# HL Topic 1 2000 - 2008

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N00/430/H(2)
SECTION A
Candidates must answer all questions in the spaces provided.
A medical physicist wishes to investigate the decay of a radioactive isotope and determine its decay
constant and half-life. A Geiger-Muller counter is used to detect radiation from a sample of the
isotope, as shown.
source
source
(a)
Voltage supply
and counter
Geiger-Muller tube
Geiger-Muller
tube
Define the activity of a radioactive sample.
[1]
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.........................................................................
Theory predicts that the activity A of the isotope in the sample should decrease exponentially with
time t according to the equation A = A0 e− λt , where A0 is the activity at t = 0 and λ is the decay
constant for the isotope.
(b)
Manipulate this equation into a form which will give a straight line if a semi-log graph is
plotted with appropriate variables on the axes. State what variables should be plotted.
[2]
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(Question A1 continued)
The Geiger-counter detects a proportion of the particles emitted by the source. The physicist
records the count-rate R of particles detected as a function of time t and plots the data as a graph of
ln R versus t, as shown below.
2 •
ln ( R / s −1 )
•
•
1
•
•
0
(c)
1
2
3
4
5
t / hr
Does the plot show that the experimental data are consistent with an exponential law?
Explain.
[1]
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(d)
The Geiger-counter does not measure the total activity A of the sample, but rather the
count-rate R of those particles that enter the Geiger tube. Explain why this will not matter in
determining the decay constant of the sample.
[1]
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(e)
[2]
From the graph, determine a value for the decay constant λ.
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(Question A1 continued)
The physicist now wishes to calculate the half-life.
(f)
Define the half-life of a radioactive substance.
[1]
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(g)
[2]
Derive a relationship between the decay constant λ and the half-life τ.
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(h)
Hence calculate the half-life of this radioactive isotope.
[1]
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Page 5
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M01/430/H(2)
SECTION A
Candidates must answer all questions in the spaces provided.
A1. Gas law experiment (data based question)
Boyle’s law states that for an ideal gas at constant temperature, pressure is inversely proportional to
volume. To test whether or not a real gas obeys Boyle’s law, three students set up the apparatus shown
below.
Bourdon
gauge
Tube
Gas
200
A
300
100
0
Oil
kPa
400
Stopcock
Air
Oil
The gas sample is enclosed in the tube and the length of the gas column can be measured against
the scale. The gas pressure in the apparatus can be adjusted by pumping air in or out through the
stopcock. The Bourdon gauge indicates ‘gauge pressure’, i.e. the difference in pressure inside and
outside the gauge.
(a)
After each gas adjustment the students wait a few minutes before reading the column length
and the Bourdon gauge. Explain why they should not take the readings immediately and
what occurs during waiting.
[2]
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(b)
Boyle’s law involves the volume V of the gas, yet the students instead measure the length L of
the gas column. Why is this acceptable?
[1]
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(c)
Show algebraically that if Boyle’s law holds, a plot of gas pressure P versus the reciprocal of
1
the column length L (i.e. P versus ) should be of straight line form through the origin.
L
[2]
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(Question A1 continued)
1
, as shown below. In order
L
1
to estimate the uncertainty in the data, the students have repeated the measurement at
= 2.8 m −1
L
three times, giving a cluster of three data points.
The students plot the pressure reading P B on the Bourdon gauge versus
400
300
200
100
0
0
1
2
3
4
1
1
=   m −1
Length of gas column  L 
-100
-200
(d)
Draw a best-fit straight line for the data points.
[1]
(e)
Determine the intercept value on the Bourdon pressure axis.
[1]
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(f)
From the three repeated measurements reflected by the cluster of points,
(i)
draw in an error bar on the graph to reflect the experimental uncertainty.
[1]
(ii)
write the pressure value and its uncertainty in the form (value ! uncertainty).
[2]
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(Question A1 continued)
(g)
The students note that their graph does not seem to go through the origin (0,0). They each
suggest different interpretations of the results of this experiment, as follows:
Student 1 concludes that since the graph does not go through the origin, this gas deviates
from Boyle’s law behaviour.
Student 2 points out that there are random uncertainties in the data. He suggests that within
experimental uncertainty the data may reasonably be fitted by a line drawn through the origin.
He concludes that the data shows that the gas obeys Boyle’s law within experimental
uncertainty.
Student 3 says there could be a systematic error somewhere in the readings or the analysis.
(i)
Discuss the reasoning of student 2 in light of the data.
[3]
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(ii)
Suggest the most likely origin of any systematic error suggested by student 3. Explain
this with reference to the particular numerical value found in question (e) of the
pressure intercept of the graph.
[2]
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(iii) If specific adjustment is made for such systematic error, are the data consistent with
Boyle’s law? Explain.
[2]
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(iv) Which student’s interpretation is best, and does the gas obey Boyle’s law?
