muhammad jawhar

1. Introduction
• Mechanical measurement and Metrology
• Definition:
• Measurement generally involves usin an
instrument as a physical means of determining
a quantity or a variable.
• Measurement work employs number of terms
which should be defined here:

2. • Instrument: a device for determining the value or
magnitude of a quantity or variable.
• Accuracy: closeness with which an instrument reading
approaches the true value of the variable being
measured.
• Precision: a measure of reproducibility of the
measurement, is a measure of the degree to which
successive measurements differ from one another.
• Sensitivity: the ratio of output signal or response of the
instrument to a change of input or measured variable.

3. Of measured variable.
• Resolution: the smallest change in measured value to which the
instrument will respond.
• Error: deviations from the true value of measured variable.
• significant figures: An indication of precision of the measurement is
obtained from number of significant figures in which the result is
expressed. Significant figures convey actual information regarding
the magnitude and measurement precision of a quantity. The more
significant figures, the grater the precision of measurement. Ex if
the force is 70 N, so that the force is between 69 N and 71N, but
when we express the force as 70.0 it means that the force is closer
to 70.0 than it is 69.9 0r 70.1, in 70 there is only two significant
figures while in 70.0 there are three. So that more significant figure
expresses a measurement of greater precision than the former.

4. • Anther example if the voltmeter reading is
indicating to 117.1+- 0,05 V, this means the
voltage reading is between 117.05 and 117.15. In
most cases we concentrate on the mean value of
the set of readings and we can find the largest
deviation from the mean.
• Ex.1
• A set of independent voltage measements taken
by four observers was recoeded as 117.02,
117.11, 117.08, and 117.03. Calculate the
average voltage, and the range of error.

5. • Solution:
• Eav=(117.02+117.11+117.08+117.03)/4=117.06
• Range =Emax-Eav=117.11-117.06=0.05
• But also
• Eav-Emin =117.06-117.02=0.04
• The average range of error=(0.05+0.04)/2=+-
0.045=+-0.05 V
• If two or more measrements are added the result is
accurate as the least accurate measurement.

6. • Ex. 2
• Two resistances R1 and R2are connected in
series. Individual resistance measurements
using Wheatstone bridge, give R1=18.7 Ω and
R2= 3.624 Ω calculate the total resistance to
the appropriate number of significant figures.
• Rt=R1 + R2= 18/7 + 3.624 = 23.6 Ω
• In multiplication significant figures increase
rapidly.

7. • Ex-3
• In calculating voltage drop, a current of 3.18 A
is recorded in a resistance of 35.68 Ω.
Calculate the voltage drop across the resistor
to the appropraite number of significant
figures.
• E=IR=35.68*3.18=113.4624=113 V

8. • Ex. 4
• Add 826 +-5 t0 628+-3
• N1 = 826 +-5 (=+- 0.605%)
• N2 = 628 +- 3 (= +- 0.477%)
• Sum=1454 +- 8 (= +- 0.55%)
• Ex. 5
• Subtract 628+-3 from 826+-5 and express the range of
doubt in the answer as a percentage.
• N1 = 826 +- 5 ( = +-0.605%)
• N2 = 628 +- 3 (= +- 0.477%)
• Difference = 198 +- 8 (= 4.04%)

9. • Ex.6
• Subtract 437 +-4 from 462+-4 and express the
range of doubt in the answer as a percetage.
• N1 = 462 +-4 ( = +- 0.87%)
• N2 = 437 +-4 (=+- 0.92%)
• Difference = 25 +-8 (= +-32%)
• Note: one should avoid measurement techniques
depending on subtraction of experimental results
because the range of doubt in the final result ma
be greatly increased.

10. • Types of Error:
• No measurement can be made with perfect
accuracy, but it is important to find out what the
accuracy actually is and how different errors have
entered into the measurement.
• Errors come from different sources:
• Gross error: largely human errors, among them
misreading of instruments, incorrect adjustment
and improper application of instruments, and
computational mistakes.

11. • Systematic errors: shortcommings of the
instruments, such as defective or worn parts,
and effects of the environment on the
equipment or the user.
• Random errors: those due to causes that can
not be directly established because of random
variations in the parameter or the system of
measurement

12. • Ex-7
• A voltmeter, having sensitivity of 1000 Ω/V, reads
100 V on its 150 –V scale when connected across
an unknown resistor in series with a
molliammeter. When the milliammeter reads 5
mA, calculate (a) apparent resistance of the
unknown resistor, (b) actual resistance of the
unknown resistor, © error due to the loading
effect of the voltmeter.

