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Updated on 23 Jun 2026, 11:01 IST
Units and Measurements Class 11 Notes help students understand how physical quantities are measured, written, compared, and checked in Physics. This is the first chapter of Class 11 Physics and forms the base for later chapters such as Motion, Laws of Motion, Work Energy and Power, Gravitation, Thermodynamics, Oscillations, and Waves.
This chapter covers SI units, fundamental and derived quantities, dimensional formulas, significant figures, errors in measurement, propagation of errors, dimensional analysis, least count, vernier caliper, screw gauge, and the principle of homogeneity of dimensions.
Units and Measurements is the chapter that explains how physical quantities are measured using standard units and how the accuracy of measurements is checked. It also teaches dimensional formulas, error analysis, significant figures, and measurement instruments like vernier caliper and screw gauge.
In Physics, every measurement is written as:
Physical Quantity = Numerical Value × Unit
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Example:
Length = 5 m
Here, 5 is the numerical value and m is the unit.
Units and Measurements is a foundational chapter in CBSE Class 11 Physics. It teaches students that Physics is not only about formulas but also about accurate measurement and correct representation of physical quantities.
A number without a unit is incomplete in Physics. For example, saying “distance is 10” is not meaningful unless we mention whether it is 10 m, 10 cm, 10 km, or 10 mm.
This chapter is important because it helps students:
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NEET
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CBSE
A physical quantity is any quantity that can be measured and expressed using a numerical value and a unit.
Examples of physical quantities include length, mass, time, speed, force, work, power, temperature, and electric current.
| Physical Quantity | Example | Unit |
| Length | 5 m | metre |
| Mass | 2 kg | kilogram |
| Time | 10 s | second |
| Speed | 20 m s-1 | metre per second |
| Force | 50 N | newton |
| Temperature | 300 K | kelvin |
A physical quantity is complete only when both numerical value and unit are written.
Correct: 4 kg
Incorrect: 4
Students often write only the numerical value in numerical answers. In Physics, the unit must be written with the final answer.
Physical quantities are classified into two main types:
Fundamental quantities are independent physical quantities that cannot be expressed in terms of other physical quantities.
They are also called base quantities.
| Fundamental Quantity | SI Unit | Symbol |
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
The SI system is based on seven base units. These base units are internationally accepted standards of measurement.
Derived quantities are physical quantities obtained by combining two or more fundamental quantities.
| Derived Quantity | Formula | SI Unit |
| Area | Length × Breadth | m2 |
| Volume | Length × Breadth × Height | m3 |
| Speed | Distance / Time | m s-1 |
| Acceleration | Velocity / Time | m s-2 |
| Force | Mass × Acceleration | N |
| Work | Force × Displacement | J |
| Power | Work / Time | W |
Speed is a derived quantity because it depends on distance and time.
Speed = Distance / Time
Therefore, the SI unit of speed is:
m / s = m s-1
The International System of Units is called the SI system. It is the most widely accepted system of measurement in science.
The SI system is used in schools, laboratories, scientific research, engineering, medicine, and international communication.
| Advantage | Explanation |
| Internationally accepted | Scientists around the world use the same system |
| Decimal-based | Conversion is easy using powers of 10 |
| Standardized | Reduces confusion in measurement |
| Scientific | Used in textbooks, experiments, and research |
In older classification, plane angle and solid angle were treated as supplementary quantities. In the SI system, they are dimensionless quantities with special names for their units.
| Quantity | Unit | Symbol | Dimension |
| Plane angle | radian | rad | [M0L0T0] |
| Solid angle | steradian | sr | [M0L0T0] |
Yes, a physical quantity can have units but no dimensions. Plane angle and solid angle are examples of quantities that have units but are dimensionless.
For plane angle:
θ = s / r
Where:
Since both arc length and radius have the dimension of length:
θ = [L] / [L] = [M0L0T0]
So, plane angle is dimensionless but has the unit radian.