[1]
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SECTION A
Candidates must answer all questions in the spaces provided.
A1. This question is about power dissipation in a resistor and the internal resistance of a battery.
In the circuit below the variable resistor can be adjusted to have known values of resistance R. The
battery has an unknown internal resistance r.
r
––––
I
A
R
The table below shows the recorded value I of the current in the circuit for different values of R.
The last column gives the calculated value of the power P dissipated in the resistor.
R/!
0
1.0
2.0
3.0
4.0
6.0
8.0
10.0
(a)
I/A
!0.01 A
1.50
1.20
1.00
0.86
0.75
0.60
0.50
0.43
P/W
0
1.4
2.0
2.2
2.3
2.2
2.0
Complete the last line of the table by calculating the power dissipated in the variable resistor
when its value is 10.0 !.
[2]
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(Question A1 continued)
(b)
If each value of R is known to !10 % determine the absolute uncertainty in the value of P
when R = 10.0 !.
[3]
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(c)
On the grid below plot a graph of power P against resistance R. (Do not include error bars).
[4]
(d)
It can be shown that the power dissipated in the external resistor is a maximum when
the value of its resistance R is equal to the value of the internal resistance r of the battery
i.e. R = r. Use this information and your graph to find the value of r.
[1]
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(e)
The manufacturer of the battery gives the value of its internal resistance as 4.50 ! ! 0.01 !.
Is the value of r that you obtained from your graph consistent with the manufacturer’s value?
Explain.
[2]
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M02/430/H(2)
SECTION A
Candidates must answer all questions in the spaces provided.
A1. This question is about the growth of an electric current in a coil.
When a coil is connected to a d.c. power supply the current in the coil does not change
instantaneously but takes a finite time to reach a steady value. For a given supply the final, steady
value of the current is determined by the resistance (R) of the coil.
In the diagram below a coil is connected to a d.c. supply of emf 4.0 V.
4.0 V
A
S
Coil
When the switch S is closed an electronic timer is started and the current I is recorded at different
values of the time t. The results are shown in the table below. (Uncertainties in measurement are
not shown).
t /s
I /A
(a)
0
0
0.2
0.8
0.6
1.6
Plot a graph of current against time.
1.0
1.9
1.4
2.0
1.8
2.0
2.0
2.0
[5]
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(Question A1 continued)
(b)
What is the steady state value of the current?
[1]
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(c)
Determine the value of the resistance R of the coil.
[1]
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(d)
By drawing a tangent to the curve at the point (0, 0) on your graph, determine the time it
would take for the current to reach its steady state value if it were to continue changing at its
initial rate. (This time is known as the time constant of the coil).
[2]
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(e)
V
where
L
V is the value of the supply potential and L is a property of the coil known as its inductance.
Show that the time constant ! for the coil is given by the expression
The initial rate at which the current in the coil changes is given by the expression
τ=
L
.
R
[3]
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(f)
Determine the value of the inductance L of the coil.
[1]
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SECTION A
Candidates must answer all questions in the spaces provided.
A1. Some students were asked to design and carry out an experiment to determine the specific latent
heat of vaporization of water. They set up the apparatus shown below.
d.c. supply
V
A
Water
Heater
g
Top-pan balance
The current was switched on and maintained constant using the variable resistor. The readings of
the voltmeter and the ammeter were noted. When the water was boiling steadily, the reading of the
top-pan balance was taken and, simultaneously, a stopwatch was started. The reading of the
top-pan balance was taken again after 200 seconds and then after a further 200 seconds.
The change in reading of the top-pan balance during each 200 second interval was calculated and
an average found. The power of the heater was calculated by multiplying together the readings of
the voltmeter and the ammeter.
(a)
Suggest how the students would know when the water was boiling steadily.
[1]
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(b)
Explain why a reading of the mass lost in the first 200 seconds and then a reading of the mass
lost in the next 200 second interval were taken, rather than one single reading of the mass lost
in 400 seconds.
[2]
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(Question A1 continued)
The students repeated the experiment for different powers supplied to the heater. A graph of the
power of the heater against the mass of water lost (the change in balance reading) in 200 seconds
was plotted. The results are shown below. (Error bars showing the uncertainties in the
measurements are not shown.)
120
100
80
power / W
60
40
20
0
0
(c)
1
2
3
4
mass / g
5
6
7
8
(i)
On the graph above, draw the best-fit straight line for the data points.
[1]
(ii)
Determine the gradient of the line you have drawn.
[3]
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(Question A1 continued)
In order to find a value for the specific latent heat of vaporization L, the students used the equation
P = mL ,
where P is the power of the heater and m is the mass of water evaporated per second.
(d)
Use your answer for the gradient of the graph to determine a value for the specific latent heat
of vaporization of water.
