Measurement and Errors

definition

Measurement of Length

A meter scale is used for lengths from

1 m m t o 100 m

. A vernier caliper is used for lengths to an accuracy of

0.1 m m

. A screw gauge and a spherometre can be used to measure lengths as small as to

0.01 m m

. To measure lengths beyond these ranges, we make use of some special indirect methods. For e.g. large distances such as the distance of a planet or a star from the earth cannot be measured directly with a meter scale. Here, we use the parallax method.

definition

Parallax Method of Measurement

Astronomers use an effect called parallax to measure distances to nearby stars. Parallax is the apparent displacement of an object because of a change in the observer's point of view.
To measure the distance

D

of a far away planet

S

by the parallax method, We observe it from two different positions (observatories)

A

and

B

on the Earth, separated by distance

A B = b

at the same time as shown in the given figure.
We measure the angle between the two directions along which the planet is viewed at these two points. The

∠ A S B

in the figure represented by symbol

θ

is called the parallax angle or parallactic angle.As the planet is very far away,

b D << 1

and therefore,

θ

is very small.
Then we approximately take

A B

as an arc of length

b

of a circle with center at

S

and the distance

D

as the radius

A S = B S

so that

A B = b = D θ

where

θ

is in radians.

D = \frac{b}{θ}

..... (1)
Having determined

D

, we can employ a similar method to determine the size or angular diameter of the planet. If

d

is the diameter of the planet and

α

the angular size of the planet (the angle subtended by

d

at the earth),
We have

α = \frac{D}{d}

..... (2)
The angle

α

can be measured from the same location on the earth. It is the angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope. Since

D

is known, the diameter

d

of the planet can be determined using

E q u a t i o n (2)

definition

Method of measurement of Time

Time is measured using a mechanical, electric or atomic clock. The cesium atomic clocks are the most accurate. Atomic clocks use the frequency of electronic transitions in certain atoms to measure the second. The unit of time is second in SI units. It is defined as

9, 192, 631, 770

cycles of the radiation that corresponds to the transition between two electron spin energy levels of the ground state of the

^{133} C s

atom.

definition

Measurement of Mass

Unified atomic mass unit (u), which has been established for expressing the mass of atoms as 1 unified atomic mass unit = 1u = (1/12) of the mass of an atom of carbon - 12 isotope including the mass of electrons =

1.66 \times 10^{- 27} k g

Mass of commonly available objects can be determined by common balance like the one used in grocery shop.

definition

Least Count Error

The smallest value that can be measured by the measuring instrument is called its least count. Measured values are good only up to this value. The least count error is the error associated with the resolution of the instrument.

Example: During Searle's experiment, zero of the Vernier scale lies between

3.20 \times 10^{- 2} m

and

3.25 \times 10^{- 2}

m of the main scale. The 20

^{t h}

division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between

3.20 \times 10^{- 2} m

and

3.25 \times 10^{- 2} m

of the main scale but now the 45

^{t h}

division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is

8 \times 10^{- 7} m^{2}

. The least count of the Vernier scale is

1.0 \times 10^{- 5} m

. What is the maximum percentage error in the Young's modulus of the wire?

Solution:

Y = \frac{F L}{l A}

since the experiment measures only change in the length of wire

∴ \frac{Δ Y}{Y} \times 100 = \frac{Δ l}{l} \times 100

From the obervation

l_{1} = M S R + 20 (L C)

(MSR-Main Scale Reading)

l_{2} = M S R + 45 (L C)

\Rightarrow

change in lengths

=

25 (LC)
and the maximum permissible percentage error in elongation is one LC

∴ \frac{Δ Y}{Y} \times 100 = \frac{(L C)}{25 (L C)} \times 100 = 4

definition

Backlash Error

Sometimes, due to wear and tear of threads of screw in instruments such as micrometer screw gauge, it is observed that on reversing the direction of rotation of the thimble, the tip of the screw does not start moving in the opposite direction at once due to slipping, but it remains stationary for a part of rotation. This causes error in observation which is called the backlash error. To avoid this, we should rotate the screw only in one direction.

definition

Random and Systematic Error

Systematic errors: The systematic errors are those errors that tend to be in one direction, either positive or negative. Systematic errors can be minimized by improving experimental techniques, selecting better instruments and removing personal bias as far as possible. Types of systematic errors are as follows:

Instrumental errors
Imperfection in experimental technique or procedure
Personal Errors

Random errors: These are the errors which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc.

definition

Accuracy of Measured Quantities

Accuracy: The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument.
For example, suppose the true value of a certain length is near

3.678 c m

. In one experiment, using a measuring instrument of resolution

0.1 c m

, the measured value is found to be

3.5 c m

, while in another experiment using a measuring device of greater resolution , say

0.01 c m

, the length is determined to be

3.38 c m

. The first measurement has more accuracy because it is closer to the true value.

definition

Precision

Precision: The closeness of agreement between replicate measurements on the same or similar objects under specified conditions.

definition

Difference between Accuracy and Precision

Accuracy: The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument.

