum speed of the oscillating plate is about 20 m s−1.
Sound of frequency 2400 Hz is emitted from a stationary source towards the oscillating plate in (b). The speed of sound is 340 m s−1.
Determine the maximum frequency of the sound that is received back at the source after reflection at the plate.
State what will happen to the angular position of the primary maxima.
State what will happen to the width of the primary maxima.
State what will happen to the intensity of the secondary maxima.
Police use radar to detect speeding cars. A police officer stands at the side of the road and points a radar device at an approaching car. The device emits microwaves which reflect off the car and return to the device. A change in frequency between the emitted and received microwaves is measured at the radar device.
The frequency change Δf is given by
where f is the transmitter frequency, v is the speed of the car and c is the wave speed.
The following data are available.
Transmitter frequency f = 40 GHz Δf = 9.5 kHz Maximum speed allowed = 28 m s–1
(i) Explain the reason for the frequency change.
(ii) Suggest why there is a factor of 2 in the frequency-change equation.
(iii) Determine whether the speed of the car is below the maximum speed allowed.
Airports use radar to track the position of aircraft. The waves are reflected from the aircraft and detected by a large circular receiver. The receiver must be able to resolve the radar images of two aircraft flying close to each other.
The following data are available.
Diameter of circular radar receiver = 9.3 m Wavelength of radar = 2.5 cm Distance of two aircraft from the airport = 31 km
Calculate the minimum distance between the two aircraft when their images can just be resolved.
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.
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.
Yellow light from a sodium lamp of wavelength 590 nm is incident at normal incidence on a double slit. The resulting interference pattern is observed on a screen. The intensity of the pattern on the screen is shown.
The double slit is replaced by a diffraction grating that has 600 lines per millimetre. The resulting pattern on the screen is shown.
Explain why zero intensity is observed at position A.
The distance from the centre of the pattern to A is 4.1 x 10–2 m. The distance from the screen to the slits is 7.0 m.
Calculate the width of each slit.
Calculate the separation of the two slits.
State and explain the differences between the pattern on the screen due to the grating and the pattern due to the double slit.
The yellow light is made from two very similar wavelengths that produce two lines in the spectrum of sodium. The wavelengths are 588.995 nm and 589.592 nm. These two lines can just be resolved in the second-order spectrum of this diffraction grating. Determine the beam width of the light incident on the diffraction grating.
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 .
There is a proposal to power a space satellite X as it orbits the Earth. In this model, X is connected by an electronically-conducting cable to another smaller satellite Y.
Satellite Y orbits closer to the centre of Earth than satellite X. Outline why
The cable acts as a spring. Satellite Y has a mass m of 3.5 x 102 kg. Under certain circumstances, satellite Y will perform simple harmonic motion (SHM) with a period T of 5.2 s.
Satellite X orbits 6600 km from the centre of the Earth.
Mass of the Earth = 6.0 x 1024 kg
Show that the orbital speed of satellite X is about 8 km s–1.
the orbital times for X and Y are different.
satellite Y requires a propulsion system.
The cable between the satellites cuts the magnetic field lines of the Earth at right angles.
Explain why satellite X becomes positively charged.
Satellite X must release ions into the space between the satellites. Explain why the current in the cable will become zero unless there is a method for transferring charge from X to Y.
The magnetic field strength of the Earth is 31 μT at the orbital radius of the satellites. The cable is 15 km in length. Calculate the emf induced in the cable.
Estimate the value of k in the following expression.
T =
Give an appropriate unit for your answer. Ignore the mass of the cable and any oscillation of satellite X.
Describe the energy changes in the satellite Y-cable system during one cycle of the oscillation.
A painting is protected behind a transparent glass sheet of refractive index nglass. A coating of thickness w is added to the glass sheet to reduce reflection. The refractive index of the coating ncoating is such that nglass > ncoating > 1.
The diagram illustrates rays normally incident on the coating. Incident angles on the diagram are drawn away from the normal for clarity.
State the phase change when a ray is reflected at B.