[3]
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(e)
The theory of the experiment would suggest that the graph line should pass through the
origin. Explain briefly why the graph does not pass through the origin.
[2]
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N03/430/H(2)
SECTION A
Candidates must answer all questions in the spaces provided.
A1. This question is about an experiment designed to investigate Newton’s second law.
In order to investigate Newton’s second law, David arranged for a heavy trolley to be accelerated
by small weights, as shown below. The acceleration of the trolley was recorded electronically.
David recorded the acceleration for different weights up to a maximum of 3.0 N. He plotted a
graph of his results.
acceleration
heavy trolley
pulley
weight
(a)
Describe the graph that would be expected if two quantities are proportional to one another.
[2]
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(Question A1 continued)
(b)
David’s data are shown below, with uncertainty limits included for the value of the weights.
Draw the best-fit line for these data.
[2]
acceleration 1.40
/ m s −2
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0.00
0.50
1.00
1.50
2.00
2.50
weight / N
(c)
Use the graph to
(i)
explain what is meant by a systematic error.
[2]
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(ii)
estimate the value of the frictional force that is acting on the trolley.
[1]
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(iii) estimate the mass of the trolley.
[2]
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SECTION A
Answer all the questions in the spaces provided.
A1. Data based question. This question is about change of electrical resistance with temperature.
The table below gives values of the resistance R of an electrical component for different values of
its temperature T. (Uncertainties in measurement are not shown.)
T / &deg;C
R/
(a)
1.2
2.0
3.5
5.2
6.8
8.1
9.6
3590
3480
3250
3060
2880
2770
2650
On the grid below, plot a graph to show the variation with temperature T of the resistance R.
Show values on the temperature axis from T = 0 &deg;C to T = 10 &deg;C .
[3]
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(Question A1 continued)
(b)
(i)
(ii)
Draw a curve that best fits the points you have plotted. Extend your curve to cover the
temperature range from 0 &deg;C to 10 &deg;C .
[1]
Use your graph to determine the resistance at 0 &deg;C and at 10 &deg;C .
[2]
Resistance at 0 &deg;C = . . . . . . . . . . . . . . . . . . . . . . .
Resistance at 10 &deg;C = . . . . . . . . . . . . . . . . . . . . . .
(c)
(d)
On your graph, draw a straight-line between the resistance values at 0 &deg;C and at 10 &deg;C . This
line shows the variation with temperature (between 0 &deg;C and 10 &deg;C ) of the resistance,
assuming a linear change.
(i)
Assuming a linear change of resistance with temperature, use your graph to determine
the temperature at which the resistance is 3060 .
[1]
[1]
Temperature = . . . . . . . . . . . . . . . . . . . . . . . . . . &deg;C
(ii)
Use your answer in (d)(i) to calculate the percentage difference in the temperature for a
resistance of 3060
that results from assuming a linear change rather than the
non-linear change.
[3]
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(e)
In a particular experiment to measure the variation with temperature of the resistance, each
measurement of resistance has an uncertainty of ! 30 and the uncertainty in the temperature
measurements is ! 0.2 &deg;C .
(i)
(ii)
On your graph in (a), show the uncertainties in the values of R and of T for
temperatures of 1.2 &deg;C, 5.2 &deg;C and 9.6 &deg;C .
[2]
State and explain whether, within the experimental uncertainties, the relationship
between resistance and temperature could be linear.
[2]
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SECTION A
Answer all the questions in the spaces provided.
A1. This question is about measuring the permittivity of free space ε 0 .
The diagram below shows two parallel conducting plates connected to a variable voltage supply.
The plates are of equal areas and are a distance d apart.
variable voltage supply
+
d
_
V
The charge Q on one of the plates is measured for different values of the potential difference V
applied between the plates. The values obtained are shown in the table below. The uncertainty in
the value of V is not significant but the uncertainty in Q is !10%.
V/V
Q / nC !10 %
10.0
30
20.0
80
30.0
100
40.0
160
50.0
180
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(Question A1 continued)
(a)
Plot the data points opposite on a graph of V (x-axis) against Q (y-axis).
[4]
(b)
By calculating the relevant uncertainty in Q, add error bars to the data points (10.0, 30) and
(50.0, 180).
[3]
On the graph above, draw the line that best fits the data points and has the maximum
permissible gradient. Determine the gradient of the line that you have drawn.
[3]
(c)
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(d)
The gradient of the graph is a property of the two plates and is known as capacitance.
Deduce the units of capacitance.
[1]
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(Question A1 continued)
The relationship between Q and V for this arrangement is given by the expression
Q=
ε0 A
d
V
where A is the area of one of the plates.
In this particular experiment A = 0.20 &plusmn; 0.05 m 2 and d = 0.50 &plusmn; 0.01mm.
(e)
Use your answer to (c) to determine the maximum value of ε 0 that this experiment yields.