13. • RT=VT/IT =100 V/ 5 mA=20 kΩ
• Neglecting the resistance of milliammeter, the
value of the unknown resistor is Rx =20 kΩ
• (b) voltmeter resistance equals
• Rv = 1000 Ω/V * 150 V=150 kΩ
• Since the voltmeter is in parallel with the
unknown resistance, we can write
• Rx=RT.Rv/(Rv-RT)=20*150/130=23.05 Ω
• % error =(actual-apparent)/actual*100=(23.05-
20)/23.05*100=13.23%

14. • Ex. 8
• Repeat ex-7 if the milliammeter reads 800 mA and the voltmeter
reads 40 V on its 150- V scale.
• RT=VT/IT =100 V/ 800 mA=50 Ω
• Neglecting the resistance of milliammeter, the value of the
unknown resistor is Rx =20 kΩ
• (b) voltmeter resistance equals
• Rv = 1000 Ω/V * 150 V=150 kΩ
• Since the voltmeter is in parallel with the unknown resistance, we
can write
• Rx=RT.Rv/(Rv-RT)=50*150/149.95=50.1 Ω
• % error =(actual-apparent)/actual*100=(50.1-50)/50.1*100=0.2%

15. • Statistical analysis
• A statistical analysis of measurement data is
common practice because it allows an
analytical determination of the uncertainty of
the final test result.
• Arithmatic mean,

16. Deviation from the mean:
D1=X1-xave, D2=X2-Xave, Dn=Xn-Xave,
Ex.9 a set of independent current measurement
was taken by six observers and recorded
as12.8 mA, 12.2 mA, 12.5 mA, 13.1 mA, 12.9
mA, and 12.4 mA. Calculate (a)the arithmatic
mean, (b) the deviation from mean equals
Xave= (12.8+12.2+12.5+13.1+12.9)/6=12.65 mA

17. • D1= 12.8-12.65 =0.15 mA
• D2=12.2-12.65=-0.45 mA
• D3=12.5-12.65=-0.15 mA
• D4= 13.1-12.65=0.45 mA
• D5=12.9-12.65=0.25 mA
• D6=12.4-12.65=-0.25 mA
• Note that algebriac sum of all the deviation
equals zero.

18. • Average deviation:
• Standard deviation of a finite number of data is
given by
• variance= (V)= mean square deviation = σ2

19. • Probability and errors
• The table below shows a tabulation of 50
voltage readings that were taken at small time
intervals and recorded to the nearest 0.1 V.
The nominal value of the measured voltage
was 100.0 V

20. • Table 1. Tabulation of voltage readings
Number of readings
Voltage reading, V
1
99.7
4
99.8
12
99.9
19
100.0
10
100.1
3
100.2
1
100.3
50

21. • The bell mouthed curve is known as Gaussian
curve between the limits -∞ and +∞, represents
the entire observations. The area under the curve
between +σ and – σ limits represents the cases
that differ from the mean by no more than
standard deviation. Integration of the area under
the curve withen +-σ limits gives the total
number of cases within these limits. For normally
dispersed data, following the Gaussian
distribution , approximately 68% of all cases lie
between the limits of + σ and –σ from the mean

22. • Table 2 shows that half of the cases are
included in the deviation limits of +-0.6745*σ.
The quantity r is called probable error and is
defined as:
• Probable error ® = +-0.6745*σ
• Table 2 area under the probability curve
Fraction of total area included
Deviation (+-) σ
0.5
0.6745
0.6828
1.0
0.9546
2
0.9972
3

23. Ex-11
Ten measurements of the resistance of a resistor
gave 101.2 Ω, 101.7 Ω, 101.3 Ω, 101.0 Ω,
101.5 Ω, 101.3 Ω, 101.2 Ω, 101.4 Ω, 101.3 Ω,
and 101.1 Ω. Assume only random errors are
present. Calculate (a) arithmetic mean, (b) the
standard deviation of the readings, © the
proper error.

24. • Solution
d2
d
Reading (x)
0.01
-0.1
101.2
0.16
0.4
101.7
0.00
.00
101.3
0.09
-0.3
101.0
0.04
0.2
101.5
0.00
0.0
101.3
0.01
-0.1
101.2
0.01
0.1
101.4
0.00
0.0
101.3
0.04
-0.2
101.1
∑d2= = 0.36
|d|
∑
X=1013.0
∑

25. • dXave = sum(x)/n=101.3
• Standard eviation = sqrt ( d2 / (n-1))=0.2 Ω
• Proper error = 0.6745*σ = 0.1349 Ω
• Limiting Errors
• In most indicating instruments the accuracy is
guaranteed to a certain percentage of full scale
reading. For example, if a resistance of a resistor
is given as 500 Ω+-10 percent, the manufacturer
guarantees that the resistance falls between the
limits 450 Ω and 550 Ω.

26. • Ex-12
• A 0-150 V voltmeter has a guaraneed accuracy of 1
percent full-scale reading. The voltage measured by
this instrument is 83 V. calculate the limiting error in
percent.
• Solution:
• Limiting error = 0,01*150 = 1.5 V
• The percentage errt the or at a meter indication of 83 V
is = 1.5/83 = 1.81 percent
• Note : if meter reads 30 V, so that the limiting
error=1.5/30= 5 percent

27. • Combining guarantee errors
• Ex-13
• The current passing through a resistor 100 +-0.2 Ω is 2 +-
0.01 A. using the relation P=I2R, calcultate the limiting error
in the computed value of power desipation.
• I = 2.0 +- 0.01A = 2.0 +- 0.5%
• R= 100 +-0.2 Ω =100 +- 0.2%
• In the worst possible combination limiting error
• P=I2.R= (2*0.5%)+0.2%=1.2%
• Power dissipation should then be writen as follows:
• P = (2.0)2*100=400+-1.2% = 400 +- 4.8W