Precision: Precision tells us to what resolution or limit the quantity is measured.

For example, suppose the true value of a certain length is near

3.678 c m

. In one experiment, using a measuring instrument of resolution

0.1 c m

, the measured value is found to be

3.5 c m

, while in another experiment using a measuring device of greater resolutio , say

0.01 c m

, the length is determined to be

3.38 c m

. The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only

0.1 c m

), while the second measurement is less accurate but more precise.

definition

Least Count

The least count of an instrument is the smallest measurement that can be taken accurately with it.

definition

Working of a simple pendulum

A simple pendulum has a heavy point mass (known as bob) suspended from a rigid support by a massless and inextensible string. When the bob from its mean position is pulled to one side and then released, the pendulum is set to motion and the bob moves alternately on either side of its mean position.

definition

Effective Length of Pendulum

It is the distance of point of oscillation (i.e. the centre of gravity of the bob) from the point of suspension.

diagram

Interpret length vs time period graphs of a simple pendulum

The following graph shows variation of time period with the length of pendulum.

definition

Factors affecting the time period of a simple pendulum

T = 2 π \sqrt{\frac{l}{g}}

where

T :

Time period,

l :

length,

g :

acceleration due to gravity
1. The time period of oscillation is directly proportional to to the square root of its effective length.
2. The time period of oscillation is inversely proportional to the square root of acceleration due to gravity.
3. The time period of oscillation does not depend on the mass or material of the body suspended.
4. The time period of oscillation does not depend on the extent of swing on either side.

definition

Vernier Constant

The Vernier Constant is equal to the difference between values of one main scale division and one vernier scale division.
Vernier Constant: Value of 1 main scale division - Value of 1 vernier scale division

definition

Calculating zero error in vernier caliper

The attached diagram shows cases of zero error in a vernier caliper.
Case (a): No zero-error
Case (b): Positive zero-error of 3 vernier scale division (3rd line coinciding). Positive zero-error correction is done by subtracting the positive zero-error from the actual reading.
Case (c): Negative zero-error of 2 vernier scale division (8th line coinciding). Negative zero-error correction is done by adding the negative zero-error from the actual reading.

definition

Measurement of length with a vernier callipers

Measurement of length with a vernier callipers:
1. Find the least count and zero error of the vernier callipers.
2. Move the jaw J2 away from the jaw J1 and place the object to be measured, between the jaws J1 and J2. Move the jaw J2 towards the jaw J1 till it touches the object. Tighten the screw S to fix the vernier scale in its position.
3. Note the main scale reading.
4. Note that division p on vernier scale which coincides or is in line with any division of the main scale. Multiply this vernier division p with the least count. This is the vernier scale reading. i.e., Vernier scale reading = p

\times

L.C.
5. Add the vernier scale reading to the main scale reading. This gives the observed length.

definition

Least count of a screw

The least count of a screw is the distance moved along the axis by it in rotating the circular scale by one division.

Least Count = \frac{Pitch}{Number of circular scale divisions}

definition

Calculating zero error in screw guage

The attached diagram shows cases of zero error in a screw guage.
Case (a): No zero-error
Case (b): Positive zero-error of 2 circular scale division. Positive zero-error correction is done by subtracting the positive zero-error from the actual reading.
Case (c): Negative zero-error of 4 circular scale division. Negative zero-error correction is done by adding the negative zero-error from the actual reading.

definition

Spherometer

A spherometer is an instrument for the precise measurement of the radius of a sphere. It is generally used for determining the radius of curvature of convex or concave mirrors and lenses. It can also be used to measure the thickness of a microscope slide or the depth of depression in a slide. The usual form consists of a fine screw moving in a nut carried on the centre of a small three-legged table or frame; the feet forming the vertices of an equilateral triangle. The lower end of the screw and those of the table legs are finely tapered and terminate in hemispheres, so that each rests on a point. If the screw has two turns of the thread to the millimetre the head is usually divided into 50 equal parts, so that differences of 0.01 millimetre may be measured without using a vernier. The radius of a spherometer is given by:

R = \frac{h}{2} + \frac{a^{2}}{6 h}

where:

h :

saggital (length along linear scale)

a :

length between two legs of spherometer
Methodology:

Place the spherometer on a flat surface and gently wind the screw downwards until it just touches the glass, as shown by one further division on the dial causing a just perceptible wobble.
The dial reading at this point could be noted, or alternatively the index may be circumferentially adjusted to zero by loosening the screw securing it to the table.
The instrument is then transferred to the lens or mirror to be measured, and the micrometer screw raised or lowered until all four points are just in contact with the glass.
The dial is then read for a second time, allowing the difference between the plane and curved settings to be found.
This procedure should be repeated in several orientations across the lens or mirror: a satisfactorily spherical shape would be proved by no change in the reading.