[4]
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M05/4/PHYSI/HP2/ENG/TZ1/XX+
SECTION A
Answer all the questions in the spaces provided.
A1. Data analysis question
At high pressures, a real gas does not behave as an ideal gas. For a certain range of pressures,
it is suggested that the relation between the pressure P and volume V of one mole of the gas at
constant temperature is given by the equation
PV = A + BP
where A and B are constants.
In an experiment to measure the deviation of nitrogen gas from ideal gas behaviour, 1 mole
of nitrogen gas was compressed at a constant temperature of 150 K. The volume V of the gas
was measured for different values of the pressure P. A graph of the product PV of pressure
and volume was plotted against the pressure P and is shown below. (Error bars showing the
uncertainties in measurements are not shown).
PV
PV //&times;102 N m
PP//&times;106 Pa
(a)
Draw a line of best fit for the data points.
[1]
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(Question A1 continued)
(b)
Use the graph to determine the values of the constants A and B in the equation
PV = A + BP .
[5]
Constant A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Constant B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
............................................................
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(c)
State the value of the constant B for an ideal gas.
[1]
......................................................................
(d)
The equation PV = A + BP is valid for pressures up to 6.0 &times;107 Pa.
(i)
[2]
Determine the value of PV for nitrogen gas at a pressure of 6.0 &times;107 Pa.
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(ii)
Calculate the difference between the value of PV for an ideal gas and nitrogen gas
when both are at a pressure of 6.0 &times;107 Pa.
[2]
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(e)
In the original experiment, the pressure P was measured to an accuracy of 5 % and the
volume V was measured to an accuracy of 2 %. Determine the absolute error in the value
of the constant A.
[3]
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SECTION A
Answer all the questions in the spaces provided.
A1. The Geiger-Nuttall theory of α -particle emission relates the half-life of the α -particle emitter
to the energy E of the α -particle . One form of this relationship is
L =
166
1
E2
− 53.5.
L is a number calculated from the half-life of the α -particle emitting nuclide and E is measured
in MeV.
Values of E and L for different nuclides are given below. (Uncertainties in the values are not
shown.)
Nuclide
(a)
E / MeV
L
1
E
1
2
/ MeV
− 12
238
U
4.20
17.15
0.488
236
U
4.49
14.87
0.472
234
U
4.82
12.89
0.455
228
Th
5.42
7.78
...........
208
Rn
6.14
3.16
0.404
212
Po
7.39
–2.75
0.368
Complete the table above by calculating, using the value of E provided, the value of
for the nuclide
228
1
1
E2
[1]
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(Question A1 continued)
1
The graph below shows the variation with 1 of the quantity L. Error bars have not been
E2
L 20
16
12
8
4
0
0.2
0.3
0.4
1
1
E2
–4
(b)
208
0.5
/ MeV
Rn. Label this point R.
(i)
Identify the data point for the nuclide
(ii)
On the graph, mark the point for the nuclide
228
Th . Label this point T.
(iii) Draw the best-fit straight-line for all the data points.
− 12
[1]
[1]
[1]
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(Question A1 continued)
(c)
(i)
[2]
Determine the gradient of the line you have drawn in (b) (iii).
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(ii)
Without taking into consideration any uncertainty in the values for the gradient and
for the intercept on the x-axis, suggest why the graph does not agree with the stated
relationship for the Geiger-Nuttall theory.
[2]
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(d)
On the graph opposite, draw the line that would be expected if the relationship for the
Geiger-Nuttall theory were correct. No further calculation is required.
(e)
U is &plusmn; 0.03 MeV. Deduce that this
1
uncertainty is consistent with quoting the value of 1 to three significant digits.
E2
......................................................................
The uncertainty in the measurement of E for
[2]
238
[3]
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SECTION A
Answer all the questions in the spaces provided.
A1. This question is about an electrostatics experiment to investigate how the force between two
charges varies with the distance between them.
A small charged sphere S hangs vertically from an insulating thread as shown below.
S
A second identically charged sphere P is brought close to S. S is repelled as shown below.
P
S
force F
r
The magnitude of the electrostatic force on sphere S is F. The separation between the two
spheres is r.
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(Question A1 continued)
(a)
On the axes below draw a sketch graph to show how, based on Coulomb’s law, you would
1
expect F to vary with 2 .
r
[2]
F
0
1
r2
0
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(Question A1 continued)
Values of F are determined for different values of r. The variation with
1
of these values is
r2
shown below. The estimated uncertainties in these values are negligible.
F /&times;10−3 N 7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
(b)
0.0
2.0
4.0
6.0
8.0
10.0
12.0
1
/&times;103 m −2
r2
(i)
Draw the best-fit line for these data points.
[2]
(ii)
Use the graph to explain whether, in the experiment, there are random errors,
systematic errors or both.
