Unit 1: Measurements — Long Questions

Physical Quantities

The quantities which are measured accurately are called physical quantities. These are such quantities in terms of which the laws of physics are expressed. These are mass, length, time, velocity, force, density, temperature, electric current, and numerous others.

Classification of Physical Quantities

Physical quantities are often divided into two categories: base quantities and derived quantities.

i. Base Quantities:

• The base quantities are the independent physical quantities in terms of which the other physical quantities can be defined.
• Base quantities are not defined in terms of other physical quantities.
• The measurement of a base quantity involves two steps: first, the choice of a standard, and second, the establishment of a procedure for comparing the quantity to be measured with the standard.

Examples

Length, mass and time etc.

ii. Derived Quantities

Derived quantities are those which depend on base quantities.

Examples

Velocity, acceleration, force, etc.

Note

Measurements must be reliable and accurate so that they can be used, easily and effectively.

International System of Units

In 1960 international committee agreed on a set of definitions and standards to describe the physical quantities.

The system that was established is called the System international (SI). SI units are used by the world's scientific community and by almost all nations. The system international (SI) consists of two kinds of units: base units and derived units.

Base Units

The base units are the independent units in terms of which the other units can be defined.

• Base units are not defined in terms of other units.
• There are seven base units in the international system of units.

Examples

For Example, meter, second, kilogram etc.

The names of base units for these physical quantities together with symbols are listed in Table 1.1.

Derived Units

Derived units are those units which depend on the base units.

Examples

N (Newton), J (Joule), m/s (meter per second) etc.

The units of plane angle and solid angle have also been included in the list of derived units since 1995.

Additional Units

In addition to base and derived units, the SI permits the use of certain additional units, including:

• The traditional mathematical units for measuring angles (degree, arcminute, and arc second).
• The traditional units for standard time are (minute, hour, day, and year).
• The logarithmic units bel (and its multiples, such as the decibel).
• Two metric units commonly used in ordinary life: the litre for volume and the tonne (metric ton) for large masses.
• Two non-metric scientific units are atomic mass unit (μ) and the electron volt (eV).
• The nautical mile and knot; units traditionally used at sea and in meteorology.
• The acre and hectare, common metric units of land area.
• The bar is a unit of pressure and it is commonly used as the milli bar in meteorology and the kilo bar in engineering.
• The angstrom and the barn, units used in physics and astronomy.

Scientific Notation

Numbers are expressed in standard form called scientific notation, which employs powers of ten. The internationally accepted practice is that there should be only one non-zero digit left of decimal.

Examples

The number 134.7 should be written as 1.347 × 10² and 0.0023 be expressed as 2.3 × 10⁻³.

Prefixes

Most prefixes indicate order of magnitude in steps of 1000 and provide a convenient way to express large and small numbers, to eliminate non-significant digits. SI also includes four of the other prefixes to accommodate usage already established before the introduction of SI (Table 1.3). They are centi (10⁻²), deci (10⁻¹), deca (10¹) and hecto (10²).

Conventions for Using SI Units

Use of SI units require special care, more particularly in writing prefixes. Some points to note are:

1. Each SI is represented by a symbol not an abbreviation. These symbols are the same in all languages. Hence, correct use of the symbol is very important.

For example For ampere, we should use "A" not "amp"; for seconds "s" not "sec", SI not S.I.

2. Full name of unit does not begin with capital letter.

For example newton, metre, etc., except Celsius.

3. Symbols appear in lower case.

For example "m" for metre, "s" for second, etc., exception "L" for litre.

4. Symbols named after scientists have initial letters capital.

For example "N" for newton, "Pa" for pascal, "W" for watt.

5. Symbols and prefixes are printed in upright (roman) style regardless of the type style in surrounding text.

For example A distance of 50 m.

6. Symbols do not take plural form.

For example 1mm, 100 mm, 1 kg, 60 kg.

7. No full stop or dot is placed after the symbol except at the end of the sentence.

8. Prefix is written before and without space to base unit.

For example "mL" not m L or "ms" not m s.

9. Base units are written one space apart. Leave a space even between the number (value) and the symbol.

For example 1 kg, 10 m s⁻¹, etc.

10. Compound prefixes are not allowed:

For example 1μμF should be 1 pF.

11. When base unit of multiple is raised to a power, the power applies to whole multiple and not to base unit alone.

For example 1 km² = 1 (km)² = 1 x(10³m)² = 1x10⁶ m².

12. Use negative index notation (m s⁻¹) instead of solidus (m/s).

13. Use scientific notation, that is, one non-zero digit left of decimal.

For example 143.7=1.437 × 10².

14. Do not mix symbols and names in the same expression.

For example metre per second or m s⁻¹, not metre/second or m/second.

