notes prepared by subhankar Karmakar
Accuracy and precision:
Accuracy: it refers to the closeness of a measurement to the true value of the the physical quantity. It indicates the relative freedom from errors. As we reduce the errors, the measurement becomes more accurate.
Precision: it refers to the resolution or the limit to which the quantity is measured. Precision is determined by the least count of the measuring instrument. The smaller the list count, greater is the precision.
Errors in measurement:
The error in a measurement is equal to the difference between the true value and the measured value of the quantity.
Error = true value - measured value
Different types of errors:
1. Constant errors
2. Systematic errors
a. Instrumental errors
b. Imperfection in experimental technique
c. Personal errors
d. Errors due to external causes
3. Random errors
4. Least count error
5. Gross errors or mistakes
1. Constant errors: the errors which affect each observation by the same amount are called constant errors.
2. Systematic errors: the errors which tend to occur in one direction, either positive or negative, are called systematic errors. Systematic errors are classified as follows:
a. Instrumental errors: These errors occur due to the inbuilt defect of the measuring instrument.
b. Imperfections in experimental technique: These errors are due to the limitations of the experimental arrangement.
c. Personal errors: These errors arise due to to individual's bias, lack of proper setting of apparatus or individual's carelessness in taking observations without observing proper precautions.
d. Errors due to external causes: These errors arise due to the the change in external conditions.
3. Random errors: The errors which occur irregularly and at random, in magnitude and direction, are called random errors.
4. Least count error: This error is due to the limitation imposed by the the least count of the measuring instrument.
5. Gross errors or mistakes: These errors are due to either carelessness of the person or due to improper adjustment of the apparatus.
Different types of error measurement:
a. True value of a physical quantity: arithmetic mean of all the measurements can be taken as the true value of the measured quantity.
If a₁, a₂, a₃, a₄, a₅ ...... aₙ be the n measured values of a physical quantity, then is true value
aₘₑₐₙ or ā = (a₁+ a₂ + a₃ + a₄ + a₅ +......+ aₙ )/n
b. Absolute Error: The magnitude of the difference between the true value of the quantity measured and the individual measured value is called absolute error.
|∆a₁| = |ā - a₁|
|∆a₂| = |ā - a₂|
|∆a₃| = |ā - a₃|
............................
|∆aₙ| = |ā - aₙ|
c. Mean or final absolute error:
The arithmetic mean of the positive magnitudes of all the absolute errors is called mean absolute error. It is given by
∆ā = (|∆a₁|+ |∆a₂| + |∆a₃| +......+ |∆aₙ| )/n
The final result of the measure of a physical quantity can be expressed as
a = ā ± ∆ā
d. Relative error:
The ratio of the mean absolute error to the true value of the measured quantity is called relative error.
Relative error, δa = ∆ā /ā
e. Percentage error:
The relative error expressed in percent is called percentage error.
Percentage Error = (∆ā/ā) x 100%
COMBINATION OF ERRORS:
a. Error in the sum of two quantities:
Let ∆A and ∆B be the absolute errors in the two quantities A and B respectively. Then,
Measured value of A = A ± ∆A
Measured value of B = B ± ∆B
Consider the sum, Z = A + B
The error ∆Z in Z is then given by
Z ± ∆Z = (A ± ∆A) + (B ± ∆B)
= (A + B) ± (∆A + ∆B)
= Z ± (∆A + ∆B)
∴ ∆Z = (∆A + ∆B)
b. Error in the difference of two quantities
Consider the difference, Z = A - B
The error ∆Z in Z is then given by
Z ± ∆Z = (A ± ∆A) - (B ± ∆B)
= (A - B) ± ∆A ∓ ∆B
= Z ± ∆A ∓ ∆B
For error ∆Z to be maximum, ∆A and ∆B must have the same sign, therefore
∴ ∆Z = (∆A + ∆B)
c. Error in the product of two quantities:
Consider the product , Z = AB
The error ∆Z in Z is given by
Z ± ∆Z = (A ± ∆A)(B ± ∆B)
= AB ± A∆B ± B∆A ± ∆A. ∆B
Dividing LHS by Z and RHS by AB [∵ Z = AB]
1 ± ∆Z/Z = 1 ± ∆B/B ± ∆A/A ± (∆A/A)(∆B/B)
As the last term is very small, it can be neglected.
± ∆Z/Z = ± (∆B/B + ∆A/A)
∴ ∆Z/Z = (∆B/B + ∆A/A)
d. Error in the division or quotient
Consider the product , Z = A/B
The error ∆Z in Z is given by
Z ± ∆Z = (A ± ∆A)/(B ± ∆B)
= A(1 ± ∆A/A)/{B(1 ± ∆B/B)}
= (A/B)(1 ± ∆A/A)(1 ± ∆B/B)⁻¹
= Z(1 ± ∆A/A)(1 ∓ ∆B/B)
[∵ (1 + x)⁻¹ ≃ 1 + nx when x <<1]
Dividing both sides by Z
1 ± ∆Z/Z = 1 ∓ ∆B/B ± ∆A/A ± (∆A/A)(∆B/B)
As the last term is very small, it can be neglected.
