Chapter 2 Units and Measurement

1. Physical Quantites
All the quantities which can be measured directly or indirectly in terms of which laws of physics are described and whose measurement is necessary are called physical quantities.
Physical quantities can be divided into two types:

• Fundamental Quantities Those physical quantities which are independent of other physical quantities and are not defined in terms of other quantities, are called fundamental or base quantities. 

e.g. Mass, Length, Time, Temperature etc.

• Derived Quantities Those quantities which can be derived from the fundamental physical quantities are called derived quantities.

e.g. Force, Velocity, Acceleration etc.

2. Physical Unit

The standard amount of a physical quantities chosen to measure the physical quantity of same kind is called a physical unit.

Physical Unit can be classified into two types:

• Fundamental Units  Those physical units which can neither be derived from one another, nor they can be further resolved into more simpler units are called fundamental units.

The units of fundamental quantities are:   length, mass, time are called fundamental units.

• Derived Units  The units of measurement of all other physical quantities, which can be obtained from fundamental units are called derived units.

e.g.  Speed, acceleration, force etc.  units of speed = m/s, can be derived from fundamental units as

speed = distance/time = m/s

3. System of Units

A system of units is the complete set of units, both fundamental and derived physical units.

the common systems which are used now days are:

• MKS System    It uses metre, kilogram and second as the fundamental units of length, mass and time respectively.
• CGS System    It is French system of units, which uses centimetre, gram and second for length, mass and time respectively.
• FPS System    It is the British engineering system of units, it uses fool as the unit of length, pound as the unit of mass and second as the unit of time.
• International System of units(SI Units)   The System of units, which is accepted internationally for measurement is the SI Units.

4. SI Fundamental Quantities

5. Supplementary Quantities and their SI Units

6. Prefixes for Power of Ten

7. Some Important Practical units

For length/Distance

• Astronomical Unit - It is the mean distance of the earth from the sun. 1 AU= 1.496x10^11 m
• Light Year - It is the distance traveled by light in vacuum in one year. 1 ly= 9.46x 10^15 m
• Parallactic Second - It is the distance at which an arc of length 1 astronomical unit subtends an angle of 1 second of arc. 1 parsec = 3.084 x 10^16 m = 3.26 ly
• Angstrom Unit - 1Å = 10^-10 m

For Mass

• Pound , 1 lb = 0.4536 kg
• Slug, 1 slug = 14.59 kg
• Quintal, 1 q = 100 kg
• Atomic mass unit( 1/12th of mass of one C atom), 1 amu = 1.66 x 10^-27 kg

For Area

• Barn,  1 barn = 10^-28 m^2
• Acre, 1 acre = 4047 m^2
• Hectare, 1 hectare = 10^4 m^2

For other quantities 

• Litre, 1 litre = 10^3 cc = 10^-3 m^3 ( where cc is cubic centimetre)
• Pascal, 1Pa = 1 Nm^-2
• Degree (for angle), 1 degree = pi/180 rad

8. Accuracy and Precision of Instruments

Accuracy of a measurement is measure of how close the measure value is to the true value.

Precsion  is measurement of how much the measured value is close to previously measured value.

9. Errors in Measurement 

Difference in the true value and the measured value of a quantity is called error of measurement.

Error = True value - Measured value

Errors can be classified as 

i. Systematic Errors

Those  errors that tend to be in one direction, either +ve or -ve are called systematic errors.

some of the sources of systematic errors are:

     (a) Instrumental Errors  They occur due to imperfect design or manufacture or calibration of the measuring instrument.

     (b) Imperfection in experiment technique  These types of errors occur due to the experimental arrangement limitations.

     (c)  Personal Errors  These errors occur due to inexperience of the observer, such as lack of proper setting of the apparatus and taking observations without observing proper precautions.

     (d) Errors due to external causes   Various parameters such as change in temperature, pressure, volume, etc during experiment may affect the reading of measurement.

ii. Random Errors

The errors which occur irregularly and at random in magnitude and direction are called Random errors.

iii. Least count Error

The smallest value that can be measured by a measuring instrument is called the least count of the instrument. 

iv. Absolute Error

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

It is denoted by |Δa|.

Suppose the measured values are a1, a2, a3,..........an.

Then, arithmetic mean of these values is 

v. Mean Absolute Error

It is the arithmetic mean of the magnitude of absolute errors in all the measurements of the quantity.

It is represented by

vi. Relative Error or Fractional Error

It is defined as the ratio of mean absolute error to the mean value of quantity measured. Thus,

Relative error or Fractional error = Mean absolute error/ Mean value

vii. Percentage error

When fractional error or relative error is expressed in percent, then it is called percentage error.

Percentage error = relative or fractional error x 100%

10. Combination of Errors

i. Error of a Sum or a Difference

Suppose two physical quantities be A and B have measured value A±ΔA, B±ΔB respectively. 

ii. Error of a product

Relative error of a Product

ΔZ/Z = ±[ΔA/A + ΔB/B]

iii. Error in case of a Measured Quantity Raised to a Power

11. Significant figures

The digits that are known reliably plus the first uncertain digit are known as significant figures.

e.g. When a measured distance is reported to be 374.5m, it has four significant figures. The figures 3, 7, 4 are certain and reliable, while digit 5 is uncertain.

Rules for determining the number of significant figures

Rule 1 -  All non zero digits are significant. eg x = 1234 has four significant figures.

Rule 2 -  All the zeroes between two non zeroes digits are significant, no matter where the decimal place is if at all. e.g. x = 10.07 also has four significant figures.

Rule 3 -  If the number is less than one, the zeros on the right of decimal point and to the left of first non zero digit are not significant. e.g. x = 0.005704 the the underlined zeroes are not significant. The zero between 7 and 4 is significant. so the no of significant no is 4.

Rule 4 -  In a number without a decimal point, the terminal of trading zeroes are not significant. e.g. x = 3210 has three significant figures.

Rule 5 -  The trading zeroes in a number with a decimal point are significant. e.g. 3.500 has four significant figures.

12. Dimensional Formulae and Dimensional Equations

The expression which shows how and which of the fundamental quantities represent the dimensionof the physical quantity is called the dimensional formula for given physical quantity. 

e.g. 

13. Dimensional Analysis and Its Applications

The dimensional analysis helps us in deducing the relations among different physical quantities and checking the accuracy, derivation and dimensional consistency or homogeneity of various numerical expressions.

i. Checking the Dimensional consistency of equations

The Principle of homogeneity of dimension states that a physical quantity equation will be dimensionally correct, if the dimensions of all the terms occurring on both sides of the equation are same.

ii. Conversion of one system of units into another

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