15. Practical work should be recorded in most convenient units depending upon the instruments being used.

For Example Measurements using screw gauge should be recorded in mm but the final results must be recorded to the appropriate base units.

16. System International do not allow the use of former CGS System units such as dyne, erg, gauss, poise, tor, etc.

Uncertainty in measurement

We can count the number of pages of a book exactly but measurement of its length needs some measuring instrument. Every instrument is calibrated to a certain smallest division mark on it and this fact puts a limit regarding its accuracy. When we take a reading with one instrument, its limit of measurement is the smallest division or graduation on its scale. Hence, every measured quantity has some uncertainty about its value. When a measurement is made, it is taken to the nearest graduation or marking on the scale. We can estimate the maximum uncertainty as being one smallest division of the instrument. This is called absolute uncertainty. It is one milli metre on a metre rule that is graduated in milli metres.

Example

For example, if one edge of the book coincides with 10.0 cm mark and the other with 33.5 cm, then the length with uncertainty is given by

(33.5 ± 0.05) cm - (10.0 ± 0.05) cm (23.5 ± 0.1) cm

It means that the true length of the book is in between 23.4 cm and 23.6 cm. Hence, the maximum uncertainty is ±0.05 cm, which is equivalent to an uncertainty of 0.1 cm. In fact, it is equal to least count of the metre rule. Uncertainty may be recorded as:

Fractional uncertainty = Absolute uncertainty / Measured value

or

Percentage uncertainty = (Absolute uncertainty / Measured value) × 100%

Uncertainty in Digital Instruments

Some modern measuring instruments have a digital scale. We usually estimate one digit beyond what is certain: with a digital scale, this is reflected in some fluctuations of the last digit. If the last digit fluctuates by 1 or 2, write down that last digit. If fluctuation is more than 2 or so in the last digit, it may mean that the reading is being influenced by some factor such as air currents. Regardless of the reason, a large fluctuation may mean that the displayed digit is not really significant.

The indication of uncertainty in a recorded value has been simplified using significant figures. If a measurement is recorded using the knowledge of significant figures, then its last digit, which is an estimation, is an indication of the accuracy of the recorded value.

Use of Significant Figures

Definition

In any measurement, the accurately known digits and the first estimated or doubtful digit are called significant figures.

Explanation

The number of digits of a measurement about which we do feel reasonably sure are called significant figures. In fact, they reflect the use of actual instrument used for that measurement. While using a calculator, the result of any calculation contains many digits after the décimal point. The additional digits may mislead another person who uses those figures into believing them. Hence, they are to be rounded off to the correct number of significant figures. This can be done by keeping in view the uncertainty or the least count of the instrument while recording observations and also quoting results of any calculations to the correct numbers of significant figures. It is better to quote the result in scientific notation to avoid any ambiguity regarding the number of significant figures.

Example

For example, weighing the same object with different balances:

Electronic balance mass = 3.145 ± 0.001 g
Lever balance : mass = 3.1 ± 0.1 g

Usually, the uncertainty ± 0.001 g or ±0.1 g is dropped, and it is understood that the number quoted has an uncertainty of at least 1 unit in the last digit. All digits which are quoted are called significant figures. Proper use of significant figures ensures that we correctly represent the uncertainty of our measurements. For example, scientists immediately realize that the reported mass 3.145 g is more accurate than a reported mass of 3.1 g, reflecting the use of a better or more precise instrument. As we improve the quality of our measuring instrument and techniques, we extend the result to more and more significant figures and correspondingly improve the experimental accuracy of the result.

Rules for significant figures

i. All digits 1,2,3,4,5,6,7,8,9 are significant.

ii. Zeros may or may not be significant. In case of zeros, the following rules may be adopted:

a) A zero between two significant figures is itself significant. For example, in 1.406, the number of significant figures is four.

b) Zeros to the left of significant figures are not significant. For example, none of the zeros in 0.00467 or 02.59 is significant.

c) Zeros to the right of a significant figure may or may not be significant. For example, the number of significant figures in 2.450 is four.

In decimal fraction, zeros to the right of a significant figure are significant. For example, all the zeros in 3.570 or 7.4000 are significant. However, in integers such as 8,000 kg, the number of significant zeros is determined by the precision of the measuring instrument. If the measuring scale has a least count of 1 kg, then there are four significant figures written in scientific notation as 8.000 x 10³ kg. If the least count of the scale is 10 kg, then the number of significant figures will be 3 written in scientific notation as 8.00 x 10³ kg and so on.

iii. When a measurement is recorded in scientific notation or standard form, the figures other than the powers of ten are significant figures. For example, a measurement recorded as 8.70 x 10⁴ kg has three significant figures.

iv. Multiplying or dividing numbers:

Keep a number of significant figures in the product or quotient not more than that contained in the least accurate factor i.e., the factor containing the least number of significant figures. For example, the computation of the following using a calculator, gives

(5.348×10⁻² × 3.64×10⁴) / 1.336 = 1.45768982×10³ = 1.46 × 10³

As the factor 3.64 x 10⁴, the least accurate in the above calculation has three significant figures, the answer should be written to three significant figures only. The other figures are insignificant and should be deleted. While deleting the figures, the last significant figure to be retained is rounded off.

