PSEB Class 11 Physics Notes For Chapter 1 Units And Measurements

Chapter 1 Units And Measurements Important Points

Fundamental Quantity: A fundamental quantity is one which is unique and freely existing. It does not depend on any other physical quantity.

Fundamental Quantity Example: Length (L), Time (T), Mass (M), etc.

Fundamental Quantities In SI System: In the 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 produced by the combination of fundamental quantities (i.e., by division or by multiplication of fundamental quantities).

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Derived Quantity Example:

Velocity = \(\frac{\text { displacement }}{\text { time }}=\frac{\mathrm{L}}{\mathrm{T}}\)

or \(\mathrm{LT}^{-1}\)

PSEB Class 11 Physics Notes Chapter 1 Units And Measurements

Acceleration = \(\frac{\text { change in velocity }}{\text { time }}\)

= \(\frac{\mathrm{LT}^{-1}}{\mathrm{~T}}=\mathrm{LT}^{-2}\) etc.

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

Fundamental Unit: The units of the fundamental quantities are called the “fundamental units”.

Fundamental Unit: 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 → meter (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”.

Derived Units Example: Area → square meter (m²), Velocity → meter/sec (m/s) etc.

International System Of Units (S.I. Units): The S.I. system consists of seven fundamental quantities and two supplementary quantities. To measure these quantities S.I. system consists of several fundamental or basic units and two auxiliary units.

Accuracy: Accuracy indicates the closeness I 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.

Precision: Precision depends on the least measurable value of the instrument. If the least measurable value is too low, then the precision of that instrument is high.

Precision Example: The least measured value of vernier calipers is 0.1 mm

The least count of the 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”.

Class 11 Physics Units And Measurements Notes

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.

Systematic Errors Example:

1. Zero error in screw gauge and
2. A faulty calibrated thermometer

Systematic Errors Are Classified As:

1. Instrumental Errors
2. The Imperfection of The Experimental Technique
3. Personal Errors.

1. Instrumental Errors: These errors arise due to the imperfect design or faulty calibration of instruments.
   • Instrumental Errors Example: Zero error in screw gauge.
2. Imperfection Of Experimental Technique: These errors are due to the procedure followed during the experiment or measurements.
   • Imperfection Of Experimental Technique Example:
     • Measurement of body temperature at armpit
     • 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.
   • Personal Errors Example: 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.

Random Errors Example:

1. Voltage fluctuations in the power supply
2. Mechanical vibrations in the experimental setup.

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 the 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 …… a_n

Then the arithmetic mean \(a_{\text {mean }}=\frac{a_1+a_2+a_3+\ldots \ldots \ldots+a_n}{n}\)

or \(a_{\text {mean }}=\sum_{i=1}^n \frac{a_1}{n}\)

PSEB Class 11 Physics Chapter 1 Notes

Absolute Error (|Δa|): The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement.

It is denoted by |Δa|

Absolute error |Δa| = |a_mean-a₁| = |True value – measured value|

Mean absolute error (Δa_mean):  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 \(\left|\Delta \mathrm{a}_1\right|,\left|\Delta \mathrm{a}_2\right|,\left|\Delta \mathrm{a}_3\right| \ldots . . \Delta \mathrm{a}_{\mathrm{n}} \mid\), then

or, \(\left|\Delta \mathrm{a}_{\text {mean }}\right|=\frac{\left|\Delta \mathrm{a}_1\right|+\left|\Delta \mathrm{a}_2\right|+\left|\Delta \mathrm{a}_3\right|+\ldots \ldots \ldots+\left|\Delta \mathrm{a}_{\mathrm{n}}\right|}{n}\)

then \(\Delta \mathrm{a}_{\text {mean }}=\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{n}} \Delta \mathrm{a}_{\mathrm{i}}\)

Relative Error: Relative error is the ratio of the mean absolute error A a to the mean value a mean of the quantity measure.

Relative Error = \(\frac{\Delta \mathrm{a}_{\text {mean }}}{\mathrm{a}_{\text {mean }}}\)

Percentage Error (δa): When relative error is expressed in percent then it is called per] centage error.

Percentage Error \((\delta \mathrm{a})=\frac{\Delta \mathrm{a}_{\text {mean }}}{\mathrm{a}_{\text {mean }}} \times 100\)

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.

Significant Figures Example: In a measurement, the length of a body is reported as 287.5 cm. Then, In that measurement. 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 For Determining Significant Numbers:

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

Rules For Arithmetic Operation With Significant Figures:

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.

In Multiplication Or Division Example: In the division \(\frac{4.327}{2.51}\) the significant figures are 4 and 3, so the least significant figures are ‘3’.

∴ \(\frac{4.327}{2.51}\) = 1.69 i.e., the final answer must have only ‘3’ significant digits.

Punjab State Board Class 11 Physics Notes Chapter 1

In Addition Or Subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

In Addition Or Subtraction Example: 436.26g + 227.2 g Here least number of significant figures after the 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.

1. The preceding significant digit is raised by one if the first non-significant digit is more] than 5.
2. The preceding significant digit is left unchanged if the first non-significant digit | is less than 5.
3. If the first non-significant figure is 5, then
   • If the preceding significant figure is an odd number, then add one to it.
   • 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.

Dimension Example: Force dimensional formula MLT^-2

Here dimensions of Mass → 1, Length → 1, Time → 2

Dimensional Formula: It is a mathematical expression giving a relation between various fundamental quantities of a derived physical quantity.

Punjab State Board Class 11 Physics Notes Chapter 1

Dimensional Formula Example: Momentum (\(\overline{\mathrm{P}}), \mathrm{MLT}^{-1}\), Energy \(\mathrm{ML}^2 \mathrm{~T}^{-2}\) etc.

Uses Of Dimensional Methods:

1. To convert units from one system to another| system.
2. To check the validity of given physical equations. For this purpose, we will use homoge-1 neity of dimensions on L.H.S and on R.H.S.
3. To derive new relations between various physical quantities.