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© Manhattan Press (H.K.) Ltd. 1
Measurements and Measurements and errorserrors
• Precision and accuracyPrecision and accuracy• Significant figuresSignificant figures
• Scientific notationcientific notation• MeasurementsMeasurements• Types of errorsTypes of errors
• Combination of errorsCombination of errors
© Manhattan Press (H.K.) Ltd. 2
Measurements and errors
Reading:
• single determination of the value of an unknown quantity
• actual reading taken during an experiment
Measurement and errors (SB p. 13)
Measurement:
• final result of the analysis of a series of readings
© Manhattan Press (H.K.) Ltd. 3
Precision and accuracy
Precision:
indicates the agreement among repeated measurements
Measurement and errors (SB p. 13)
• take readings repeatedly
• calculate mean ( )
• deviation (d) =
• measure the precision: by the mean deviation
x
ixx
© Manhattan Press (H.K.) Ltd. 4
Precision and accuracy
Improvement:
e.g.
• Use a hand lens when reading the scale of a meter
• Use a plane mirror behind the pointer
• Read the scale when the pointer is directly on top of its image
Measurement and errors (SB p. 14)
© Manhattan Press (H.K.) Ltd. 5
Accuracy
Measurement and errors (SB p. 14)
Precision and accuracy
A measurement is said to be accurate if it is close to the actual value.
Accuracy:
Indicates how correct the result is
© Manhattan Press (H.K.) Ltd. 6
Accuracy
Measurement and errors (SB p. 14)
Precision and accuracy
e.g.
1. Using a metre rule
Length recorded = 34.7 cm
length accurate to 0.1 cm
34.7 0.1 cm
© Manhattan Press (H.K.) Ltd. 7
Accuracy
Measurement and errors (SB p. 14)
Precision and accuracy
2. Using a micrometre screw gauge
Length recorded = 3.62 mm
3.620 0.005 mm maximum possible error
Fractional error = 62030050
Mean valueerror possible Maximum
.
.
© Manhattan Press (H.K.) Ltd. 8
Accuracy
Measurement and errors (SB p. 14)
Precision and accuracy
3. Percentage error
%.%..
%
2760100623010
100Mean value
error possible Maximum
Smaller the percentage error,
higher the accuracy
Go to
More to Know 4More to Know 4
Go to
Common Error
© Manhattan Press (H.K.) Ltd. 9
(a) Most significant digit:
the leftmost non-zero digit
(b) Least significant digit:
the rightmost non-zero digit for the number without decimal point, or
the rightmost digit (including zero) for the number with decimal point
Significant figures:
No. of digits except for zeros at the beginning
Measurement and errors (SB p. 15)
Significant figures
© Manhattan Press (H.K.) Ltd. 10
(a) ruler of smallest division 0.1 cm
Length = 8.074 6 cm is not consistent
Length = 8.1 0.1 cm
Significant figures:
consistent with the accuracy of the measurement
Measurement and errors (SB p. 15)
Significant figures
© Manhattan Press (H.K.) Ltd. 11
(b) Mass of X = 0.376 kg (3 sig. fig.)
Mass of Y = 0.056 2 kg (3 sig. fig.)
Combined weight
mg = (0.376 + 0.056 2) ×9.8
= 0.432 2 ×9.8
= 4.235 56 N
Combined weight = 4.2 N (2 sig. fig.)
Measurement and errors (SB p. 15)
Significant figures
g (2 sig. fig.)
© Manhattan Press (H.K.) Ltd. 12
No. of significant figures
= no. of significant figures in the quantity which has the least no. of significant figures
Measurement and errors (SB p. 16)
Significant figures
© Manhattan Press (H.K.) Ltd. 13
Measurement and errors (SB p. 16)
Significant figures
No. of significant figures reflects a measurement’s order of accuracy
e.g.
Length of classroom = 8 m (1 sig. fig.)
Actual measured length
= 7.5 m to 8.5 m
© Manhattan Press (H.K.) Ltd. 14
Measurement and errors (SB p. 16)
Significant figures
e.g.
Length of classroom = 8.0 m (2 sig. fig.)