SI prefixes are used to express very large or very small quantities conveniently.
| Prefix | Symbol | Power of 10 |
| tera | T | 1012 |
| giga | G | 109 |
| mega | M | 106 |
| kilo | k | 103 |
| centi | c | 10-2 |
| milli | m | 10-3 |
| micro | μ | 10-6 |
| nano | n | 10-9 |
| pico | p | 10-12 |
1 km = 103 m
1 cm = 10-2 m
1 mm = 10-3 m
1 μm = 10-6 m
Students often confuse m and M.
m can mean metre or milli depending on context.M means mega.Very large distances in astronomy cannot be measured using ordinary scales. For such distances, special units are used.
| Unit | Meaning |
| Astronomical Unit | Average distance between Earth and Sun |
| Light year | Distance travelled by light in one year |
| Parsec | Distance corresponding to a parallax angle of one second of arc |
Parallax is the apparent shift in the position of an object when it is viewed from two different positions.
The parallax method is used to measure large distances, especially the distance of stars from Earth.
Hold your finger in front of your eyes. Close one eye and observe it. Then open the other eye and close the first one. The finger appears to shift its position. This apparent shift is called parallax.
D = b / θ
Where:
For small angles, the angle θ must be measured in radians.
Very small distances such as atomic size, nuclear size, thickness of thin sheets, and diameter of thin wires cannot be measured using ordinary scales.
For school-level practicals, instruments like vernier caliper and screw gauge are used to measure small lengths accurately.
Examples of very small distances include:
Accuracy means how close a measured value is to the true or actual value.
Example:
If the actual length of an object is 10.00 cm and the measured value is 9.99 cm, the measurement is accurate.
Precision means how close repeated measurements are to one another.
Example:
If repeated measurements are 9.81 cm, 9.82 cm, and 9.81 cm, the measurements are precise because they are close to each other.
| Accuracy | Precision |
| Closeness to the true value | Closeness among repeated readings |
| Related to correctness | Related to consistency |
| Affected by systematic error | Affected by random error |
| A measurement can be accurate but not precise | A measurement can be precise but not accurate |
Students often use accuracy and precision as the same term. Accuracy is about closeness to the true value, while precision is about closeness among repeated readings.
Error is the difference between the measured value and the true or accepted value of a physical quantity.
No measurement is perfectly exact. Every measurement has some uncertainty.
The main types of errors are:
Systematic errors are errors that occur due to faulty instruments, wrong calibration, or incorrect experimental method.
Systematic errors usually affect all readings in the same direction.
Random errors are unpredictable errors caused by irregular and unknown factors during measurement.
Random errors can be reduced by taking repeated readings and calculating the mean value.
Gross errors are errors caused by carelessness or mistakes made by the observer.
Gross errors are not experimental uncertainty. They are avoidable mistakes and can be reduced by careful work.
Absolute error is the magnitude of the difference between an individual measured value and the mean value.
If a1, a2, a3, ..., an are measured values, then the mean value is:
amean = (a1 + a2 + a3 + ... + an) / n
The absolute errors are:
Δa1 = |a1 - amean|
Δa2 = |a2 - amean|
Δa3 = |a3 - amean|
In general:
Δai = |ai - amean|
Mean absolute error is the average of all absolute errors.
Δamean = (Δa1 + Δa2 + Δa3 + ... + Δan) / n
Mean absolute error gives an estimate of uncertainty in repeated measurements.
Relative error is the ratio of mean absolute error to the mean measured value.
Relative Error = Δa / a
Relative error has no unit because it is the ratio of two similar quantities.
Percentage error is the relative error multiplied by 100.
Percentage Error = (Δa / a) × 100
If measured value = 50 cm and absolute error = 0.5 cm:
Percentage Error = (0.5 / 50) × 100
Percentage Error = 1%
Propagation of errors means calculating the error in a final result when measured quantities are added, subtracted, multiplied, divided, or raised to powers.
If:
Z = A + B
or
Z = A - B
Then maximum absolute error is:
ΔZ = ΔA + ΔB
For addition and subtraction, absolute errors are added.
If:
Z = A × B
or
Z = A / B
Then:
ΔZ / Z = ΔA / A + ΔB / B
When physical quantities are multiplied or divided, the percentage error in the final result is the sum of the percentage errors of the individual quantities.
For multiplication and division, relative errors or percentage errors are added.
If:
Z = An
Then:
ΔZ / Z = n(ΔA / A)
If:
Z = A2
Then:
ΔZ / Z = 2(ΔA / A)
Significant figures are the meaningful digits in a measured value. They show the precision of measurement.
| Rule | Example | Number of Significant Figures |
| All non-zero digits are significant | 245 | 3 |
| Zeros between non-zero digits are significant | 1005 | 4 |
| Leading zeros are not significant | 0.0045 | 2 |
| Trailing zeros after decimal are significant | 2.500 | 4 |
| Trailing zeros without decimal may be ambiguous | 1500 | Depends on notation |
0.0056 has 2 significant figures.
