TS Inter 1st Year Physics Notes Chapter 2 Units and Measurements

Here students can locate TS Inter 1st Year Physics Notes 2nd Lesson Units and Measurements to prepare for their exam.

TS Inter 1st Year Physics Notes 2nd Lesson Units and Measurements

→ Fundamental Quantity : A fundamental quantity is one which is unique and freely existing. It does not depend on any other physical quantity. Ex: Length (L), Time (T), Mass (M) etc.

→ Fundamental quantities in SI System : In SI system length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity are taken as fundamental quantities.

→ Derived quantity: A derived quantity is pro-duced by the combination of fundamental quantities (i.e., by division or by multiplica-tion of fundamental quantities).
Ex: Velocity = \(\frac{\text { displacement }}{\text { time }}=\frac{\mathrm{L}}{\mathrm{T}}\) or LT-1
Acceleration = \(\frac{\text { change in velocity }}{\text { time }}\)
= \(\frac{\mathrm{LT}^{-1}}{\mathrm{~T}}\) = LT2

→ Unit: The standard which is used to measure the physical quantity is called the Unit’.

→ Fundamental unit: The units of the funda-mental quantities are called the “fundamental units”.
Ex : Length → Meter (m), Mass → Kilogram (kg), Time Second (sec) etc.

→ Basic units or fundamental units of SI system : The basic units in S.I. system are Length → metre (L), Mass → kilogram (kg), Time second (s); electric current → ampere (amp), Thermodynamic temperature → Kelvin (K); Amount of substance → mole (mol); Luminous intensity → candela (cd); Auxilliary units : Plane angle → Radian (rad); Solid angle → steradian (sr)

→ Derived units: The units of derived quantities are known as “derived units”.
Ex: Area → square meter (m2),
Velocity → meter/sec (m/s) etc.

→ International system of units (S.I. units) :
S.I. system consists of seven fundamental quantities and two supplementary quantities. To measure these quantities S.I. system consists of seven fundamental or basic units and two auxiliary units.

→ Accuracy: Accuracy indicates the closeness of a measured value to the true value of the quantity. If we are very close to the true value then our accuracy is high.

TS Inter 1st Year Physics Notes Chapter 2 Units and Measurements

→ Precision : Precision depends on the least measurable value of the instrument. If the least measurable value is too less, then precision of that instrument is high.
Ex : Least measured value of vernier callipers is 0.1 mm
Least count of screw gauge is 0.01 mm.
Among these two, the precision of the screw gauge is high.

→ Error: The uncertainty of measurement of a physical quantity is called “error”.
→ Systematic errors : Systematic errors always tend to be in one direction i.e., positive or negative. For systematic errors, we know the reasons for the error. They can be reduced by proper correction or by proper care. Ex:

  • Zero error in screw gauge and
  • A faulty calibrated thermometer

Note :
Systematic errors are classified as

  • Instrumental errors
  • Imperfection of experimental technique
  • Personal errors.

1) Instrumental errors: These errors arise due to the imperfect design or faulty calibration of instruments.
Ex : Zero error in screw gauge.

2) Imperfection of experimental technique: These errors are due to the procedure followed during experiment or measurements. Ex : 1) Measurement of body temperature at armpit 2) Simple pendulum oscillations with high amplitude.

3) Personal errors: These errors arise due to an individual’s approach or due to lack of proper setting of apparatus.
Ex : Parallax error is a personal error.

→ Methods To Reduce Systematic Errors :
Systematic errors can be minimized by improving experimental techniques, by selecting better instruments and by removing personal errors.

→ Random errors :
These errors will occur irregularly. They may be positive (or) negative in sign. We cannot predict the presence of these errors.
Ex:

  • Voltage fluctuations of power supply
  • Mechanical vibrations in experimental set up.

→ Least count error: This is a systematic error. It depends on the smallest value that can be measured by the instrument.
Least count error can be minimized by using instruments of highest precision.

