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05 AL/Structural Question/P.1
HONG KONG ADVANCED LEVEL EXAMINATION
AL PHYSICS
2005 Structural Question
1. A block of mass m (in kg) rests on a smooth horizontal surface inside a container as
shown in Figure 1.1. The block is connected to the container with two identical
light springs, each with force constant k (in N m-1). Initially both springs are under
tension. The block is displaced horizontally as shown and then released from rest.
Assume that the force constant of each spring remains unchanged when being
stretched or compressed.
x
Figure 1.1
(a) The displacement of the block from its equilibrium position is x (in m).
(i) Draw a labelled diagram to show all the forces acting on the block.
(2 marks)
(ii) Express the net force F acting on the block in terms of k and x.
(1 mark)
(iii) Hence, state the equivalent force constant of the system in terms of k.
(1 mark)
(b) A student suggests modifying the device shown in Figure 1.1 in order to
measure a constant acceleration. The block is connected to the light, movable
contact of the rheostat of the circuit below such that they can move together
freely to both ends of the rheostat. When the whole set-up is at rest, the
movable contact is at the mid-point of the rheostat, as shown in Figure 1.2.
Given: m = 0.1 kg and k = 5 N m-l.
6V
0.2 m
Figure 1.2
x
0.1 kg
V
direction of
acceleration
05 AL/Structural Question/P.2
(i) The whole set-up moves with a horizontal acceleration a. What is the
relationship between the acceleration a and the displacement x? (The
displacement x towards the left is taken to be positive.)
(2 marks)
(ii) The rheostat has length 0.2 m. Referring to Figure 1.2, express the
voltmeter reading V (in V) in terms of the displacement x of the movable
contact from the mid-point. Hence find the relationship between a and V.
(2 marks)
(iii) State an advantage of connecting the block to the mid-point rather than to
one end of the rheostat.
(1 mark)
(iv) Another student constructed the set-up shown in Figure 1.2 and fixed it to
a bridge to monitor its vibration. In an earthquake, the block oscillated
along the direction of the springs and was found to resonate. State the
condition for resonance to occur and determine the approximate frequency
of the oscillation.
(3 marks)
2. A student uses the following experimental set-up to measure the speed of sound in
air.
Figure 2.1
Microphones A and B are connected respectively to the inputs, Y1 and Y2 , of a dual
trace CRO. Microphone A is fixed while microphone B can be moved along the
line PQ. Sound waves generated by the signal generator are detected by both
microphones A and B.
05 AL/Structural Question/P.3
(a) When A and B are switched on, the waveform obtained by adding up the signals
of Y1 and Y2 is observed on the screen of the CRO, as shown in Figure 2.2.
Figure 2.2
The time base of the CRO is set to 0.2 ms div-1. Calculate the frequency of the
sound waves.
(2 marks)
(b) (i) Microphone B is now moved towards P. A series of successive maximum
and minimum amplitudes of the trace is observed on the CRO screen as B
is being moved along PQ. Explain briefly.
(2 marks)
(ii) Four positions of microphone B at which successive maxima occur are
marked along the line PQ as shown in Figure 2.3. Their corresponding
distances from P are indicated in brackets.
Figure 2.3
(26.3 cm)
(58.2 cm)
(89.3 cm)
P
(120.8 cm)
Q
Calculate the speed of sound in air.
(3 marks)
(c) Another student proposes that she could use microphone A alone to find the
wavelength of the sound waves. Her experimental set-up is shown in Figure
2.4.
With microphone B removed and the display mode of the CRO changed to X-Y
mode, the sinusoidal signal produced by the signal generator is also fed into the
X-input of the CRO. Describe, with the aid of a series of diagrams, what will
appear on the CRO screen as microphone A is moved slowly along PQ. Also
describe how this student can determine the wavelength of the sound waves.
Explain briefly.
(4 marks)
Figure 2.4
05 AL/Structural Question/P.4
With microphone B removed and the display mode of the CRO changed to X-Y
mode, the sinusoidal signal produced by the signal generator is also fed into the
X-input of the CRO. Describe, with the aid of a series of diagrams, what will
appear on the CRO screen as microphone A is moved slowly along PQ. Also
describe how this student can determine the wavelength of the sound waves.