4.080 has 4 significant figures.
300.0 has 4 significant figures.
Rounding off is used to write a result with the correct number of significant figures.
| Situation | Rule |
| Digit to be dropped is less than 5 | Leave the preceding digit unchanged |
| Digit to be dropped is greater than 5 | Increase the preceding digit by 1 |
| Digit to be dropped is exactly 5 | Follow standard rounding convention |
3.246 rounded to 3 significant figures = 3.25
8.432 rounded to 2 significant figures = 8.4
In addition or subtraction, the final answer should have the same number of decimal places as the quantity with the least number of decimal places.
12.11 + 18.0 + 1.013 = 31.123
The least number of decimal places is 1, so the answer is:
31.1
In multiplication or division, the final answer should have the same number of significant figures as the quantity with the least number of significant figures.
2.5 × 3.42 = 8.55
Here, 2.5 has 2 significant figures and 3.42 has 3 significant figures.
So, the final answer should have 2 significant figures:
8.6
Dimensions show how a physical quantity depends on fundamental quantities such as mass, length, and time.
| Fundamental Quantity | Dimensional Symbol |
| Mass | [M] |
| Length | [L] |
| Time | [T] |
| Electric current | [A] |
| Temperature | [K] |
| Amount of substance | [mol] |
| Luminous intensity | [cd] |
A dimensional formula is an expression that represents a physical quantity in terms of fundamental dimensions.
Speed = Distance / Time
Distance has dimension:
[L]
Time has dimension:
[T]
Therefore:
Speed = [L] / [T]
Speed = [LT-1]
So, the dimensional formula of speed is:
[M0L1T-1]
or simply:
[LT-1]
| Physical Quantity | Formula | Dimensional Formula |
| Area | Length × Breadth | [L2] |
| Volume | Length3 | [L3] |
| Density | Mass / Volume | [ML-3] |
| Speed | Distance / Time | [LT-1] |
| Velocity | Displacement / Time | [LT-1] |
| Acceleration | Velocity / Time | [LT-2] |
| Momentum | Mass × Velocity | [MLT-1] |
| Force | Mass × Acceleration | [MLT-2] |
| Impulse | Force × Time | [MLT-1] |
| Work | Force × Displacement | [ML2T-2] |
| Energy | Work done | [ML2T-2] |
| Power | Work / Time | [ML2T-3] |
| Pressure | Force / Area | [ML-1T-2] |
| Stress | Force / Area | [ML-1T-2] |
| Strain | Change in dimension / Original dimension | [M0L0T0] |
| Torque | Force × Distance | [ML2T-2] |
| Frequency | 1 / Time period | [T-1] |
| Gravitational constant G | Fr2 / m1m2 | [M-1L3T-2] |
| Planck’s constant h | Energy × Time | [ML2T-1] |
| Surface tension | Force / Length | [MT-2] |
| Coefficient of viscosity | From viscous force relation | [ML-1T-1] |
The Principle of Homogeneity of Dimensions states that the dimensions of all terms on both sides of a physical equation must be identical.
This means only physical quantities with the same dimensions can be added, subtracted, or compared.
Consider the equation:
v = u + at
Dimensions of velocity:
[v] = [LT-1]
Dimensions of initial velocity:
[u] = [LT-1]
Dimensions of acceleration multiplied by time:
[at] = [LT-2] [T] = [LT-1]
Therefore:
[LT-1] = [LT-1] + [LT-1]
So, the equation is dimensionally correct.
The three main applications of dimensional analysis are checking the correctness of equations, converting units, and deriving relationships between physical quantities.
A physical equation is dimensionally correct if both sides have the same dimensions.
Example:
s = ut + 1/2 at2
Dimensions of displacement:
[s] = [L]
Dimensions of ut:
[ut] = [LT-1] [T] = [L]
Dimensions of at2:
[at2] = [LT-2] [T2] = [L]
So, the equation is dimensionally correct.
Dimensional analysis can be used to find relations among physical quantities.
Example: The time period T of a simple pendulum may depend on length l and acceleration due to gravity g.