Actual measured length
= 7.95 m to 8.05 m
© Manhattan Press (H.K.) Ltd. 15
Measurement and errors (SB p. 16)
Scientific notation
Scientific notation:
represent extremely large or extremely small numbers
e.g.
800 000 (up to 1, 2 or 3 sig. fig?)
8.00 x 105 (3 sig. fig)
© Manhattan Press (H.K.) Ltd. 16
Measurement and errors (SB p. 16)
Measurements
1. Metre rule
Maximum possible error
= Half the smallest division
=
= 0.05 cm210.
© Manhattan Press (H.K.) Ltd. 17
Measurement and errors (SB p. 17)
Measurements
e.g.
At “8 cm” mark, reading = 8.00 0.05 cm
7.95 cm – 8.05 cm
1. the first reading from “0 cm”
2. the second reading from “8 cm”
Length = 8.00 0.1 cmMaximum
possible error
2 reading errors
© Manhattan Press (H.K.) Ltd. 18
Measurement and errors (SB p. 17)
Measurements
2. Vernier caliper
external diameter
internal diameter
depth of container
© Manhattan Press (H.K.) Ltd. 19
2. Vernier caliper
Maximum possible error
= Half the smallest division
=
= 0.05 mm210.
Measurement and errors (SB p. 17)
Measurements
© Manhattan Press (H.K.) Ltd. 20
Measurement and errors (SB p. 18)
Measurements
e.g.1.30 cm to 1.40 cm
The 7th mark is exactly opposite a mark on the main scale. Reading = 1.370 cm
Reading
= 1.370 0.005 cm
Go to
More to Know 5More to Know 5
© Manhattan Press (H.K.) Ltd. 21
Measurement and errors (SB p. 18)
Measurements
3. Micrometre screw gauge
Measure outer dimension of object up to accuracy of at least 0.01 mm
© Manhattan Press (H.K.) Ltd. 22
Measurement and errors (SB p. 18)
Measurements
Length of division in main scale = 0.5 mm
Each smallest division = = 0.01 mm5050.
Maximum possible error
= Half the smallest division
=
= 0.005 mm2010.
© Manhattan Press (H.K.) Ltd. 23
Measurement and errors (SB p. 19)
Measurements
e.g.
Reading
= 2.480 0.005 mm
Go to
More to Know 6More to Know 6
© Manhattan Press (H.K.) Ltd. 24
Measurement and errors (SB p. 19)
Measurements
4. Computer data-logging system
Data collection and storage for data processing later
InterfaceSensor
© Manhattan Press (H.K.) Ltd. 25
Measurement and errors (SB p. 20)
Types of errors
1. Systematic errors
Systematic errors are errors in the measurement of physical quantities due to instruments, faults in the surrounding conditions or mistakes made by the observer. An experiment with small systematic error is said to be accurate.
© Manhattan Press (H.K.) Ltd. 26
Measurement and errors (SB p. 21)
Types of errors
Sources of systematic error
(a) Zero errors
reading on instrument is not zero when it is not used
Go to
More to Know 7More to Know 7
© Manhattan Press (H.K.) Ltd. 27
Measurement and errors (SB p. 21)
Types of errors
(b) Personal errors of the observer
From physical constrains or limitation of an individual (reaction time)
© Manhattan Press (H.K.) Ltd. 28
Measurement and errors (SB p. 21)
Types of errors
(c) Errors due to instruments
e.g.
(i) A watch which is fast
(ii) An ammeter which is used under different conditions from which it had have calibrated
© Manhattan Press (H.K.) Ltd. 29
Measurement and errors (SB p. 21)
Types of errors
(d) Errors due to wrong assumption
e.g.
acceleration due to gravity (g) is assumed to be 9.81 m s-2
Systematic errors cannot be reduced by repeating measurements using the same method, same instrument and by the same observer.
© Manhattan Press (H.K.) Ltd. 30
Measurement and errors (SB p. 22)
Types of errors
Systematic errors can reduced by taking measurements carefully, or varying conditions of measurements.
More accurate Less accurate
© Manhattan Press (H.K.) Ltd. 31
Measurement and errors (SB p. 22)
Types of errors
2. Random errors
Random errors are errors in a measurement made by the observer or person who takes the measurement. An experiment with small random error is said to be precise.