Explain briefly.
(4 marks)
3. (a) Mobile radars are installed in police cars to check the speed of cars on
highways. A student simulates the situation using an ultrasound source
representing the mobile radar, and trolleys A and B representing the police car
and the car suspected of speeding respectively. Trolleys A and B are travelling
at speeds v1 and v2 respectively along the same straight line. The ultrasound
source attached to trolley A emits ultrasonic waves at a frequency f towards
trolley B.
Ultrasound
source
(Mobile radar)
Trolley A
(Police car)
v1
Trolley B
(Car suspected of speeding)
v2
Express the wavelength  of the ultrasonic waves emitted from the moving
source and the frequency f ’ of the waves observed by trolley B in terms of the
physical quantities given and the speed v0 of sound in air. Explain briefly.
(3 marks)
(b) In a real situation, mobile radars emitting microwaves of frequency 10 GHz are
used. The reflected signal from the car suspected of speeding together with the
original signal are fed into the mobile radar in the police car to find the beat
2(v2  v1 )
f , where c is the speed of
frequency. This beat frequency is given by
c
light in air. Suppose v1 = 50 km h-1 and the beat frequency registered is 560 Hz.
The uncertainty in measuring the beat frequency is 5% and the speed limit on
this highway is 80 km h-1. Can you deduce whether the car driver is exceeding
the speed limit? Explain.
(3 marks)
(c) The new method employed in detecting speeding cars is the use of portable
‘LIDAR’ (LIght Detection And Ranging). A policeman on a footbridge across
a highway uses a LIDAR to check the speed of a car. The LIDAR emits a pulse
of infra-red waves towards the car. The pulse is reflected off the car and travels
back to the LIDAR. Using the time difference t between transmitting the
pulse and detecting the reflected pulse, the LIDAR determines how far away the
05 AL/Structural Question/P.5
car is. Thus by emitting two consecutive pulses separated by a certain time
interval T, the speed of the car can be found.
Time difference t between transmitting the
pulse and detecting the reflected pulse
First pulse
t1
Second pulse
t2
2(t2  t1 )
, where c is the
2T
speed of light in air. Given t2 - t1 = 0.78 ns and T = 5.0 ms, find the
car speed.
(3 marks)
(i) Show that the speed of the car is given by
(ii) The overall uncertainty of the LIDAR in measuring car speed is 3%.
Calculate the corresponding speed range of the car in km h-1.
(2 marks)
4. Figure 4.1 shows a home-made anemometer (an instrument for measuring wind
speed) which is made by attaching four plastic bowls to the ends of two identical
metal rods. The rods are fixed perpendicularly to each other and are free to rotate
about a vertical column. A vertical rectangular coil of area A, made of N turns of
fine insulated copper wire, is attached to the column as shown. The working
principle of this anemometer is that, as the wind blows, the coil will rotate about
the vertical column. The e.m.f. induced in the coil due to the earth’s magnetic field
can indicate the wind speed.
Figure 4.1
(a) In Hong Kong, the earth’s magnetic field makes an angle of 30 with the
horizontal, as shown in Figure 4.2. The magnetic flux density B of the earth is
50 T.
magnetic field lines
Figure 4.2
B
30 
horizontal
05 AL/Structural Question/P.6
Calculate the horizontal component (BH) of the earth’s magnetic flux density in
Hong Kong.
(1 mark)
(b) (i) The anemometer rotates at a constant angular velocity . At time t the
normal of the coil makes an angle  with the horizontal component BH of
the earth's magnetic field.
Figure 4.3
Write down an expression for the total flux linkage of the coil in terms of
BH, A, N and . Hence show that the induced e.m.f. in the coil is given by
 = BHAN sin t. (You may assume that  = 0 at t = 0.)
(3 marks)
(ii) In order to measure the induced e.m.f., the two ends of the coil are
connected to a CRO via a device. State what device is needed and sketch
the corresponding waveform.
(2 marks)
(iii) It is known that the dimensions of the coil are 40 cm  20 cm and it has
1000 turns. The wind speed v (in ms-1) and the peak value of the induced
e.m.f. peak (in V) are related by the expression v = 120peak + 2.5.