Let:
T ∝ lagb
Writing dimensions:
[T] = [L]a[LT-2]b
[T] = [La+bT-2b]
Comparing powers of T:
1 = -2b
b = -1/2
Comparing powers of L:
0 = a + b
a = 1/2
Therefore:
T ∝ l1/2g-1/2
T ∝ √(l / g)
Dimensional analysis helps convert a physical quantity from one system of units to another.
If the dimensions of a quantity are:
[MaLbTc]
then unit conversion can be done by comparing the units of mass, length, and time in two systems.
Dimensional analysis is useful, but it has some limitations.
| Limitation | Explanation |
| Cannot find numerical constants | It cannot determine constants like 2, 1/2, or π |
| Cannot handle trigonometric functions | It cannot derive formulas involving sin θ, cos θ, log x, or ex |
| Cannot distinguish quantities with same dimensions | Work and torque have the same dimensions |
| Cannot prove full correctness | An equation may be dimensionally correct but physically wrong |
| Difficult with many variables | It becomes limited when variables exceed fundamental dimensions |
Work and torque both have the dimensional formula:
[ML2T-2]
But work is a scalar quantity and torque is a vector quantity. Dimensional analysis cannot show this difference.
A vernier caliper is an instrument used to measure small lengths, internal diameter, external diameter, and depth accurately.
The least count of a vernier caliper is the smallest distance it can measure accurately. It is commonly 0.1 mm or 0.01 cm.
Least Count = 1 MSD - 1 VSD
Another common formula is:
Least Count = Value of 1 MSD / Number of Vernier Scale Divisions
Where:
Students often forget to convert mm to cm or cm to m in final answers.
A screw gauge is used to measure very small thicknesses and diameters more accurately than a vernier caliper.
The least count of a screw gauge is:
Least Count = Pitch / Number of divisions on circular scale
Here, pitch is the distance moved by the screw in one complete rotation.
If pitch = 1 mm and the circular scale has 100 divisions:
LC = 1 mm / 100
LC = 0.01 mm
| Feature | Vernier Caliper | Screw Gauge |
| Used for | Length, diameter, depth | Very small thickness or diameter |
| Accuracy | Less than screw gauge | More accurate |
| Least count formula | 1 MSD - 1 VSD | Pitch / Circular scale divisions |
| Common error | Zero error | Zero error, backlash error |
| Example use | Diameter of a coin | Diameter of a wire |
Zero error occurs when the zero mark of an instrument does not coincide with the reference zero when no object is being measured.
| Type | Meaning | Correction |
| Positive zero error | Instrument shows a positive reading when actual reading is zero | Subtract zero error |
| Negative zero error | Instrument shows a negative reading when actual reading is zero | Add correction |
Students often forget to apply zero correction in vernier caliper and screw gauge questions.
| Topic | Formula |
| Physical quantity | Numerical Value × Unit |
| Relative error | Δa / a |
| Percentage error | (Δa / a) × 100 |
| Error in sum or difference | ΔZ = ΔA + ΔB |
| Error in product or division | ΔZ / Z = ΔA / A + ΔB / B |
| Error in power | ΔZ / Z = n(ΔA / A) |
| Least count of vernier caliper | 1 MSD - 1 VSD |
| Least count of screw gauge | Pitch / Circular scale divisions |
| Speed | Distance / Time |
| Acceleration | Velocity / Time |
| Force | Mass × Acceleration |
| Work | Force × Displacement |
| Power | Work / Time |
Question: A length is measured as 20.0 cm with an absolute error of 0.1 cm. Find the percentage error.
Given:
a = 20.0 cm
Δa = 0.1 cm
Formula:
Percentage Error = (Δa / a) × 100
Solution:
Percentage Error = (0.1 / 20.0) × 100
Percentage Error = 0.5%
Answer: The percentage error is 0.5%.
Question: Find the dimensional formula of force.
Formula:
F = ma
Mass has dimension:
[m] = [M]
Acceleration has dimension:
[a] = [LT-2]
Therefore:
[F] = [M][LT-2]
[F] = [MLT-2]
Answer: The dimensional formula of force is [MLT-2].
Question: A screw gauge has pitch 0.5 mm and 50 divisions on its circular scale. Find its least count.