[3]
.................................................................
.................................................................
.................................................................
.................................................................
.................................................................
(iii) Calculate the gradient of the line drawn in (b) (i).
[2]
.................................................................
.................................................................
.................................................................
.................................................................
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(Question A1 continued)
(iv) The magnitude of the charge on each sphere is the same. Use your answer to (b) (iii)
to calculate this magnitude.
[4]
.................................................................
.................................................................
.................................................................
.................................................................
(c)
Explain how a graph showing the variation with lg r of lg F can be used to verify the
relation between r and F.
[3]
......................................................................
......................................................................
......................................................................
......................................................................
......................................................................
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sEction a
Answer all the questions in the spaces provided.
a1. This question is about the rise of water in a capillary tube.
A capillary tube is a tube that is open at both ends and has a very narrow bore. A capillary
tube is supported vertically with one end immersed in water. Water rises up the tube due to
a phenomenon called capillary action. The water in the bore of the tube forms a column of
height h as shown below.
narrow bore
glass wall
glass wall
h
water
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(Question A1 continued)
(a)
The height h, for a particular capillary tube was measured for different temperatures of
the water. The variation with temperature  of the height h is shown below. Uncertainties
in the measurements are not shown.
17
16
15
14
13
h / cm
12
11
10
9.0
8.0
0
10
20
30
40 50
 / &deg;C
60
70
80
90
(i)
On the graph above, draw a best-fit line for the data points.
[1]
(ii)
Determine the height h0 of the water column at temperature  = 0 &deg;C.
[1]
..................................................................
..................................................................
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(Question A1 continued)
(b)
Explain why the results of this experiment suggest that the relationship between the
height h and temperature  is of the form
h = h0(1 − k)
where k is constant.
[4]
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
(c)
Deduce that the value of k is approximately 4.8  10−3 deg C−1.
[3]
.......................................................................
.......................................................................
.......................................................................
.......................................................................
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(Question A1 continued)
(d)
The experiment is repeated using tubes with bores of different radii r but keeping the
1
water temperature constant. The graph below shows the variation with of the height h
r
for capillary tubes of different radii r for a water temperature of 20 &deg;C.
0.35
0.30
0.25
0.20
h/m
0.15
0.10
0.05
0
0
5.0
10.0
15.0
1
/ &times;103 m −1
r
20.0
25.0
It is suggested that capillary action is one of the means by which water moves from the
roots of a tree to the leaves. A particular tree has a height of 25 m.
Use the graph to estimate the radius of the bore of the tubes that would enable water to
be raised by capillary action from ground level to the top of the tree. Comment on your
Estimate:
[4]
.............................................................
.............................................................
.............................................................
.............................................................
.............................................................
Comment: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.............................................................
.............................................................
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sEction a
Answer all the questions in the spaces provided.
a1. A hot object may be cooled by blowing air past it. This cooling process is known as forced
convection. In order to investigate forced convection, hot oil was placed in a metal can. The
can was placed on an insulating block and air was blown past the can, as shown below.
stirrer
thermometer
lid
hot oil
current of air
metal can
insulating block
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(Question A1 continued)
The hot oil was stirred continuously and its temperature was taken every minute as it cooled.
The graph below shows the variation with time of the temperature of the cooling oil.
120
100
temperature / &deg;C
80
60
40
20
0
0
2
4
6
8
time / minutes
10
12
14
It is thought that the rate R of decrease of temperature depends on the temperature difference
between the oil and its surroundings (the excess temperature θ E). The temperature of the
surroundings was 26 &deg;C.
(a)
On the graph above,
(i)
(ii)
draw a straight-line parallel to the time axis to represent the temperature of the
surroundings.
[1]
by drawing a suitable tangent, calculate the rate of decrease of temperature, in &deg;C s–1,
for an excess temperature of 50 Celsius degrees (&deg;C).
[4]
..................................................................
..................................................................
..................................................................
..................................................................
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(Question A1 continued)
(b)
In order to investigate the variation with R of θ E, a graph of R against θ E is plotted.
The graph below shows four plotted data points. Uncertainties in the points are not
included.
0.24
0.20
0.16
R / &deg;C s–1
0.12
0.08
0.04
0.00
0
20
40
60
θE /
80
100
&deg;C
(i)
Using your answer to (a)(ii), plot the data point corresponding toθθEE = 50 &deg;C.
[1]
(ii)
The uncertainty in the measurement of R at each excess temperature is &plusmn;10 %.
On the graph, draw error bars to represent the uncertainties in R at excess temperatures
of 20 &deg;C and 81 &deg;C.
[2]
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(Question A1 continued)
(c)
(i)
Explain why the graph in (b) supports the conclusion that the excess temperature θ E
is related to the rate of cooling R by the expression
R = kθ E ,
where k is a constant.
[3]
..................................................................