definition

Reporting Numbers

In scientific notation all numbers are written in the form:

m \times 10^{n}

where,

n

is the power.

definition

Significant Digits

Every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first digit are known as significant digits or significant figures.
Example: If we say that the period of oscillation of a simple pendulum is

1.62

s, the digits 1 and 6 are reliable and certain, while the digit 2 is uncertain. Thus, the measured value has three significant figures.

definition

Rules for determining number of significant digits

Rules are as follows:

All the non-zero digits are significant
All the zeros between 2 non-zero digits are significant, no matter where the decimal point is.
If the number is less than 1, the zeros on the right side of decimal point but to the left of the first non-zero digit are not significant (i.e leading zeros are never significant).
In a number with a decimal point, trailing zeros, those to the right of the last non-zero digit, are significant.
The trailing zeros in a number without a decimal point, are not significant.
The trailing zeros in a number with a decimal point, are significant.

For example in 1.001 number of significant digits is 4.

definition

Rules for arithmetic operations with significant figures

As there are rules for determining the number of significant figures in directly measured quantities, there are rules for determining the number of significant figures in quantities calculated from these measured quantities.
Only measured quantities figure into the determination of the number of significant figures in calculated quantities. Exact mathematical quantities like in the formula for the area of a circle with radius

r

r^{2}

has no effect on the number of significant figures in the final calculated area. Similarly the in the formula for the kinetic energy of a mass

m

with velocity

v

m v^{2}

, has no bearing on the number of significant figures in the final calculated kinetic energy. The constants and are considered to have an infinite number of significant figures.
For quantities created from measured quantities by multiplication and division, the calculated result should have as many significant figures as the measured number with the least number of significant figures.

For example,

1.001 + 2.03 = 3.031

Number of significant figures are

4

and

3

in the numbers added and after addition of numbers it's

4

definition

Rules of Rounding off

The rule by convention is that the preceding digit is raised by

1

if the insignificant digit to be dropped is more than

5

, and is left unchanged if the latter is less than

5

. In the case where the insignificant digit is

5

, the convention is that if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by

1

definition

Absolute Error

The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement.

definition

Mean Absolute Error

The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. The arithmetic mean of all the absolute error is taken as the mean absolute error of the value of the physical quantity.

definition

Relative Error

The relative error is the ratio of the mean absolute error (

δ a_{m e a n}

) to the mean value (

a_{m e a n}

) of the quantity measured.
Relative error

= \frac{δ a_{m e a n}}{a_{m e a n}}

definition

Percentage Error

When the relative error is expressed in percent, it is called the percentage error (

δ a

∴

Percentage error (

δ a

)

= \frac{δ a_{m e a n}}{a_{m e a n}} \times 100

formula

Absolute, Relative or Percentage Error for a quantity that is sum of measured quantities

y = y_{1} + y_{2} + y_{3} . . y_{n}

Then absolute error will be given by:

Δ y = Δ y_{1} + Δ y_{2} + Δ y_{3} + . . Δ y_{n}

And its relative error will be given by

\frac{Δ y}{y}

example

Absolute, Relative or Percentage Error for a quantity that is product of measured quantities

y = y_{1} \times y_{2}

Relative error is given by:

\frac{Δ y}{y} = \frac{Δ y_{1}}{y_{1}} + \frac{Δ y_{2}}{y_{2}}

Absolute error will be given by measuring

Δ y

value.

example

Absolute, Relative or Percentage Error for a quantity which is a measured quantity raised to a power

y = y_{1}^{a}

Relative error is given by:

\frac{Δ y}{y} = a \frac{Δ y_{1}}{y_{1}}

Absolute error will be given by measuring

Δ y

value.

definition

Absolute, Relative or Percentage Error

y = y_{1}^{a} \times y_{2}^{b}

Relative error is given by:

\frac{Δ y}{y} = a \frac{Δ y_{1}}{y_{1}} + b \frac{Δ y_{2}}{y_{2}}

Absolute error will be given by measuring

Δ y

value.

definition

Reporting errors in measured quantities using rules of significant digits and rounding off

Example: Each side of a cube is measured to be

7.203 m

. What are the total surface area and the volume of the cube to appropriate significant figures?
Solution:
Surface area of the cube

= 6 (7.203)^{2} m^{2} = 311.299254 m^{2} = 311.3 m^{2}

Volume of the cube

= (7.203)^{3} m^{3} = 373.714754 m^{3} = 373.7 m^{3}

TB - UG

Monday, July 31, 2017