Given:
Pitch = 0.5 mm
Number of circular scale divisions = 50
Formula:
LC = Pitch / Number of circular scale divisions
Solution:
LC = 0.5 / 50 mm
LC = 0.01 mm
Answer: The least count is 0.01 mm.
| Mistake | Correct Approach |
| Forgetting units in final answer | Always write the unit with the answer |
| Confusing accuracy and precision | Accuracy means closeness to true value; precision means closeness among readings |
| Adding relative errors in addition | Add absolute errors in addition/subtraction |
| Adding absolute errors in multiplication | Add relative errors in multiplication/division |
| Forgetting zero correction | Apply correction in vernier and screw gauge questions |
| Writing wrong significant figures | Follow significant figures rules |
| Treating radian as dimensional | Radian has unit but is dimensionless |
| Thinking dimensional correctness proves formula is correct | It only checks dimensional consistency |
This is a complete, SEO-optimized question bank with detailed answers, formulas, and diagrams for CBSE Class 11 Physics Chapter 1: Units and Measurements. It is formatted to help students score maximum marks in school board exams and lay a solid foundation for competitive exams like JEE and NEET.
Answer: A physical quantity is any property of a material or system that can be measured quantitatively using a standard unit and expressed numerically. Examples include mass, length, time, temperature, and force.
Answer: A unit is a chosen standard of reference of the same physical nature used to measure and compare any physical quantity. For instance, the meter (m) is the standard unit of length.
Answer: The SI (International System of Units) unit of length is the meter, represented by the symbol m.
Answer: The least count is the smallest value of a physical quantity that can be measured accurately using a given measuring instrument. It represents the limit of resolution of the instrument.
Answer:Significant figures are the digits in a measured value that are known with certainty plus one first digit that is estimated or uncertain.
Answer: The absolute error of a measurement is the magnitude of the difference between the true value (usually the mean value of multiple measurements) and the individual measured value.
Formula: Δai = |amean - ai|
Answer:Relative error (or fractional error) is defined as the ratio of the mean absolute error to the mean (true) value of the measured quantity.
Formula: Relative Error = Δamean / amean
Answer: The dimensional formula of force is [M1 L1 T-2].
Answer: The SI unit of power is the watt, represented by the symbol W (where 1 W = 1 J/s).
Answer:Zero error occurs when the zero mark of the measuring scale of an instrument does not coincide with the reference zero mark of the main scale when the measuring jaws or surfaces are completely closed.
Answer: Although accuracy and precision are often used interchangeably, in physics they have distinct meanings:
| Feature | Accuracy | Precision |
| Definition | How close the measured value is to the true or accepted value. | How close successive measurements of the same quantity are to each other. |
| Dependence | Depends on minimizing systematic errors. | Depends on the resolution/least count of the measuring instrument. |
Answer: Two major limitations of dimensional analysis are:
π, or e) in a physical equation.sin θ), exponential (e.g., ex), or logarithmic (e.g., log x) functions.Answer:
Answer: The least count of a Vernier Caliper (also called the vernier constant) is the difference between the value of one main scale division (MSD) and one vernier scale division (VSD).
Formula:Least Count (LC) = 1 MSD - 1 VSD = 1 MSD × (1 - 1/N)
(where N is the total number of divisions on the vernier scale. Typically, LC = 0.1 mm or 0.01 cm).
Answer: The radian is the unit used to measure plane angle. By definition, a plane angle (θ) is the ratio of the length of the arc (s) to the radius (r) of the circle: θ = s / r
Since both arc length (s) and radius (r) have the dimension of length [L], the ratio is: [θ] = [L] / [L] = [L0] = [M0 L0 T0]
Thus, the radian is a ratio of two identical physical quantities, making it **dimensionless**, despite having a unit.
Answer: In multiplication and division, the final calculated result must be rounded off to have the same number of significant figures as the original measurement with the least number of significant figures.
Example: If you multiply 3.8 (2 significant figures) by 0.125 (3 significant figures), the calculator gives 0.475. The result must be rounded to 2 significant figures: 0.48.
Answer: The Principle of Homogeneity of Dimensions states that the dimensions of all the terms on both sides of a physically correct equation must be the same. This means you can only add, subtract, or compare physical quantities if they have the identical dimensional formulas.
Example: Let us verify the second equation of motion: s = ut + ½at2
[s] = [L][ut] = [L T-1] × [T] = [L]½) are dimensionless: [at2] = [L T-2] × [T2] = [L]Since the dimensions of the term on the Left Hand Side are equal to the dimensions of each individual term on the Right Hand Side (all equal to [L]), the equation is **dimensionally correct** under the principle of homogeneity.
