Ds 1 è suddivisibile e conserva i seguenti testi:
3.2. Il testo di riferimento
There are two simple rules for obtaining an estimate of the overall uncertainty in a final result. The rules are:
1 For quantities which are added or subtracted to give a final result, add the actual uncertainties.
2 For quantities which are multiplied together or divided to give a final result, add the fractional uncertainties.
Suppose that we wish to obtain the value of a physical quantity x by measuring two other quantities, y and z. The relation between x, y and z is known, and is
x = y + z
If the uncertainties in y and z are ∆y and ∆z respectively, the uncertainty ∆x in x is given by
∆x = ∆y + ∆z
If the quantity x is given by x = y − z
the uncertainty in x is again given by
∆x = ∆y + ∆z
Examples
1 I1 and I2 are two currents coming into a junction in a circuit. The current I going out of the junction is given by
I = I1 + I2
In an experiment, the values of I1 and I2 are determined as 2.0 ± 0.1 A and 1.5 ± 0.2 A respectively. What is the value of I? What is the uncertainty in this value?
Using the given equation, the value of I is given by I = 2.0 + 1.5 = 3.5 A. The rule for combining the uncertainties gives ∆I = 0.1 + 0.2 = 0.3 A. The result for I is thus (3.5 ± 0.3) A.
2 In an experiment, a liquid is heated electrically, causing the temperature to change from 20.0 ± 0.2 °C to 21.5 ± 0.5 °C. Find the change of temperature, with its associated uncertainty.
2.2 Errors and uncertainties
The change of temperature is 21.5 − 20.0 = 1.5 °C. The rule for combining the uncertainties gives the uncertainty in the temperature change as 0.2 + 0.5 = 0.7 °C.
The result for the temperature change is thus (1.5 ± 0.7) °C.
Note that this second example shows that a small difference between two quantities may have a large uncertainty, even if the uncertainty in measuring each of the quantities is small. This is an important factor in considering the design of experiments, where the difference between two quantities may introduce an unacceptably large error.
Now it’s your turn
12 Two set-squares and a ruler are used to measure the diameter of a cylinder. The cylinder is placed between the set-squares, and the set-squares are aligned with the ruler, in the manner of the jaws of calipers. The readings on the ruler at opposite ends of a diameter are 4.15 cm and 2.95 cm. Each reading has an uncertainty of ±0.05 cm.
(a) What is the diameter of the cylinder?
(b) What is the uncertainty in the diameter?
Now suppose that we wish to find the uncertainty in a quantity x, whose relation to two measured quantities, y and z, is
x = Ayz
where A is a constant. The uncertainty in the measurement of y is ±∆y, and that in z is
±∆z. The fractional uncertainty in x is given by
∆xx = ∆y y + ∆zz
To combine the uncertainties when the quantities are raised to a power, for example x = Ayazb
where A is a constant, the rule is
∆x x = a
∆y
y
+ b ∆z
z
Example
A value of the acceleration of free fall g was determined by measuring the period of oscillation T of a simple pendulum of length l. The relation between g, T and l is
g = 4π2
(
Tl2)
In the experiment, l was measured as 0.55 ± 0.02 m and T was measured as 1.50 ± 0.02 s.
Find the value of g, and the uncertainty in this value.
Substituting in the equation, g = 4π2(0.55/1.502) = 9.7 m s−2. The fractional uncertainties are ∆l/l = 0.02/0.55 = 0.036 and ∆TlT = 0.02/1.50 = 0.013.
Applying the rule to find the fractional uncertainty in g
∆ g
g = ∆ll + 2∆l
T = 0.036 + 2 × 0.013 = 0.062
The actual uncertainty in g is given by (value of g) × (fractional uncertainty in g)
= 9.7 × 0.062 = 0.60 m s−2. The experimental value of g, with its uncertainty, is thus (9.7 ± 0.6) m s−2.
Note that it is not good practice to determine g from the measurement of the period of a pendulum of fixed length. It would be much better to take values of T for a number of different lengths l, and to draw a graph of T 2 against l. The gradient of this graph is 4π2/g.
Now it’s your turn
13 A value of the volume V of a cylinder is determined by measuring the radius r and the length L. The relation between V, r and L is
V = πr 2L
In an experiment, r was measured as 3.30 ± 0.05 cm, and L was measured as 25.4 ± 0.4 cm. Find the value of V, and the uncertainty in this value.
2 Measurement techniques
If you find it difficult to deal with the fractional uncertainty rule, you can easily estimate the uncertainty by substituting extreme values into the equation.
For x = Ay az b, taking account of the uncertainties in y and z, the lowest value of x is given by
xlow = A(y − ∆y)a(z − ∆z)b and the highest by xhigh = A(y + ∆y)a(z + ∆z)b
If xlow and xhigh are worked out, the uncertainty in the value of x is given by (xhigh − xlow)/2.
Example
Apply the extreme value method to the data for the simple pendulum experiment in the Example on page 37.