Calculate the wind speed when the anemometer is rotating at 12.5 rev s-1.
(3 marks)
05 AL/Structural Question/P.7
5. The count rate C of a mixture of two radioactive substances X and Y is measured at
different times. The variation of ln C with time t is plotted in the graph below. It is
known that the half-life of X is much longer than that of Y. (Note: all count rates
are corrected ones.)
(a) Explain why the graph approaches a straight line eventually. Write down the
equation of the straight line in terms of the following physical quantities.
CX
CXo
kX
t
= count rate of X at time t
= initial count rate of X
= decay constant of X
= time elapsed
(3 marks)
(b) (i) Use the graph to find the initial count rate of X and its half-life (in days).
Show your working.
(4 marks)
(ii) Estimate the count rate of the mixture at t = 19 days.
(2 marks)
(iii) Find the initial count rate of Y.
(2 marks)
(iv) Estimate the half-life of Y (in days) from the graph.
(2 marks)
(c) One of the radioactive substances in the mixture can be used as a tracer to
check whether the thyroid gland is absorbing iodine normally from the blood.
The tracer decays by emitting beta particles and gamma rays, and the gamma
rays are monitored from outside the body close to the thyroid. A dose of iodine
with the tracer is injected into a patient’s blood, and 20% of that dose is
05 AL/Structural Question/P.8
absorbed by the thyroid gland. The detector records 0.4% of the gamma rays
emitted.
(i) Why are the beta particles not monitored?
(1 mark)
(ii) Suggest a reason why the detector records only 0.4% of the gamma rays
emitted from the radioactive nuclei inside the thyroid.
(1 mark)
(iii) Whenever radioactive nuclides are used, the risks and benefits must be
carefully considered. Apart from the half-life of the source, suggest TWO
other factors that should be considered.
(2 marks)
6. (a) A spacecraft with an astronaut on board is launched on a
rocket booster. The rocket with the spacecraft has a total
initial mass of 4.80  105 kg at take-off. The rocket engine
propels hot exhaust gas at a constant speed of 2600 ms-1
relative to the rocket in a backward direction. Assume that
2.30  103 kg of gas is expelled in the first second.
(Neglect air resistance.)
(i) Calculate the average thrust (the upward force) acting
on the rocket due to the exhaust gas during the first
second.
(2 marks)
(ii) Assuming that the change in mass of the rocket during
the first second is negligible, estimate the acceleration
of the rocket.
(2 marks)
Figure 6.1
(iii) If the rocket keeps expelling exhaust gas at the same rate for the first 20 s,
explain how the rocket's acceleration will change.
(2 marks)
(b) The spacecraft of mass 7.80  103 kg now enters a circular orbit around the
earth at a height of 3.43  105 m above the earth's surface. (The radius of the
earth is 6.37  106 m.)
(i) Calculate the speed of the spacecraft in the orbit.
(4 marks)
(ii) How long does it take for the spacecraft to orbit the earth 14 times?
(2 marks)
(iii) Show that the total mechanical energy of the spacecraft in the orbit can be
expressed solely in terms of its mass and speed. Calculate this energy and
explain the physical meaning of its sign.
(4 marks)
(c) Give TWO reasons why an aircraft is unable to fly in space like a rocket.
(2 marks)
05 AL/Structural Question/P.9
7. A student uses the RC circuit shown in Figure 7.1 to investigate the charging of a
capacitor. The battery has an e.m.f. of 3 V and negligible internal resistance.
3V
C
S
R
Figure 7.1
(a) The capacitance of C is set at 0.5 F and the resistance of R is set at 1 k.
Initially, C is uncharged and the switch S is closed at time t = 0.
(i) Calculate the time constant  of the circuit and the voltage across R at t = .
(3 marks)
(ii) In the space provided below, sketch the variation with time t of the
voltages VR and VC across R and C respectively after the switch S is closed.
Indicate their voltage values at t = .
(2 marks)
VR/V
VC/V
3
3
2
2
1
1
0
0.5
1.0
t/ms
0
0.5
1.0
(iii) The student repeats the experiment using a smaller capacitance of 0.1 F.