Explain the condition for w that eliminates reflection for a particular light wavelength in air .
State the Rayleigh criterion for resolution.
The painting contains a pattern of red dots with a spacing of 3 mm. Assume the wavelength of red light is 700 nm. The average diameter of the pupil of a human eye is 4 mm. Calculate the maximum possible distance at which these red dots are distinguished.
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.
Monochromatic light of wavelength λ is normally incident on a diffraction grating. The diagram shows adjacent slits of the diffraction grating labelled V, W and X. Light waves are diffracted through an angle θ to form a second-order diffraction maximum. Points Z and Y are labelled.
State the effect on the graph of the variation of sin θ with n of:
State the phase difference between the waves at V and Y.
State, in terms of λ, the path length between points X and Z.
The separation of adjacent slits is d. Show that for the second-order diffraction maximum .
Monochromatic light of wavelength 633 nm is normally incident on a diffraction grating. The diffraction maxima incident on a screen are detected and their angle θ to the central beam is determined. The graph shows the variation of sinθ with the order n of the maximum. The central order corresponds to n = 0.
Determine a mean value for the number of slits per millimetre of the grating.
using a light source with a smaller wavelength.
increasing the distance between the diffraction grating and the screen.
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.
An elastic climbing rope is tested by fixing one end of the rope to the top of a crane. The other end of the rope is connected to a block which is initially at position A. The block is released from rest. The mass of the rope is negligible.
The unextended length of the rope is 60.0 m. From position A to position B, the block falls freely.
In another test, the block hangs in equilibrium at the end of the same elastic rope. The elastic constant of the rope is 400 Nm–1. The block is pulled 3.50 m vertically below the equilibrium position and is then released from rest.
An elastic climbing rope is tested by fixing one end of the rope to the top of a crane. The other end of the rope is connected to a block which is initially at position A. The block is released from rest. The mass of the rope is negligible.
The unextended length of the rope is 60.0 m. From position A to position B, the block falls freely.
At position C the speed of the block reaches zero. The time taken for the block to fall between B and C is 0.759 s. The mass of the block is 80.0 kg.
For the rope and block, describe the energy changes that take place
At position B the rope starts to extend. Calculate the speed of the block at position B.
Determine the magnitude of the average resultant force acting on the block between B and C.
Sketch on the diagram the average resultant force acting on the block between B and C. The arrow on the diagram represents the weight of the block.
Calculate the magnitude of the average force exerted by the rope on the block between B and C.
between A and B.
between B and C.
The length reached by the rope at C is 77.4 m. Suggest how energy considerations could be used to determine the elastic constant of the rope.
Calculate the time taken for the block to return to the equilibrium position for the first time.
Calculate the speed of the block as it passes the equilibrium position.
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.
Monochromatic light from two identical lamps arrives on a screen.
The intensity of light on the screen from each lamp separately is I0.
On the axes, sketch a graph to show the variation with distance x on the screen of the intensity I of light on the screen.
Monochromatic light from a single source is incident on two thin, parallel slits.
The following data are available.
The intensity I of light at the screen from each slit separately is I0. Sketch, on the axes, a graph to show the variation with distance x on the screen of the intensity of light on the screen for this arrangement.
The slit separation is increased. Outline one change observed on the screen.
Two equal positive fixed point charges Q = +44 μC and point P are at the vertices of an equilateral triangle of side 0.48 m.
Point P is now moved closer to the charges.
A point charge q = −2.0 μC and mass 0.25 kg is placed at P. When x is small compared to d, the magnitude of the net force on q is F ≈ 115x.
An uncharged parallel plate capacitor C is connected to a cell of emf 12 V, a resistor R and another resistor of resistance 20 MΩ.
Show that the magnitude of the resultant electric field at P is 3 MN C−1
State the direction of the resultant electric field at P.
Explain why q will perform simple harmonic oscillations when it is released.
Calculate the period of oscillations of q.
At t = 0, the switch is connected to X. On the axes, draw a sketch graph to show the variation with time of the voltage VR across R.
