Units and Measurement Class 11th Notes - Free NCERT Class 11 Physics Chapter 2 notes - Download PDF

The NCERT Class 11 Physics chapter 2 notes covers all the topics in a systematic way. The main topics covered in Units and measurement of NCERT Class 11 Physics chapter 2 notes are as follows: Fundamental physical quantities, Derived physical quantities, System of units, Definition of fundamental units, Supplementary units, Importance of SI unit system, Parallax methods, Errors, etc.

The NCERT Class 11 Physics chapter 2 notes notes are so prepared to give the idea about each and every topic in detail so to overcome the doubts and solve the numerical easily. The NCERT CBSE Class 11 Physics chapter 2 notes are easily downloadable as they are free of cost and can be accessed anytime anywhere.

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NCERT Class 11 Physics Chapter 2 Notes

Fundamental Quantities and Fundamental Units

The units of quantities ( length, mass, and time) are independent of one another; no one can be changed or related to any other unit. These quantities are called fundamental quantities” and their units are called “fundamental units”

• Seven fundamental quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminescent intensity.

Derived Unit

The units of all physical quantities other than the seven fundamental quantities are called “derived units”. Some examples are

Volume = length ✕ breadth ✕ height = m ✕ m ✕ m = m³

Density = mass/volume = kgm⁻³

System of Units

• CGS system:- It is known as the Gaussian system and it is based on ‘centimeter’, “gram”, and “second” units of length, mass, and time. It is a metric or decimal system.

• FPS System:-It is the British system that uses “foot” (ft), “pound” (lb), “second” (s).

• MKS System:- It is a metric system that uses “meter” (m), “kilogram” (kg), “second” (s).

• International System:- In 1960, this system was internationally accepted.

Basic Units

Supplementary Units

Some Supplementary Fundamental units

• Meter: It is the fundamental unit of length

1 kilometer(km) = 10³m

1 centimeter(cm) = 10⁻²m

1 millimeter (mm) = 10⁻³m

1 micron(μ) = 10⁻⁶m

1 nanometer (nm) = 10⁻⁹m

1 Angstrom (Å) = 10⁻¹ºm

1 fermometer = 1 fermi (fm) = 10⁻¹⁵m

For measuring astronomical distance

• Astronomical unit (AU): 1AU is the average distance of the sun from the earth.

1 AU = 1.496 ✕ 10¹¹m ≅ 1.5 ✕ 10¹¹m

• Light Year (ly): 1 light-year is the distance traveled by light in one year.

1 light-year = (3 ✕ 10⁸ms⁻¹) ✕ (365 ✕24✕60✕60)s = 9.46 ✕ 10¹⁵m

• Parallactic second or parsec(pc): 1 parsec is the distance at which the average radius of earth’s orbit around the sun would subtend an angle of 1” (second of arc)

• Kilogram: It is the unit of mass.

1 metric ton = 10³kg

1 quintal = 10²kg

1 gram = 10⁻³kg

1miligram = 10⁻³g = 10⁻⁶kg

To measure the mass of an atom or molecule we use a unified atomic mass unit(u) defined as 1/12th mass of an atom of the carbon-12 isotope.

1u = 1.66 ✕ 10⁻²⁷kg

• Second(s): It is the unit of time.

1 millisecond (ms) = 10⁻³s

1 microsecond (*s) = 10⁻⁶s

1 nanosecond (ns) = 10⁻⁹s

1 picosecond (ps) = 10⁻¹²s

Shake is the unit of time used in microscopic physics

1 shake = 10⁻⁸s

Accuracy and Precision

Accuracy is defined as‘’ The degree to which an observed value happens to agree with the true value of a quantity (or the closeness of measured value to the true value) is referred to as accuracy.’’

we can define precisely as the precision of an observed value which informs us at what resolution the quantity is measured. Thus, measured values that are really close to each other (and may or may not be close to the true value) are precise values.

Error in the Measurement

There are two types of errors in measurements taken during a laboratory experiment: errors of observation and errors in the instrument. Technically, measurement uncertainty is referred to as an error. It's expressed in percentage

Errors are classified into two types

Systematic errors: These errors occur consistently across repeated experiments under identical conditions. There is a rule for finding these errors. Therefore, once the rule that governs these errors is identified, these errors can be eliminated by applying proper corrections to the result obtained. It is subclassified as

1. Instrumental Errors: These errors are due to shortcomings in the instruments like errors due to defective alignment of the instrument, zero error in vernier calipers, screw gauges,s, and spherometers.

2. Error due to imperfection:- This error occurs due to some imperfection of the apparatus used in an experiment. E.g., in heat experiments, the loss of heat due to radiation is an unwanted phenomenon.

3. Gross Errors:- These are errors introduced due to the casual approach of the experiment such as improper adjustment of the experimental setup in the observation part and computational mistakes etc. in the deduction part.

Random errors:-These occur as a result of

(i) minor modifications to the experiment's conditions and

(ii) In taking readings, the observer's judgment is incorrect.

Methods of minimizing random errors:- By taking the arithmetic mean of a large number of readings of the same quantity, the random error can be reduced. The mean will be extremely close to the true value.

Absolute error:-

Absolute error is the magnitude of the difference between the true value of the measured quantity and the individual measured value.