Limitations of Significant Figures

Significant figures deal with only one source of uncertainties that inherent in reading the scale. Real experimental uncertainties have many contributions, including personal errors and sometimes hidden systematic errors. One cannot do better than that what the scale reading allows, but the total uncertainty may well be more than what the significant figure of the measurements would suggest.

Rules for rounding of figures

Following rules are followed for rounding off figures:

(i) If the first digit dropped is less than 5, the last digit retained should remain unchanged.

e.g; 3.42 is rounded off as 3.4.

(ii) If the first digit dropped is more than 5, the digit to be retained is increased by one.

(iii) If the first digit dropped is more than 5, then there are two cases:

(a) If digit next to 5 is zero or there is no digit next to 5 then we follow odd-even rule. i.e. if retained digit is even, it remains unchanged. But if retained digit is odd, it is increased by 1.

(b) If digit next to 5 is a non-zero digit, the digit to be retained is increased by 1.

Example

For example, the following numbers are rounded off to three significant figures as follows. The digits are deleted one by one.

43.75 is rounded off as 43.8
56.8546 is rounded off as 56.9
73.650 is rounded off as 73.6
64.350 is rounded off as 64.4

(iv) In adding or subtracting numbers:

The number of decimal places retained in the answer should be equal to the smallest number of decimal places in any of the quantities being added or subtracted. In this case, the number of significant figures is not important. It is the position of decimal that matters. For example, suppose we wish to add the following quantities expressed in metres.

Examples

(i) 72.1 (ii) 2.7543
3.42 4.10
0.003 1.273
75.523 8.1273

Correct answer 75.5 m 8.13 m

In case (i), the number 72.1 has the smallest decimal places, thus the answer is rounded off to the same position which is then 75.5 m. In case (ii), the number 4.10 has the smallest number of decimal places and hence, the answer is rounded off to the same decimal positions which is then 8.13 m.

Limitations of Significant Figures

Significant figures deal with only one source of uncertainties that inherent in reading the scale. Real experimental uncertainties have many contributions, including personal errors and sometimes hidden systematic errors. One cannot do better than that what the scale reading allows, but the total uncertainty may well be more than what the significant figure of the measurements would suggest.

Precision and accuracy

The terms precision and accuracy are frequently used in physics measurements. They should be distinguished clearly. The precision of a measurement is determined by the instrument or device being used.

Precision

"Precision of a measurement is defined as the least count of the measuring instrument." Precision is also called the Absolute Uncertainty. The smaller the least count the more precise is the measurement.

Accuracy

Accuracy is defined as the closeness of a measurement of a physical quantity. It is expressed by the fractional or percentage uncertainty. The smaller the fractional or percentage uncertainty, the more accurate is the measurement.

Explanation

For example, the length of an object is recorded as 25.5 cm by using a metre rule having smallest division in milli metre. Its precision or absolute uncertainty (least count) = ± 0.1 cm.

Fractional uncertainty = 0.1 cm / 25.5 cm = 0.004

Percentage uncertainty = (0.1 cm / 25.5 cm) × 100 = 0.4%

Another measurement taken by Vernier Callipers with least.count 0.01 cm is recorded as 0.45 cm. it has precision or absolute uncertainty (least count) = ± 0.01 cm.

Fractional uncertainty = 0.01 cm / 0.45 cm = 0.02

Percentage uncertainty = (0.01 cm / 0.45 cm) × 100 = 2%

Thus, the reading 25.5 cm taken by metre rule is although less precise but is more accurate having less percentage uncertainty or error.

Whereas the reading 0.45 cm taken by Vernier Callipers is more precise but is less accurate. In fact, it is the relative measurement which is important. The smaller the physical quantity, the more precise instrument should be used. Here the measurement 0.45 cm demands that a more precise instrument, such as micrometer screw gauge, with least count 0.001 cm, should have been used. Hence, we can conclude that:

Precise measurement and accurate measurement:

A precise measurement is the one which has less precision or absolute uncertainty and an accurate measurement is the one which has less fractional or percentage uncertainty.

Assessment of Total Uncertainty in The Final Result

Knowing the uncertainties in all the factors involved in a calculation, the maximum possible uncertainty or error in the final result can be found as follows:

(i) For addition and Subtraction:

Absolute uncertainties are added in addition and subtraction.