Sketch on the graphs used in (ii) the corresponding curves VR’ and VC’.
Label your curves.
(2 marks)
(b) The student replaces the battery with a signal generator which generates square
waves of frequency 1 kHz. A dual trace CRO is used to observe the input
waveform and the waveform across components R or C. The student selects
one of the following combinations of R and C.
Combination
1
2
3
R
1 k
1 k
5 k
C
0.5 F
0.05 F
0.5 F
t/ms
05 AL/Structural Question/P.10
The time base and Y-gain of the CRO are set at 0.2 ms div-1 and 1 V div-1
respectively. The waveforms produced are shown on page 5.
(i) Deduce the peak-to-peak voltage of the square waves.
(1 mark)
(ii) Waveform X of the voltage across R is observed when using one of the RC
combinations.
(I) State which RC combination has been used.
(1 mark)
(II) Describe and explain briefly the change in magnitude and direction of
the current in the circuit within the first 1 ms.
(3 marks)
05 AL/Structural Question/P.11
(iii) Without changing the connections of the CRO, waveform Y is obtained
when using another RC combination.
(I) State which RC combination has been used. Explain why waveform Y
differs in shape from waveform X.
(2 marks)
(II) Sketch the corresponding waveform of the voltage across capacitor C.
(2 marks)
8. Figure 8.1 shows the essential structures of an X-ray tube. Electrons emitted from
the filament strike a target after being accelerated through a potential difference of
75 kV. A current of 20 mA is recorded by the milliammeter connected in series
with the X-ray tube.
Figure 8.1
Given: Planck constant = 6.6  10-34 J s
Charge on electron = -1.6  10-19 C
Mass of an electron = 9.1  10-31 kg
(a) Calculate
(i) the speed of the electrons striking the target, assuming that they are emitted
with negligible speed from the filament.
(2 marks)
(ii) the shortest wavelength, min, of the X-rays emitted by the tube.
(2 marks)
(b) (i) Estimate the electrical power input of the X-ray tube.
(2 marks)
(ii) Calculate the number of electrons which strike the target per second.
(2 marks)
(iii) Calculate the useful power output (for producing X-rays) if on average 80
bombarding electrons produce one X-ray photon which has an average
wavelength of 2.2  10-11 m. Referring to the input and output powers
calculated, account for the low efficiency of the X-ray tube.
(3 marks)
05 AL/Structural Question/P.12
(c) The resulting X-ray spectrum is shown below.
Suggest one change in the setting of the X-ray tube so as to increase the
maximum energy of the X-ray photons. Explain your answer. On the diagram
above, sketch the corresponding X-ray spectrum.
(4 marks)
9. Most electronic instruments such as electronic calculators will switch off
automatically if not used for a few minutes so as to conserve battery life. The
circuit in Figure 9.1 contains a module designed by a student in order to
automatically switch off an electronic device 3 minutes after the switch S is pressed
and released.
Transistorized relay module
A
+
S
Figure 9.1
C
9V
D
100 k
input
Electronic
device
B
E
_
(a) A second student found that one component in the module is connected
wrongly. Identify the wrongly-connected component and state how it should be
connected.
(1 mark)
(b) The circuit is now connected correctly. Explain why the electronic device can
be switched on once the switch S is pressed and state the function of the diode
D in the circuit.
(3 marks)
(c) Assume that, whenever there is a base current flowing through the transistor,
the voltage VBE across the base B and the emitter E is always equal to +0.7 V.
05 AL/Structural Question/P.13
(i) State the voltage across the capacitor C when the switch S is pressed.
Calculate the corresponding base current.
(3 marks)
(ii) In the corrected circuit, the voltage at the input must be at least +5.0 V for
the electromagnetic relay to be energized. This voltage is dependent on the
charging of the capacitor C through the resistor of resistance 100 k.
Calculate the base current when the voltage at the input is +5.0 V. Also,
calculate the capacitance of C so that the electronic device will switch off 3
minutes after switch S is opened. (There is no need to consider the input
resistance of the transistor.)
(4 marks)
- END OF PAPER -