© Manhattan Press (H.K.) Ltd. 32
Measurement and errors (SB p. 22)
Types of errors
Sources
(a) Due to parallax
Go to
More to Know 8More to Know 8
© Manhattan Press (H.K.) Ltd. 33
Measurement and errors (SB p. 23)
Types of errors
(b) Due to temperature change
Go to
More to Know 9More to Know 9
More precise Less precise
© Manhattan Press (H.K.) Ltd. 34
Measurement and errors (SB p. 23)
Combination of errors
1. Addition or subtraction
(a) If U = x + y
U = (x + y)
(b) If V = x - y
V = (x + y)
x, y are errors
Go to
Example 3Example 3
© Manhattan Press (H.K.) Ltd. 35
2. Product
If U = xyz
][zz
yy
xx
UU
Measurement and errors (SB p. 24)
Combination of errors
x, y, z are errors
Go to
Example 4Example 4
© Manhattan Press (H.K.) Ltd. 36
3. Quotient
If U =
][yy
xx
UU
yx
Measurement and errors (SB p. 25)
Combination of errors
Go to
Example 5Example 5
© Manhattan Press (H.K.) Ltd. 37
Measurement and errors (SB p. 26)
Combination of errors
4. Constant power
If U = xp (p is constant)
xxp
UU
© Manhattan Press (H.K.) Ltd. 38
5. General case
If U = (c, p, q, r are constant)r
qp
z
ycx
][zzr
yy
qxxp
UU
Measurement and errors (SB p. 26)
Combination of errors
Go to
Example 6Example 6Go to
Example 7Example 7
© Manhattan Press (H.K.) Ltd. 39
End
© Manhattan Press (H.K.) Ltd. 40
Maximum possible error
If a metre rule is graduated in mm, the maximum possible error of a measurement is equal to the half of the smallest division (0.05 cm or 0.5 mm). Why should the maximum possible error of the measurement of the metal rod be 0.1 cm? For details, please refer to the Section D of Metre rule.
Return to
TextText
Measurement and errors (SB p. 14)
© Manhattan Press (H.K.) Ltd. 41
A common zero error arisesfrom using a metre rule fromone end, which may be worn.It is a better practice to usethe centre of the rule, insteadof measuring from one end.
Return to
TextText
Measurement and errors (SB p. 15)
© Manhattan Press (H.K.) Ltd. 42
Most vernier calipers will read zero when the jaws are closed, without an object in place. However, as a result of misuse or wear, the instrument may not read zero. In these cases, a zero reading error must be added or subtracted. The maximum possible error becomes ± 0.05 mm × 2 = ± 0.1 mm.
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TextText
Measurement and errors (SB p. 18)
© Manhattan Press (H.K.) Ltd. 43
Choice of measuring instrument
When we choose a measuring instrument, we should consider:
1. its convenience to be used,2. its precision, and3. the range of the reading we need.
Return to
TextText
Measurement and errors (SB p. 19)
© Manhattan Press (H.K.) Ltd. 44
Zero reading error and zero error
Zero reading error is due to the limitation of measuring device at the “0” mark and is equal to the half of the smallest division of the device.Zero error of the device is due to the deviation from “0” value at the “0” mark.
Return to
TextText
Measurement and errors (SB p. 21)
© Manhattan Press (H.K.) Ltd. 45
1. Reading can be taken with great precision but not accurate if there is a systematic error.
2. Reading can be accurate but not precise when there is a random error.
Return to
TextText
Measurement and errors (SB p. 22)
© Manhattan Press (H.K.) Ltd. 46
Random errors can be reduced by repeated measurements while systematic errors cannot.
Return to
TextText
Measurement and errors (SB p. 23)
© Manhattan Press (H.K.) Ltd. 47
Q: Q: The internal diameter d1 and the external
diameter d2 of a metal tube are d1 = 45 ± 1
mm and d2 = 60 ± 2 mm. What is the
maximum percentage error in the total thickness of the tube when it is pressed together?