Example

For example, the distance between two positions x₁ = 15.4 ± 0.1 cm and x₂ = 25.6 cm ± 0.1 cm is recorded as:

x = x₂ - x₁,= 10.2 ± 0.2 cm

and addition of two lengths is:

ℓ₁ = 8.5 ± 0.1 cm and ℓ₂ = 12.6± 0.1 cm recorded as:

ℓ = ℓ₁ + ℓ₂ = 21.1 ± 0.2 cm

(ii) For Multiplication and division:

Percentage uncertainties are added in multiplication and division.

Example

For example, the maximum possible uncertainty in the value of resistance R of a conductor determined by the potential difference applied across the conductor resulting in current flowing through it is estimated as under:

Let V = 3.4 ± 0.1 V
I = 0.68 ± 0.05 A
using R = V/I

Percentage uncertainty in V = (0.1V / 3.4 V) × 100% = 3%

Percentage uncertainty in I = (0.5 A / 0.68 A) × 100% = 7%

Hence, total percentage uncertainty in the value of R is 3 + 7 = 10%

The value of R will be written as:

R = (3.4 V) / (0.68 V) = 5.0 ohm

So, 5.4 ohm ± 10%

Hence, R = 5.0 ± 0.5 ohms, uncertainty being an estimate only, is recorded by one significant figure

(iii) For Power Factor

The percentage uncertainty is multiplied by the power factor in the formula.

Example

For example, the calculation of cross-sectional area of a cylinder of radius r =1.25 cm using formula for Area A = πr² is given by the %age uncertainty which is A = 2 × %age uncertainty in radius r. As uncertainty is multiplied by power factor, it increases the precision demand of measurement. When the radius of a small sphere is measured as 1.25 cm by Vernier Callipers with least count 0.01 cm, then

The radius r is recorded as r = 1.25 ± 0.01 cm

%age uncertainty in radius r is r = (0.01 / 1.25) ×100 = 0.8%

Total percentage uncertainty in area A = 2 × 0.8 = 1.6%

Thus A = πr²

= 3.14 (1.25)² = 4.906 cm² with 1.6% uncertainty

Thus, the result should be recorded as A = 4.91 ± 0.08 cm²

Dimensions of Physical quantities

Definition

"The representation of a quantity in base units in the form of symbols within square brackets."

Any physical quantity can be described by certain familiar properties such as length, mass, time, temperature, electric current, etc. These measurable properties are called dimensions. Dimensions deal with the qualitative nature of a physical quantity in terms of fundamental quantities.

Examples

The quantities such as length, depth, height, diameter, light year are all measured in metre and denoted by the same dimension, basically known as length given by symbol L written within square bracket [L]. Similarly, the other fundamental quantities, mass, time, electric current and temperature are denoted by specific symbols [M], [T], [A] and [θ], respectively. These five dimensions have been chosen as being basic because they are easy to measure in experiments.

Explanation

The dimensions of other quantities indicate how they are related to the basic quantities are combination of fundamental dimensions. For example, speed v is measured in metres per second, so it has the dimensions of length [L] divided by time [T].

[v] = [L] / [T] = [L][T⁻¹] = [LT⁻¹]

As the acceleration a = Δv / Δt

Dimensions of acceleration are

[a] = [v] / [T] = [LT⁻¹][T⁻¹] = [LT⁻²]

Also, dimensions of force can be written as

[F] = [m][a] = [M][LT⁻²] = [MLT⁻²]

Benefits (Uses of dimensional analysis):

By the use of dimensionality, we can check the homogeneity (correctness) of a physical equation, and also, we can derive formula for a physical quantity.

(i) Homogeneity of Physical Equations

Correctness of an equation can be checked by showing that the dimensions of quantities on both sides of the equation are the same. This is known as principle of homogeneity.

Illustration with example

Suppose a car starts from rest (v_i = 0) and covers a distance S in time t moving with an acceleration a. The equation of motion is given by,

S = v_i t + ½ at²

As v_i = 0

or S = ½ at²

Numerical factors like ½ have no dimensions, so they can be ignored. By putting the dimensions of both sides of the equations:

[S] = [a][t²]

Writing the symbols of dimensions of both sides

[L] = [LT⁻²][T²]

[L] = [LT⁰]

or [L] = [L]

This shows that dimensions on both sides of equation are the same, therefore, the equation is dimensionally correct.

Derivation of a Formula

Dimensionality can be used to derive a possible formula for a physical quantity by correct estimation of various factors on which the quantity depends.

Limitations in Dimensional Analysis

(i) The dimensional method cannot identify where an equation is wrong. Even if an equation is proved correct, we can only say the equation might be correct.

(ii) The method does not provide a check on any numerical factor or constant. That can only be determined by experiments or plotting some suitable graph between the variables.

(iii) We cannot find dimension of exponential function and trigonometric functions.