The switch is then connected to Y and C discharges through the 20 MΩ resistor. The voltage Vc drops to 50 % of its initial value in 5.0 s. Determine the capacitance of C.
The diagram shows the position of the principal lines in the visible spectrum of atomic hydrogen and some of the corresponding energy levels of the hydrogen atom.
A low-pressure hydrogen discharge lamp contains a small amount of deuterium gas in addition to the hydrogen gas. The deuterium spectrum contains a red line with a wavelength very close to that of the hydrogen red line. The wavelengths for the principal lines in the visible spectra of deuterium and hydrogen are given in the table.
Light from the discharge lamp is normally incident on a diffraction grating.
Determine the energy of a photon of blue light (435nm) emitted in the hydrogen spectrum.
Identify, with an arrow labelled B on the diagram, the transition in the hydrogen spectrum that gives rise to the photon with the energy in (a)(i).
Explain your answer to (a)(ii).
The light illuminates a width of 3.5 mm of the grating. The deuterium and hydrogen red lines can just be resolved in the second-order spectrum of the diffraction grating. Show that the grating spacing of the diffraction grating is about 2 × 10–6 m.
Calculate the angle between the first-order line of the red light in the hydrogen spectrum and the second-order line of the violet light in the hydrogen spectrum.
The light source is changed so that white light is incident on the diffraction grating. Outline the appearance of the diffraction pattern formed with white light.
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.
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.
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 small metal pendulum bob is suspended at rest from a fixed point with a length of thread of negligible mass. Air resistance is negligible.
The pendulum begins to oscillate. Assume that the motion of the system is simple harmonic, and in one vertical plane.
The graph shows the variation of kinetic energy of the pendulum bob with time.
When the 75 g bob is moving horizontally at 0.80 m s–1, it collides with a small stationary object also of mass 75 g. The object and the bob stick together.
Calculate, in m, the length of the thread. State your answer to an appropriate number of significant figures.
Label on the graph with the letter X a point where the speed of the pendulum is half that of its initial speed.
The mass of the pendulum bob is 75 g. Show that the maximum speed of the bob is about 0.7 m s–1.
Calculate the speed of the combined masses immediately after the collision.
Show that the collision is inelastic.
Sketch, on the axes, a graph to show the variation of gravitational potential energy with time for the bob and the object after the collision. The data from the graph used in (a) is shown as a dashed line for reference.
The speed after the collision of the bob and the object was measured using a sensor. This sensor emits a sound of frequency f and this sound is reflected from the moving bob. The sound is then detected by the sensor as frequency f′.
Explain why f and f′ are different.
The lens of an optical system is coated with a thin film of magnesium fluoride of thickness d. Monochromatic light of wavelength 656 nm in air is incident on the lens. The angle of incidence is θ. Two reflected rays, X and Y, are shown.
The following refractive indices are available.
Air = 1.00
Magnesium fluoride = 1.38
Lens = 1.58
The thickness of the magnesium fluoride film is d. For the case of normal incidence (θ = 0),
Light from a point source is incident on the pupil of the eye of an observer. The diameter of the pupil is 2.8 mm.
Predict whether reflected ray X undergoes a phase change.
state, in terms of d, the path difference between the reflected rays X and Y.
calculate the smallest value of d that will result in destructive interference between ray X and ray Y.
discuss a practical advantage of this arrangement.
Draw, on the axes, the variation with diffraction angle of the intensity of light incident on the retina of the observer.
Estimate, in rad, the smallest angular separation of two distinct point sources of light of wavelength 656 nm that can be resolved by the eye of this observer.
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.
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 .
A mass–spring system oscillates horizontally on a frictionless surface. The mass has an acceleration when its displacement from its equilibrium position is .
The variation of with is modelled in two different ways, A and B, by the graphs shown.
Outline two reasons why both models predict that the motion is simple harmonic when is small.
Determine the time period of the system when is small.
Outline, without calculation, the change to the time period of the system for the model represented by graph B when is large.
The graph shows for model A the variation with of elastic potential energy Ep stored in the spring.
Describe the graph for model B.
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.