..................................................................
..................................................................
..................................................................
(ii)
At high excess temperatures, the equation in (i) is thought to become invalid.
Discuss whether the graph in (b) provides any evidence for this suggestion.
[2]
..................................................................
..................................................................
..................................................................
(d)
In a second experiment, the data is analysed by plotting a graph of lgR against lgθ E.
(lg is the logarithm to the base 10.)
(i)
On the axes below, draw a sketch graph to show the line that would be obtained.
(Note that this is a sketch graph. No data points or values on the axes are required.)
[1]
lgR
lgθ E
(ii)
Assuming the expression in (c)(i) is correct, state the gradient of the line of the
graph. Also, explain how the value of k is obtained.
[2]
..................................................................
..................................................................
..................................................................
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sEcTion a
Answer all the questions in the spaces provided.
a1. This question is about thermal energy transfer through a rod.
A student designed an experiment to investigate the variation of temperature along a copper rod
when each end is kept at a different temperature. In the experiment, one end of the rod is placed
in a container of boiling water at 100 &deg;C and the other end is placed in contact with a block of
ice at 0.0 &deg;C as shown in the diagram.
temperature sensors
boiling water
100 &deg;C
ice
0 &deg;C
copper rod
not to scale
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(Question A1 continued)
Temperature sensors are placed at 10 cm intervals along the rod. The final steady state
temperature θ of each sensor is recorded, together with the corresponding distance x of each
sensor from the hot end of the rod.
The data points are shown plotted on the axes below.
θ / &deg;C 110
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80 90
x / cm
The uncertainty in the measurement of θ is &plusmn;2 &deg;C. The uncertainty in the measurement of x
is negligible.
(a)
(b)
On the graph above, draw the uncertainty in the data points for x = 10 cm, x = 40 cm
and x = 70 cm.
[2]
On the graph above, draw the line of best-fit for the data.
[1]
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(Question A1 continued)
(c)
Explain, by reference to the uncertainties you have indicated, the shape of the line you
have drawn.
[2]
.......................................................................
.......................................................................
.......................................................................
(d)
(i)
Use your graph to estimate the temperature of the rod at x = 55 cm.
[1]
..................................................................
(ii)
Determine the magnitude of the gradient of the line (the temperature gradient)
at x = 50 cm.
[3]
..................................................................
..................................................................
..................................................................
..................................................................
(e)
The rate of transfer of thermal energy R through the cross-sectional area of the rod is
∆θ
along the rod. At x = 10 cm, R = 43 W and
∆x
∆θ
= 1.81&deg;C cm −1. At x = 50 cm the value
the magnitude of the temperature gradient is
∆x
of R is 25 W.
Use these data and your answer to d(ii) to suggest whether the rate R of thermal energy
transfer is in fact proportional to the temperature gradient.
[3]
.......................................................................
.......................................................................
.......................................................................
.......................................................................
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(Question A1 continued)
(f)
It is suggested that the variation with x of the temperature θ is of the form
θ = θ 0 e − kx
where θ 0 and k are constants.
State how the value of k may be determined from a suitable graph.
[2]
.......................................................................
.......................................................................
.......................................................................
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sEction a
Answer all the questions in the spaces provided.
a1. The question is about investigating a fireball caused by an explosion.
When a fire burns within a confined space, the fire can sometimes spread very rapidly in the
form of a circular fireball. Knowing the speed with which these fireballs can spread is of
great importance to fire-fighters. In order to be able to predict this speed, a series of controlled
experiments was carried out in which a known amount of petroleum (petrol) stored in a can
was ignited.
The radius R of the resulting fireball produced by the explosion of some petrol in a can was
measured as a function of time t. The results of the experiment for five different volumes of
petroleum are shown plotted below. (Uncertainties in the data are not shown.)
25
Key:
30  10–3 m3
25  10–3 m3
15  10–3 m3
10  10–3 m3
5.0  10–3 m3
20
15
R/m
10
5
0
0
(a)
10
20
30 40
t / ms
50
60
70
The original hypothesis was that, for a given volume of petrol, the radius R of the fireball
would be directly proportional to the time t after the explosion. State two reasons why
the plotted data do not support this hypothesis.
1.
[2]
..................................................................
..................................................................
2.
..................................................................
..................................................................
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(Question A1 continued)
(b)
The uncertainty in the radius is  0.5 m. The addition of error bars to the data points would
show that there is in fact a systematic error in the plotted data. Suggest one reason for this
systematic error.
[2]
.......................................................................
.......................................................................
.......................................................................
.......................................................................
(c)
Since the data do not support direct proportionality between the radius R of the fireball
and time t, a relation of the form
R = kt n
is proposed, where k and n are constants.
In order to find the value of k and of n, lg(R) is plotted against lg(t). The resulting graph,
for a particular volume of petrol, is shown below. (Uncertainties in the data are not shown.)
