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LAB NUMBER 1
VERNIERS, MICROMETERS AND UNCERTAINITIES
INSTRUCTIONS FOR EXPERIMENTS
ANAND M. SHARAN SEPT 7, 2010
VERNIER CALIPER
MICROMETER READINGS
2 of 6
1
CALCULATING UNCERTAINTIES IN EXPERIMENTS
The algebra is as follows:
A Quality MeasuredUA Uncertainty of A
Suppose there are several measurements - A, B, C, D.....then
Addition
1) A B U B∠ ± ∠UA
= ( ) ( )A B U UA B+ ∠ +
SUBTRACTION
2) ( ) ( )A U B UA B U U
A B
A B
∠ − ∠
= − ∠ +
MULTIPLICATION
3) ( ) ( )( ) ( )
A U B U
AB U UA B
A B
∠ × ∠
= ∠ +
DIVISION
4) ( )A UB U
AB
U UA
BA B
∠∠
= ⎛⎝⎜
⎞⎠⎟
∠ +
5) A constant has zero uncertainty. For example, the acceleration due to gravity,
( )g m s
g Ug
=
∠ =
9 81 2
0
. / ^
3 of 6
2
HOW TO EXPRESS UNCERTAINTY
Let us look at the following instruments for measurements.
Lab #1 Uncertainties
Metrology: The science of measurement.
Length Measurements:
RangeResolution
Smallest Main ScaleGraduation
RepeatabilityUncertainty
Rule 0-300 mm 1 mm 1 mm±
Vernier Caliper 0-150 mm 0.05 mm 0.05 mm±
Micrometer 0-25 mm 0.002 mm 0.002 mm±
Examples:
length of 10 mm (ruler) = 10 mm 1 mm or 10 mm 10%± ±
EXAMPLELet us look at the application of the above in the measurement of Stress-Strain curve. Thereadings are as follows:
4 of 6
3
d0 d1 Load Instrument Instrument
1.2 1.1 4000kg MicrometerResolution
(0.002)
BalanceResolution
(1kg)
strain = ( ) ( )d dd
o
o
−=
−=
−11 1212
0112
. ..
..
11 0 00211
12 0 00212
12 0 00212
. ..
. ..
. ..
∠⎛⎝⎜
⎞⎠⎟
⎧⎨⎩
⎫⎬⎭
− ∠⎛⎝⎜
⎞⎠⎟
∠⎛⎝⎜
⎞⎠⎟
( )( )
11 0 0001818 12 0 00166612 0 001666
. . . .. .
∠ − ∠
∠
( )= ∠0 08333 0 005138. .
stress = forceArea d
Lo=
400 ∠1
400⎛⎝⎜
⎞⎠⎟ × 9.81∠
= ×
0
431416
412 2 0 00166
22π
.( . ( * . )
=( )
( )( ){ }3924 0 0025 0
0 7854 12 0 0016 12 0 0016
∠ +
× ∠ ∠
.
. . . . .
= ( )4996 0 0025 0 0 0016 0 0016∠ + + +. . .
= 4996 0 0057∠ . Nm2
Modulus of Elasticity = StressStrain
=∠∠
4996 0 00570 0833 0 005738
.. .
5 of 6
= (4996/0.08333) ( 0.0057 + 0.005138 ) = 59954.39818 0.010838 It can also be expressed as = 59954.39818 ± [ (59954.39818 ) x( 0.010838 ) ] = 59954.39818 ± [ 649.7857675 ] = 60604.18395 Maximum ; 59304.61241 Minimum
VERNIER CALIPER
MICROMETER READINGS
2 of 6
1
CALCULATING UNCERTAINTIES IN EXPERIMENTS
The algebra is as follows:
A Quality MeasuredUA Uncertainty of A
Suppose there are several measurements - A, B, C, D.....then
Addition
1) A B U B∠ ± ∠UA
= ( ) ( )A B U UA B+ ∠ +
SUBTRACTION
2) ( ) ( )A U B UA B U U
A B
A B
∠ − ∠
= − ∠ +
MULTIPLICATION
3) ( ) ( )( ) ( )
A U B U
AB U UA B
A B
∠ × ∠
= ∠ +
DIVISION
4) ( )A UB U
AB
U UA
BA B
∠∠
= ⎛⎝⎜
⎞⎠⎟
∠ +
5) A constant has zero uncertainty. For example, the acceleration due to gravity,
( )g m s
g Ug
=
∠ =
9 81 2
0
. / ^
3 of 6
2
HOW TO EXPRESS UNCERTAINTY
Let us look at the following instruments for measurements.
Lab #1 Uncertainties
Metrology: The science of measurement.
Length Measurements:
RangeResolution
Smallest Main ScaleGraduation
RepeatabilityUncertainty
Rule 0-300 mm 1 mm 1 mm±
Vernier Caliper 0-150 mm 0.05 mm 0.05 mm±
Micrometer 0-25 mm 0.002 mm 0.002 mm±
Examples:
length of 10 mm (ruler) = 10 mm 1 mm or 10 mm 10%± ±
EXAMPLELet us look at the application of the above in the measurement of Stress-Strain curve. Thereadings are as follows:
4 of 6
3
d0 d1 Load Instrument Instrument
1.2 1.1 4000kg MicrometerResolution
(0.002)
BalanceResolution
(1kg)
strain = ( ) ( )d dd
o
o
−=
−=
−11 1212
0112
. ..
..
11 0 00211
12 0 00212
12 0 00212
. ..
. ..
. ..
∠⎛⎝⎜
⎞⎠⎟
⎧⎨⎩
⎫⎬⎭
− ∠⎛⎝⎜
⎞⎠⎟
∠⎛⎝⎜
⎞⎠⎟
( )( )
11 0 0001818 12 0 00166612 0 001666
. . . .. .
∠ − ∠
∠
( )= ∠0 08333 0 005138. .
stress = forceArea d
Lo=
400 ∠1
400⎛⎝⎜
⎞⎠⎟ × 9.81∠
= ×
0
431416
412 2 0 00166
22π
.( . ( * . )
=( )
( )( ){ }3924 0 0025 0
0 7854 12 0 0016 12 0 0016
∠ +
× ∠ ∠
.
. . . . .
= ( )4996 0 0025 0 0 0016 0 0016∠ + + +. . .
= 4996 0 0057∠ . Nm2
Modulus of Elasticity = StressStrain
=∠∠
4996 0 00570 0833 0 005738
.. .
5 of 6
= (4996/0.08333) ( 0.0057 + 0.005138 ) = 59954.39818 0.010838 It can also be expressed as = 59954.39818 ± [ (59954.39818 ) x( 0.010838 ) ] = 59954.39818 ± [ 649.7857675 ] = 60604.18395 Maximum ; 59304.61241 Minimum