Because of the form of the equation for g, the lowest value for g will be obtained if the lowest value of l and the highest value for T are substituted. This gives
glow = 4π2(0.53/1.522) = 9.1 m s−2
The highest value for g is obtained by substituting the highest value for l and the lowest value for T. This gives
ghigh = 4π2(0.57/1.482) = 10.3 m s−2
The uncertainty in the value of g is thus (ghigh − glow)/2 = (10.3 − 9.1)/2 = 0.6 m s−2, as before.
Now it’s your turn
14 Apply the extreme value method to the data for the volume of the cylinder, on page 37.
If the expression for the quantity under consideration involves combinations of products (or quotients) and sums (or differences), then the best approach is the extreme value method.
Summary
● Methods available for the measurement of length include:
metre rule (range 1 m, reading uncertainty 1 mm)
micrometer screw gauge (range 50 mm, reading uncertainty 0.01 mm) vernier caliper (range 100 mm, reading uncertainty 0.1 mm).
● Methods available for the measurement of mass include:
top-pan balance spring balance lever balance.
● Methods available for the measurement of time include:
stopclock (reading uncertainty 0.2 s) stopwatch (reading uncertainty 0.01 s) cathode-ray oscilloscope.
● Methods available for the measurement of temperature include:
liquid-in-glass thermometer thermocouple thermometer.
● Methods available for the measurement of current and potential difference include:
analogue meter digital meter multimeter
cathode-ray oscilloscope.
● Methods available for the measurement of magnetic flux density include the Hall probe.
● Accuracy is concerned with how close a reading is to its true value.
● Precision is determined by the size of the random error and can be controlled by the experimenter.
Examination style questions
Examination style questions
1 You are asked to measure the internal diameter of a glass capillary tube (diameter about 2 mm). You are also to investigate the uniformity of the tube along its length.
Suggest suitable methods.
2 The value of the acceleration of free fall varies slightly at different places on the Earth’s surface. Discuss whether this means that
a a top-pan balance, b a spring balance, c a lever balance,
should be re-calibrated when they are moved to different locations.
If you needed to, how would you calibrate a balance?
3 The shutter on a particular camera has settings which allow it to be open for (nominally) 1 s, 0.5 s, 0.25 s, 0.125 s, 0.067 s, 0.033 s, 0.017 s, 0.008 s, 0.004 s, 0.002 s and 0.001 s. Suggest a method (or methods) of calibrating the exposure times over this range.
4 Explain the factors you would consider when deciding whether to use a liquid-in-glass or a thermocouple thermometer in particular experimental situations.
5 Summarise the advantages and disadvantages of analogue and digital ammeters.
6 Explain how to use a cathode-ray oscilloscope to measure the characteristics of the sinusoidal output from a signal generator.
7 A metal wire has a cross-section of diameter of approximately 0.8 mm.
a State what instrument should be used to measure the
diameter of the wire. [1]
b State how the instrument in a is
i checked so as to avoid a systematic error in the
measurements, [1]
ii used so as to reduce random errors. [2]
Cambridge International AS and A Level Physics, 9702/22 May/June 2010 Q 1
8 a State the most appropriate instrument, or instruments, for the measurement of the following.
i the diameter of a wire of diameter about 1 mm [1]
ii the resistance of a filament lamp [1]
iii the peak value of an alternating voltage [1]
b The mass of a cube of aluminium is found to be 580 g with an uncertainty in the measurement of 10 g. Each side of the cube has a length of (6.0 ± 0.1) cm.
Calculate the density of aluminium with its uncertainty.
Express your answer to an appropriate number of
significant figures. [5]
Cambridge International AS and A level Physics, 9702/21 May/June 2009 Q 1 9 A simple pendulum may be used to determine a value for
the acceleration of free fall g. Measurements are made of the length L of the pendulum and the period T of oscillation.
The values obtained, with their uncertainties, are as shown.
T = (1.93 ± 0.03) s L = (92 ± 1) cm
a Calculate the percentage uncertainty in the measurement of
i the period T, [1]
ii the length L. [1]
b The relationship between T, L and g is given by g = 4π2L
T 2
Using your answers in a, calculate the percentage
uncertainty in the value of g. [1]
c The values of L and T are used to calculate a value of g as 9.751 m s–2.
i By reference to the measurements of L and T, suggest why it would not be correct to quote the value of g as
9.751 m s–2. [1]
ii Use your answer in b to determine the absolute uncertainty in g.
Hence state the value of g, with its uncertainty, to an appropriate number of significant figures. [2]
Cambridge International AS and A level Physics, 9702/22 Oct/Nov 2009 Q 1
● Uncertainty indicates the range of values within which a measurement is likely to lie.
● A systematic uncertainty (or systematic error) is often due to instrumental causes, and results in all readings being above or below the true value. It cannot be eliminated by averaging.
● A random uncertainty (or random error) is due to the scatter of readings around the true value. It may be reduced by repeating a reading and averaging, or by plotting a graph and taking a best-fit line.
● Combining uncertainties:
for expressions of the form x = y + z or x = y − z, the overall uncertainty is
∆x = ∆y + ∆z
for expressions of the form x = Ay az b, the overall fractional uncertainty is
∆x/x = a(∆y/y) + b(∆z/z)
Starting points
● Kinematics is a description of how objects move.
● The motion of objects can be described in terms of quantities such as position, speed, velocity and acceleration.