1.3
1.2
1.1
lg(R)
1.0
0.9
0.8
1
1.1
1.2
1.3
1.4 1.5
lg(t)
1.6
1.7
1.8
1.9
Use this graph to deduce that the radius R is proportional to t0.4. Explain your reasoning.
[4]
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
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(Question A1 continued)
(d)
It is known that the energy released in the explosion is proportional to the initial volume
of petrol. A hypothesis made by the experimenters is that, at a given time, the radius of
the fireball is proportional to the energy E released by the explosion. In order to test this
hypothesis, the radius R of the fireball 20 ms after the explosion was plotted against the
initial volume V of petrol causing the fireball. The resulting graph is shown below.
15
10
R/m
5
0
0
5
10
15 20 25
V /  10–3 m3
30
35
The uncertainties in R have been included. The uncertainty in the volume of petrol is
negligible.
(i)
Describe how the data for the above graph are obtained from the graph in (a).
[1]
..................................................................
..................................................................
(ii)
Draw the line of best-fit for the data points.
[2]
(iii) Explain whether the plotted data together with the error bars support the hypothesis
that R is proportional to V.
[2]
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
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(Question A1 continued)
(e)
Analysis shows that the relation between the radius R, energy E released and time t is in
fact given by
R5 = Et2.
Use data from the graph in (d) to deduce that the energy liberated by the combustion of
1.0  10–3 m3 of petrol is about 30 MJ.
[4]
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
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sEction a
Answer all the questions in the spaces provided.
a1. As part of a road-safety campaign, the braking distances of a car were measured.
A driver in a particular car was instructed to travel along a straight road at a constant
speed v. A signal was given to the driver to stop and he applied the brakes to bring the
car to rest in as short a distance as possible. The total distance D travelled by the car
after the signal was given was measured for corresponding values of v. A sketch-graph
of the results is shown below.
v
0
0
(a)
D
State why the sketch graph suggests that D and v are not related by an expression of
the form
D = mv + c,
where m and c are constants.
[1]
.......................................................................
.......................................................................
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(Question A1 continued)
(b)
It is suggested that D and v may be related by an expression of the form
D = av + bv2,
where a and b are constants.
In order to test this suggestion, the data shown below are used. The uncertainties in the
measurements of D and v are not shown.
v / m s–1
D/m
D
/ (i
..........
v
10.0
14.0
1.40
13.5
22.7
1.68
18.0
36.9
2.05
22.5
52.9
27.0
74.0
2.74
31.5
97.7
3.10
D
.
v
[1]
(i)
In the table above, state the unit of
(ii)
D
Calculate the magnitude of , to an appropriate number of significant digits,
v
for v = 22.5 m s–1.
[1]
..................................................................
..................................................................
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(Question A1 continued)
(c)
D
(y-axis) against v (x-axis). Some of the
Data from the table are used to plot a graph of
v
data points are shown plotted below.
3.50
3.00
D
/ (S.I. units)
v
2.50
2.00
1.50
1.00
0.50
0.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
–1
v / ms
On the graph above,
(i)
plot the data points for speeds corresponding to 22.5 m s–1 and to 31.5 m s–1.
[2]
(ii)
draw the best-fit line for all the data points.
[1]
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(Question A1 continued)
(d)
Use your graph in (c) to determine
(i)
the total stopping distance D for a speed of 35 m s–1.
[2]
..................................................................
..................................................................
..................................................................
(ii)
the intercept on the
D
axis.
v
[1]
..................................................................
(iii) the gradient of the best-fit line.
[2]
..................................................................
..................................................................
..................................................................
(e)
Using your answers to (d)(ii) and (d)(iii), deduce the equation for D in terms of v.
D=
(f)
[1]
..................................................................
The uncertainty in the measurement of the distance D is &plusmn;0.3 m and the uncertainty in
the measurement of the speed v is &plusmn;0.5 m s–1.
(i)
For the data point corresponding to v = 27.0 m s–1, calculate the absolute uncertainty
D
in the value of .
v
[2]
..................................................................
..................................................................
..................................................................
(ii)
Each of the data points in (b) was obtained by taking the average of several values
of D for each value of v. Suggest what effect, if any, the taking of averages will
have on the uncertainties in the data points.
[2]
..................................................................
..................................................................
..................................................................
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Section a
Answer all the questions in the spaces provided.
a1. This question is about data analysis.
Data for the refractive index n of a type of glass and wavelength  of the light transmitted
through the glass are shown below.
Only the uncertainties in the values of n are significant and these uncertainties are shown by
error bars.
1.6065
1.6060
1.6055
1.6050
1.6045
n
1.6040
1.6035
1.6030
1.6025
1.6020
1.6015
300
350
400
450
500
550
600
650
/nm
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(Question A1 continued)
(a)
State why the data do not support the hypothesis that there is a linear relationship between
refractive index and wavelength.