→ Arithmetic mean: The average value of all the measurements is taken as arithmetic mean.
Let the number of observations be a1, a2, a3 ……….. an

Then the arithmetic mean
amean = \(\frac{\mathbf{a}_1+\mathbf{a}_2+\mathbf{a}_3+\ldots \ldots \ldots .+\mathbf{a}_{\mathbf{n}}}{\mathbf{n}}\)
or amean = \(\sum_{i=1}^n \frac{a_i}{n}\)

→ Absolute error (|Δa|): The magnitude of the difference between the individual measurement and true value of the quantity is called absolute error of the measurement. It is denoted by |Δa|
Absolute error
|Δa| = |amean – ai|
= |True value – measured value|

→ Mean absolute error ( Aa[nrnnl: The arithmetic mean value of all absolute errors is known as mean absolute error.
Let ‘n’ measurements are taken, then their absolute errors are, say |Δa1|, |Δa2|, |Δa3| …….. ||Δan|, then
|Δamean| = \(\frac{\left|\Delta a_1\right|+\left|\Delta a_2\right|+\left|\Delta a_3\right|+\ldots \ldots \ldots+\left|\Delta a_n\right|}{n}\)
or
Δamean = \(\frac{1}{n} \sum_{i=1}^n \Delta a_i\)

→ Relative error: Relative error is the ratio of the mean absolute error A amean to the mean value a mean of the quantity measure.
Relative error = \(\frac{\Delta \mathbf{a}_{\text {mean }}}{\mathbf{a}_{\text {mean }}}\)

TS Inter 1st Year Physics Notes Chapter 2 Units and Measurements

→ Percentage error (δa): When relative error is expressed in percent then it is called percentage error.
Percentage error (δa) = \(\frac{\Delta \mathbf{a}_{\text {mean }}}{\mathbf{a}_{\text {mean }}}\) × 100

→ Significant figures: The scientific way to report a result must always have all the reliably known (measured) values plus one uncertain digit (first digit). These are known as “significant figures”.
This additional digit indicates the uncertainty of measurement.
Ex: In a measurement, the length of a body is reported as 287.5 cm. Then, in that measu-rement, the length is believable up to 287 cm
i. e., the digits 2, 8 and 7 are certain. The first digit (5) is uncertain. Its value may change.

→ Rules in determining significant numbers

  • All the non-zero digits are significant.
  • All the zeros in between two non-zero digits are significant.
  • If the number is less than one, the zeros on the right of decimal point to the first nonzero digit are not significant.
    Ex : In a result 0.002308 the zeros before the digit ‘2’ are non significant.
  • The terminal or trailing zeros in a number without decimal point are not significant. Ex: In the result 123 m = 12300 cm = 123000 mm the zeros after the digit ‘3’ are not significant.
  • The trailing zeros in a number with a decimal point are significant.
    Ex : In the result 3.500 or 0.06900 the last zeros are significant. So number of significant figures are four in each case.

→ Rules for arithmetic operation with sig-nificant figures
1. In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
Ex : In the division \(\frac{4.237}{2.51}\) the significant figures are 4 and 3, so least significant figures are ‘3’.
\(\frac{4.237}{2.51}\) = 1.69 i.e., final answer must have only ‘3’ significant digits.

2. In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.
Ex: 436.26g + 227.2 g Here least number of significant figures after decimal point is one.
436.26 + 272.2 = 708.46 must be expressed as 708.5 (after rounding off the last digit).

→ Rounding off the uncertain digits Rules for rounding off procedure : In rounding off the numbers to the required number of significant digits the following rules are followed.

  • The preceding significant digit is raised by one if the first non-significant digit is more than 5.
  • The preceding significant digit is left unchanged if the first non-significant digit is less than 5.
  • If the first non-significant figure is 5 then
    (a) If the preceding significant figure is an odd number then add one to it.
    (b) If the preceding significant figure is an even number then it is unchanged and 5 is discarded.

→ Dimension: The power of a fundamental quantity in the given derived quantity is called
dimension.
Ex: Force dimensional formula MLT-2 Here dimensions of Mass → 1, Length → 1, Time → 2

→ Dimensional formula: It is a mathematical expression giving relation between various fundamental quantities of a derived physical quantity.
Ex : Momentum (P),MLT-1,
Energy ML2T-2 etc.

→ Uses of dimensional methods :

  • To convert units from one system to another system.
  • To check the validity of given physical equations. For this purpose, we will use homogeneity of dimensions on L.H.S and on R.H.S.
  • To derive new relations between various physical quantities.

→ Dimensional formulae of physical quantities:
TS Inter 1st Year Physics Notes Chapter 2 Units and Measurements 1
TS Inter 1st Year Physics Notes Chapter 2 Units and Measurements 2
TS Inter 1st Year Physics Notes Chapter 2 Units and Measurements 3

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