Solution
Measurement and errors (SB p. 24)
© Manhattan Press (H.K.) Ltd. 48
Solution:Solution:
Return to
TextText
Thickness of the tube (t)
= d2 – d1 = 60 – 45 = 15 mm
Error in t (δt)
= ±(δd2 + δd1) = ±2 + 1 = ±3 mm
∴ Thickness of the tube = 15 ± 3 mm
∴ Maximum percentage error in t :
= 20%
%%tt 100
153100
Measurement and errors (SB p. 24)
© Manhattan Press (H.K.) Ltd. 49
Q:Q: The dimensions of a box are recorded as follows:Length () = 5.0 ± 0.2 cmWidth (b) = 4.0 ± 0.1 cmHeight (h) = 8.0 ± 0.2 cmWhat is the maximum percentage error in the volume of the box?
Solution
Measurement and errors (SB p. 24)
© Manhattan Press (H.K.) Ltd. 50
Solution:Solution:
Return to
TextText
Volume of box ( V )
= bh = 5.0 4.0 8.0 = 160 cm3
∴Maximum percentage error in V:
%hh
bb%
VV 100][100
%..
.
... 100]
0820
0410
0520[
= 9%
Measurement and errors (SB p. 25)
© Manhattan Press (H.K.) Ltd. 51
Q:Q: The mass of a metal block is 11.5 ± 0.5 kg and its volume is 1 000 ± 20 cm3. How would you express the density of the metal?
Solution
Measurement and errors (SB p. 25)
© Manhattan Press (H.K.) Ltd. 52
Solution:Solution:
Return to
TextText
Density ()
32 cm kg 101511000
511)( Volume
)( Mass
..V
M
Maximum fractional error in :
)10(1.15]1000
2051150[
]1000
2051150[
][
2-
....
VV
MM
= 0.07 x 10-2 kg cm-3
Density of metal () = (1.15 0.07) x 10-2 kg cm-3
Note: The error is calculated to only one sig. fig.
Measurement and errors (SB p. 25)
© Manhattan Press (H.K.) Ltd. 53
Q:Q: In an experiment, the external diameter D and internal diameter d of a metal tube werefound to be 64 ± 2 mm and 47 ± 1 mm respectively. What is the maximum percentage error in the cross-sectional area of the metal tube as shown in the figure?
Solution
Measurement and errors (SB p. 26)
© Manhattan Press (H.K.) Ltd. 54
Solution:Solution:
Return to
TextText
Area of the shaded region (A)
%
%
%dDdD
dDdD
%AA
20
100]4764
34764
3[
100])()(
)()(
[100
))((4
)(4
22
dDdD
dD
Maximum error in (D + d):
δ(D + d) = δD + δd = 2 + 1 = 3 mm
Maximum error in (D – d):
δ(D – d) = δD + δd = 2 + 1 = 3 mm
Maximum percentage error in the cross-sectional area:
Measurement and errors (SB p. 26)
© Manhattan Press (H.K.) Ltd. 55
Q:Q: (a) Explain the principle of vernier calipers.(b) The figure shows a steel vernier caliper which was used to measure the diameter of a cylinder. What is the reading for the diameter?
(c) Assuming that the zero reading is correct, what is the percentage error in this measurement?(d) Calculate the percentage error in the cross-sectional area of the cylinder.Solution
Measurement and errors (SB p. 27)
© Manhattan Press (H.K.) Ltd. 56
Solution:Solution:
(a) Vernier calipers consist of two scales.1. The main scale has 1.0 cm divided into 10 equal divisions.
Size of each division = 0.10 cm2. The vernier scale has the length of 10 divisions adding up to 0.9 cm.
Size of each division = 0.09 cm∴Difference in size of one division on the main scale and one division on the vernier scale is:
(0.10 – 0.09) cm = 0.01 cmIn the figure above, the distance between the 1.2 cm mark on the main scale and the 0 mark on the vernier scale is 3 0.01 = 0.03 cm.∴The vernier reading is 1.23 cm.
Measurement and errors (SB p. 27)
© Manhattan Press (H.K.) Ltd. 57
Solution (cont’d) :Solution (cont’d) :
Return to
TextText
(b) From the figure, diameter of the cylinder (d) = 2.53 cm
(c) Error in d (δd) = 0.005 cm
∴ Percentage error:
%
%.
.%dd
.20
100532
0050100
(d) Cross-sectional area (A) =
∴ Percentage error:4
2d
0.4% .202
1002100
%
%dd%
AA
Measurement and errors (SB p. 27)