[1]
.......................................................................
.......................................................................
(b)
Draw a best-fit line for the data points.
(c)
The rate of change of refractive index D with wavelength is referred to as the dispersion.
λ
At any particular value of wavelength, D is defined by
λ
[2]
∆n
D =
λ ∆λ
Use the graph to determine the value of D at a wavelength of 380 nm.
λ
[4]
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
(d)
It is suggested that the relationship between n and  is of the form
n = kλ p
where k and p are constants.
[3]
State and explain the graph that you would plot in order to determine the value of p.
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
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(Question A1 continued)
(e)
A second suggestion is that the relationship between n and  is of the form
n = A+
B
λ2
where A and B are constants.
1
To test this suggestion, values of n are plotted against values of 2 . The resulting graph
λ
with the line of best fit is shown below.
1.6065
1.6060
1.6055
1.6050
1.6045
n
1.6040
1.6035
1.6030
1.6025
1.6020
1.6015
1.6010
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1
/&times;10−15 m −2
λ2
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(Question A1 continued)
(i)
Use the graph to determine the value of the constant A.
[3]
..................................................................
..................................................................
..................................................................
(ii)
State the significance of the constant A.
[1]
..................................................................
..................................................................
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Section a
Answer all the questions in the spaces provided.
a1. Some data for the resistance R of an electrical component at different temperatures are
shown below.
t / &deg;c
R/Ω
10.0
15.0
25.0
30.0
35.0
40.0
2600
2150
1510
1280
1080
925
A graph of the variation with temperature t of the resistance R of the component is
shown below. Error bars have been included.
3400
3200
3000
2800
2600
2400
R/Ω
2200
2000
1800
1600
1400
1200
1000
800
0
5
10
15
20 25
t / &deg;C
30
35
40
45
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(Question A1 continued)
(a)
Estimate the
(i)
uncertainty range in the temperature measurements.
[1]
..................................................................
(ii)
percentage uncertainty in the resistance at 10.0 &deg;C.
[2]
..................................................................
..................................................................
..................................................................
(b)
Use the graph to determine the
(i)
(ii)
most probable resistance of the component at 20.0 &deg;C and at 5.0 &deg;C.
Resistance at 20.0 &deg;C
...............................................
[1]
Resistance at 5.0 &deg;C
...............................................
[2]
rate of change of resistance with temperature at 20.0 &deg;C.
[3]
..................................................................
..................................................................
..................................................................
(c)
The relationship between resistance and temperature is not linear. Describe, and explain,
the evidence for a non-linear relationship that is provided by the graph.
[2]
.......................................................................
.......................................................................
.......................................................................
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(Question A1 continued)
(d)
A student suggests that the relationship between the resistance R and temperature is of
the form
c
R=
T
where c is a constant and T is the thermodynamic (absolute) temperature.
Use data from the table to determine, without drawing a graph, whether this suggestion
is correct.
[3]
.......................................................................
.......................................................................
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Section a
Answer all the questions in the spaces provided.
a1. This question is about the mass-radius relation for a certain type of star.
The radius R and mass M of ten different stars were measured and the results are shown
plotted below.
R / R S 2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.0
0.0
0.4
0.8
1.2
1.6
2.0
M / MS
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(Question A1 continued)
The radius is expressed in terms of the Sun’s radius R S and the mass in terms of the
Sun’s mass M S.
The uncertainty in the measurement of the mass is negligible.
measurement of the radius is &plusmn;0.05 R S.
(a)
Draw error bars for the first and the last data points.
(b)
(i)
The uncertainty in the
[1]
suggest why there might be a linear relationship between R and M for these stars.
[2]
..................................................................
..................................................................
(ii)
determine the equation for this linear relationship.
[3]
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
(iii) estimate the maximum value for the mass of this type of star.
[1]
..................................................................
..................................................................
(c)
Suggest why no star of this type can in fact have a mass equal to your answer to (b)(iii).
[1]
.......................................................................
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(Question A1 continued)
(d)
Additional data show that the relation between R and M is in fact not linear, as suggested
by the graph below.
R / R S 2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.0
0.0
0.4
0.8
1.2
1.6
2.0
M / MS
Uncertainties in the data are not shown.
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(Question A1 continued)
(i)
Draw a line of best-fit for the data.
[1]
(ii)
The new data suggests that the maximum value for the mass of this type of star is
[1]
..................................................................
..................................................................
[1]
..................................................................
(e)
It is hypothesized that the mass–radius relation for a different type of star is R=kM n
where k and n are constants.
Explain how a graph may be used to
(i)
verify this hypothesis.
[2]
..................................................................
..................................................................
..................................................................
..................................................................
(ii)
determine the constant n.
[1]
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