Answer: To derive the dimensional formula of pressure, we break down its formula into base physical quantities:
Step 1: Write the primary formula of Pressure.Pressure = Force / Area
Step 2: Find the dimensions of Area.Area = Length × Breadth = [L] × [L] = [L2]
Step 3: Find the dimensions of Force.Force = Mass × Acceleration = Mass × (Velocity / Time)Force = [M] × ([L T-1] / [T]) = [M L T-2]
Step 4: Substitute the values back into the pressure equation.[Pressure] = [Force] / [Area] = [M L T-2] / [L2][Pressure] = [M L1-2 T-2] = [M L-1 T-2]
Thus, the dimensional formula of pressure is [M1 L-1 T-2].
Answer: When physical quantities containing errors are multiplied or divided, the fractional (or percentage) errors add up. Here is the mathematical proof:
Let two quantities be measured as A ± ΔA and B ± ΔB. Let the calculated value be Z = A × B.
Applying errors, the equation becomes:Z ± ΔZ = (A ± ΔA)(B ± ΔB) = AB ± AΔB ± BΔA ± ΔAΔB
Divide both sides by Z (where Z = AB):1 ± (ΔZ/Z) = 1 ± (ΔB/B) ± (ΔA/A) ± (ΔAΔB / AB)
Since the term (ΔAΔB / AB) is a product of two very small numbers, it is negligible and can be ignored. Therefore, the maximum relative error is:ΔZ/Z = ΔA/A + ΔB/B
Important Rule: The same rule applies for division (Z = A / B). In both multiplication and division, the **maximum relative error in the result is the sum of the relative errors of the individual components**.
Answer: Here are the fundamental rules to determine the number of significant figures in a measurement:
143.85 g has 5 significant figures.200.08 m has 5 significant figures.0.0034 kg has only 2 significant figures (3 and 4).4.500 m has 4 significant figures.4700 can be written as 4.7 × 103 (2 sig figs) or 4.700 × 103 (4 sig figs).Answer: Both Vernier Calipers and Screw Gauges are precision instruments used in laboratories to measure small dimensions with higher accuracy than a standard meter ruler.
Total Reading = Main Scale Reading (MSR) + [Vernier Scale Coincidence (VSD) × Least Count].Total Reading = Pitch Scale Reading (PSR) + [Circular Scale Reading (CSR) × Least Count].Answer:
1. What is Dimensional Analysis?
Dimensional analysis is a tool in physics that studies the relationships between physical quantities by identifying their fundamental dimensions (Mass [M], Length [L], Time [T], etc.) and analyzing how they combine to form derived units.
2. Applications of Dimensional Analysis (With Examples):
v2 = u2 + 2as. [v2] = [L2T-2]. [u2] = [L2T-2], and [as] = [LT-2][L] = [L2T-2]. LHS and RHS match.n1[u1] = n2[u2], or: n2 = n1 × [M1/M2]a × [L1/L2]b × [T1/T2]c.T = k · lx · gy, and solving for exponents to find T = 2π√(l/g).3. Limitations of Dimensional Analysis:
M, L, T).s = ut + at2 is dimensionally correct but lacks the correct factor ½).Answer: Every measurement contains uncertainties called errors. These errors are broadly classified as follows:
1. Systematic Errors
These errors consistently skew measurements in one direction (always positive or always negative) and can be trace to specific causes. They include:
2. Random Errors
These errors occur irregularly and unpredictably due to minor variations in laboratory environments (temperature, vibrations, or voltage). They do not have a uniform sign or magnitude. They are minimized by taking a large number of trials and finding the average value.
3. Least Count Errors
This is the error associated with the resolution of the instrument. It is equal to the least count of the scale used. For instance, a standard ruler has a least count error of 1 mm.