∴ ∆Z/Z = (∆B/B + ∆A/A)
e. 1. Error in the power of a quantity:
Consider. Z = Aⁿ
The error ∆Z in Z is given by
Z ± ∆Z = (A ± ∆A)ⁿ = Aⁿ (1 ± ∆A/A)ⁿ
= Z (1 ± n∆A/A)
[∵ (1 + x)⁻¹ ≃ 1 + nx when x <<1]
Dividing both sides by Z, we get
1 ± ∆Z/Z = 1 ± n(∆A/A)
or ∆Z/Z = n(∆A/A)
2. General rule:
Consider. Z = Pᵃ Qᵇ / Rᶜ
Then ∆Z/Z = a(∆P/P) + b(∆Q/Q) + c(∆R/R)
Numericals :
Q1. The length of a rod as measured in an experiment was found to be 2.48 m, 2.46 m, 2.49 m, 2.50 m, 2.48 m. Find the (a) average length, (b) the absolute error in each observation and (c) the percentage error.
Soln. (a) Average length
= (2.48 + 2.46 + 2.49 + 2.50 + 2.48)/5
= 12.41/5 = 2.482 = 2.48
∴ true length, ā = 2.48 m
(b) The absolute errors in different measurements are:
|∆a₁| = |ā - a₁| = |2.48 - 2.48| = 0.00 m
|∆a₂| = |ā - a₂| = |2.48 - 2.46| = 0.02 m
|∆a₃| = |ā - a₃| = |2.48 - 2.49| = 0.01 m
|∆a₄| = |ā - a₄| = |2.48 - 2.50| = 0.02 m
|∆a₅| = |ā - a₅| = |2.48 - 2.48| = 0.00 m
(c) the absolute error, |∆ā|
= (0.00 + 0.02 + 0.01+ 0.02 + 0.00)/5
= 0.01 m
∴ correct length, ā ± |∆ā| = 2.48 ± 0.01 m
∴ percentage error = (0.01/2.48)x 100%
= 0.40%
Q2. In successive measurements, the readings of the period of oscillation of a simple pendulum were found to be 2.63 s, 2.56 s, 2.42 s, 2.71 s and 2.80 s in an experiment. Calculate (a) mean value of the period of oscillation
(b) absolute error in each measurement
(c) mean absolute error
(d) relative error
(e) percentage error and
(f) Express the result in proper form.
Soln. (a) mean period of oscillation
= (2.63 + 2.56 + 2.42 + 2.71 + 2.80)/5
= 13.12/5 = 2.624 s ≃ 2.62 s
(b) absolute errors in different measurement,
|∆a₁| = |ā - a₁| = |2.62 - 2.63| = 0.01 s
|∆a₂| = |ā - a₂| = |2.62 - 2.56| = 0.06 s
|∆a₃| = |ā - a₃| = |2.62 - 2.42| = 0.20 s
|∆a₄| = |ā - a₄| = |2.62 - 2.71| = 0.09 s
|∆a₅| = |ā - a₅| = |2.62 - 2.80| = 0.18 s
(c) mean absolute error, |∆ā|
= (0.01 + 0.06 + 0.20 + 0.09 + 0.18)/5
= 0.11 s
(d) relative error δā = |∆ā|/ā
= 0.11/2.62 = 0.04
(e) percentage error = 0.04 x 100% = 4%
(f) in terms of absolute error,
(2.62 ± 0.11) s
In terms of percentage error,
(2.62 ± 4%) s.
Homework:
Q3. In an experiment, refractive index of glass was observed to be 1.45, 1.56, 1.54, 1.44, 1.54 and 1.53. Calculate (a) mean value of refractive index, (b) mean absolute error, (c) fractional error aur relative error, (d) percentage error,
(e) express the result in terms of absolute error and percentage error.
Q4. In an experiment to measure focal length of a concave mirror, the value of focal length in successive observations turns out to be 17.3 cm, 17.8 cm, 18.3 cm, 18.2 cm, 17.9 cm and 18.0 cm. Calculate the mean absolute error and percentage error. Also, express the result in a proper way.
Numericals on combination of errors:
Q5. Two resistances R₁ = 100 ± 3 Ω and R₂ = 200 ± 4 Ω are connected in series. What is their equivalent resistance?
Q6. Two different masses are determined as (23.7 ± 0.5) g and (17.6 ± 0.3) g. What is the sum of their masses?
Q7. The initial and final temperatures of a water bath are (18 ± 0.5)°C and (40 ± 0.3)°C. What is the rise in temperature of the bath?
Q8. The resistance R =V/I, where V = 100 ± 5 V and I = 10 ± 0.2 A. Find the percentage error in R.
Q9. The percentage errors in the measurement of mass and speed are 2% and 3% respectively. How much will be the maximum error in the estimate of kinetic energy obtained by measuring mass and speed?
Q10. The length, breadth and height of a rectangular block of wood were measured to be :
l = 12.13 ± 0.02 cm;
b = 8.16 ± 0.01 cm and
h = 3.46 ± 0.01 cm
Determine the percentage error in the volume of the block.
Q11. The period of oscillation of a simple pendulum is T = 2π √(L/g). Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g?
Q12. A physical quantity X is given by
X = (a²b³)/(c√d). If the percentage errors of measurement in a, b, c and d are 4%, 2%, 3% and 1% respectively, then calculate the percentage error in X.
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