∴ An absolute error = True value ~ measured value

Let a1, a2, a3,...an represent the measured values of the physical quantity, the absolute error.

Where is a true value or a mean value.

Mean absolute error:-

The overall arithmetic mean of the errors of different measurements is the mean absolute error.

Mean absolute error,

Relative Error or Fractional Error:-

The ratio of the mean absolute error to the true value of the measured quantity is known as relative or fractional error.

Percentage Error:-

If the fractional error is multiplied by 100, it is known as a percentage error.

The least Count of Measuring instruments

The smallest value, that the measuring instrument can measure is referred to as its least count. All readings or measured values are only valid up to this point. The error associated with the instrument's resolution is the least count error.

For example, a vernier caliper has the least count of 0.01 cm; a spherometer may have the least count of 0.001 cm. In both systematic and random errors, the least count error is a type of random error., but within a determinate size. If we use a meter scale to measure length, the graduations may be spaced at 1 mm division scale spacing or interval.

Combination of Errors

When a number of quantities are involved in the final calculation, then errors associated with the measurement of all quantities will affect the end result. In this way, the error in the final result depends upon:

1. The error in the individual measurement

2. The nature of the mathematical operation is done on them to arrive at the final result.

There are some rules for calculating the combined error

1. Sum and Difference: Let limiting errors in two physical quantities x and y are ±Δx and ±Δy respectively.

z = x + y

Let the limiting error in the sum z be ±Δz

z ± Δz = (x ± Δx) + (y ± Δy)

± Δz = ± Δx ± Δy

The max. Possible error in z, is given by

Δz = Δx + Δy

Consider the next difference

z = x - y

z ± Δz = (x ± Δx) - (y ± Δy)

± Δz = ± Δx ∓ Δy

The max. Possible error in z, is given by

Δz = Δx + Δy

Thus, when two quantities are added or subtracted the limiting error in the final result is the sum of limiting errors in the quantities involved.

2. Product and Quotient:

Let

Δx.Δy is very small compared to other terms and so it is neglected.

Dividing both sides by z(= xy),

The max. Possible error in z,

Where, and are the relative errors in y and x respectively.

And the percentage error,

Consider next division

Expanding with the help of the binomial theorem and neglecting the higher terms

But

Thus,

The last term is small in comparison to other terms, so it is neglected

The maximum percentage error in z

Thus, when two quantities are multiplied or divided by, the fractional error in the final result is the sum of the fractional errors in the quantities to be multiplied or to be divided.

Significant Figures

In the measurement of a quantity, the digits are measured accurately, and the first doubtful digits are called the significant figure. For example, the length of the side of the cube read by a vernier caliper is 2.58cm. The last digit 8 is doubtful because the length can be anywhere between 2.57 and 2.59cm; it will be read as 2.58cm. So there are three significant figures.

For example, the length of 2.308 cm has four significant figures. But in different units, the same value can be written as 0.02308 m or 23.08mm or 23080 μm. There are four significant figures in all of these numbers (digits 2, 3, 0, 8). This shows that the location of the In determining the number of significant figures, the decimal point has no bearing. The rules in the example are as follows:

• All of the non-zero digits are significant (important).

• Zeros between two non-zero digits are significant.

• If the number is zero(s) on the right of the decimal point but to the left of the first non-zero digit are not significant. [In 0.00 2308, zeroes are not significant].

• Numbers without a decimal point do not have meaning if the number has one or more trailing zeros.

[Thus 113 m = 11300 cm = 113000 mm has three significant figures.]

• A number with a decimal point has significant trailing zero(s).

[The numbers 3.500 or 0.06900 both have four significant figures each.]

• To avoid such confusion in determining the number of significant figures, it is best to report all measurements in scientific notation (to the power of 10).

• It is ideal for reporting measurements in scientific notation.

• The trailing zero(s) are not significant for a number greater than one without a decimal.

• The trailing zero(s) of a decimal number are significant

Dimensional Analysis

The dimensions of a physical quantity are calculated by raising the fundamental units to the powers that are needed to derive its derived units.

The length, mass, and time are denoted by [L], [M], and [T] respectively.

Uses of Dimensional Equations

• To convert units of one system into units of another system:

The product of a physical quantity's numerical value and its corresponding unit is a constant. Let the numerical value of a physical quantity p are n₁ and n₂ in two different systems and the corresponding units are u₁ and u₂, then

……(i)

If the dimensions of the physical quantity are a in mass, b in length, and c in time, then its dimensional formula will be . If the fundamental units in one system are M₁, L₁, and T₁

Similarly, if the fundamental units in the second system are M₂, L₂, and T₂, then

According to eqn (i), we have

Using this formula we can convert the numerical value of a physical quantity from one system of units into the other system.

• To check the correctness of an equation:

The dimensions of all the terms on both sides of a physical equation must be the same.

This is called the principle of homogeneity of dimensions.

E.g.,

m = [M], v = [LT⁻¹], g = [LT⁻²] and h = [L]

[M] [LT⁻¹]² = [M] [LT⁻²][L]

[ML²T⁻²] = [ML²T⁻²]

Both sides are the same.

• To establish the relationship among various physical quantities: If we know the factors on which a given physical quantity may possibly depend then, using dimensions, we can find a formula relating the quantity with those factors.

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