4. Mathematical Calculation of Errors
In analyzing experimental data, we calculate three distinct error values:
Δamean = (Σ|Δai|) / nΔamean / amean(Δamean / amean) × 100%Answer:
Part A: Determining Significant Figures
Significant figures represent the digits in a number that carry meaningful contributions to its measurement resolution. The key rules are:
12.3 has 3 sig figs).10.02 has 4 sig figs).0.005 has 1 sig fig).8.200 has 4 sig figs).Part B: Rounding Off Rules (With Examples)
When dropping non-significant digits from a calculated result, follow these rules:
| Rule Condition | Action | Example |
| Digit to be dropped is **greater than 5** | Preceding digit is increased by 1. | 7.86 rounds to 7.9 |
| Digit to be dropped is **less than 5** | Preceding digit is left unchanged. | 3.94 rounds to 3.9 |
| Digit to be dropped is **exactly 5**, followed by non-zero digits | Preceding digit is increased by 1. | 16.351 rounds to 16.4 |
| Digit to be dropped is **exactly 5**, followed by zero and preceding digit is **odd** | Preceding digit is increased by 1. | 4.350 rounds to 4.4 |
| Digit to be dropped is **exactly 5**, followed by zero and preceding digit is **even** | Preceding digit is left unchanged. | 4.250 rounds to 4.2 |
Answer: Here is a reference table showing the relationship and dimensional formulas of the most important physical quantities in the Class 11 and 12 physics curriculum:
| Physical Quantity | Relation Formula | SI Unit | Dimensional Formula |
| Area | Length × Breadth | m2 | [M0 L2 T0] |
| Velocity | Displacement / Time | m/s | [M0 L1 T-1] |
| Acceleration | Velocity / Time | m/s2 | [M0 L1 T-2] |
| Force | Mass × Acceleration | N (Newton) | [M1 L1 T-2] |
| Work / Energy | Force × Distance | J (Joule) | [M1 L2 T-2] |
| Pressure | Force / Area | Pa (Pascal) | [M1 L-1 T-2] |
| Frequency | 1 / Time Period | Hz | [M0 L0 T-1] |
| Gravitational Constant (G) | F · r2 / (m1 · m2) | N·m2/kg2 | [M-1 L3 T-2] |
Answer:
1. Least Count in Instruments
The **least count** is the smallest division on the scale of a measuring instrument. It represents the ultimate precision boundary of the instrument.
For example:
1 mm0.1 mm0.01 mmTaking measurements smaller than the least count introduces uncertainty, which contributes to least count error.
2. Zero Error in Instruments
Zero error is a systematic error that occurs when the instrument's indicator does not point to exactly zero when no measurement is being taken (e.g., when jaws of calipers are shut). It is classified into two types:
Correct Value = Observed Reading - Positive Zero Error.Correct Value = Observed Reading + |Negative Zero Error|.Students can download the Units and Measurements Class 11 Notes PDF for quick revision before school exams, unit tests, JEE, and NEET preparation.
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Also Check: Class 11 Physics Notes Chapter Wise
Units and Measurements is the foundation chapter of Class 11 Physics. It teaches how physical quantities are measured, how units are written, how errors are calculated, and how dimensions are used to check formulas.
Students should focus on:
This chapter is useful for school exams, practicals, JEE, NEET, and all future Physics chapters.
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The least count of a vernier caliper is the smallest distance it can measure accurately. It is commonly 0.1 mm or 0.01 cm.
It is calculated using:
LC = 1 MSD - 1 VSD
or:
LC = Value of 1 MSD / Number of Vernier Scale Divisions
Yes, a quantity can have units but no dimensions. Plane angle and solid angle are common examples.
Plane angle is measured in radians and solid angle is measured in steradians. Both are ratios of similar quantities, so they are dimensionless.
The three main applications of dimensional analysis are checking the correctness of equations, converting units, and deriving relations between physical quantities.
Dimensional analysis is useful, but it cannot find numerical constants or prove complete physical correctness of an equation.
In multiplication and division, percentage errors are added.
If:
Z = A × B
or:
Z = A / B
then:
ΔZ / Z = ΔA / A + ΔB / B
From Newton’s law of gravitation:
F = Gm1m2 / r2
Therefore:
G = Fr2 / m1m2
Writing dimensions:
[G] = [MLT-2][L2] / [M][M]
[G] = [M-1L3T-2]
Planck’s constant has the dimensions of energy multiplied by time.
h = E × t
Energy has dimension:
[E] = [ML2T-2]
Time has dimension:
[t] = [T]
Therefore:
[h] = [ML2T-2][T]
[h] = [ML2T-1]
Yes, Units and Measurements is important for JEE and NEET because it includes dimensional analysis, error calculation, significant figures, and measurement-based concepts. Questions from this chapter are usually formula-based and scoring.