"Units, Dimensions and Vectors"

About

This is the first chapter of the NIOS Senior Secondary Physics course (Paper Code: 312). It lays the foundation for all of physics by introducing the language of measurement — units, dimensions, and vectors. You will learn how physical quantities are measured and expressed, how to check the correctness of equations using dimensional analysis, and how to work with vectors which are essential for describing motion, force, and fields in later chapters.

Key Concepts

1.1 Physics and Measurement

Physics is the study of the fundamental laws of nature. These laws are:

• Universal — they apply everywhere in the universe
• Precise — they give accurate, reproducible results
• Consistent — they do not change with time or place

The application of physics has transformed human life through electricity, electronics, communication, transportation, medical technology, and space exploration.

1.2 Units and Systems of Units

A unit is a standard reference used to express a physical quantity. The SI system (Système International) is the modern metric system with seven base units:

Derived units are combinations of base units (e.g., velocity = m/s, force = kg⋅m/s² = newton).

Unit prefixes:

1.3 Significant Figures

Significant figures in a measurement are the digits that carry meaningful information about the precision of the quantity. They include all certain digits plus the first uncertain digit.

Rules for counting significant figures:

1. All non-zero digits are significant
2. Zeros between non-zero digits are significant
3. Leading zeros are NOT significant
4. Trailing zeros in a number with a decimal point ARE significant
5. Trailing zeros in a number without a decimal point are NOT significant (ambiguous)

Examples:

• 426.69 → 5 significant figures
• 4200304.002 → 10 significant figures
• 0.3040 → 4 significant figures
• 4050 m → 3 significant figures (no decimal point)
• 5000 → 1 significant figure (no decimal point)

Changing units does NOT change the number of significant figures. E.g., 3.486 m and 348.6 cm both have 4 significant figures.

1.4 Dimensional Analysis

The dimension of a physical quantity shows how it relates to the base quantities. Common dimensional formulas:

Four applications of dimensional analysis:

1. Checking the correctness of physical equations
2. Converting units from one system to another
3. Deriving relations between physical quantities
4. Identifying dimensions of unknown quantities

These are all based on the principle of dimensional homogeneity: all terms in a physical equation must have the same dimensional formula.

1.5 Vectors and Scalars

Vector representation: A vector $\vec{A}$ has magnitude $|\vec{A}|$ and direction $\hat{A}$.

Types of vectors:

• Unit vector: Magnitude = 1, e.g., $\hat{i},\hat{j},\hat{k}$
• Null vector: Magnitude = 0
• Equal vectors: Same magnitude and direction
• Negative vector: Same magnitude, opposite direction

1.6 Vector Operations

Addition (Triangle Law): $\vec{R}=\vec{A}+\vec{B}$

If $\vec{A}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$ and $\vec{B}=b_x\hat{i}+b_y\hat{j}+b_z\hat{k}$: $\vec{A}+\vec{B}=(a_x+b_x)\hat{i}+(a_y+b_y)\hat{j}+(a_z+b_z)\hat{k}$

Scalar (Dot) Product: $\vec{A}\cdot \vec{B}=|\vec{A}||\vec{B}|\cos \theta =a_x{b}_x+a_y{b}_y+a_z{b}_z$

Vector (Cross) Product: $\vec{A}\times \vec{B}=|\vec{A}||\vec{B}|\sin \theta \cdot \hat{n}$

In component form: $\vec{A}\times \vec{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_x& a_y& a_z\\ b_x& b_y& b_z\end{array}\right|$

INTEXT QUESTIONS 1.1

Q1. Discuss the nature of laws of physics.

Ans: The laws of physics are universal, precise, and consistent across space and time. They describe the behavior of the natural world, such as motion, gravity, and energy, and are based on experimental observations and reasoning. These laws apply to everything in the universe, from subatomic particles to galaxies.

Q2. How has the application of the laws of physics led to better quality of life?

Ans: Applications of physical laws have enabled the development of technologies like electricity, communication, transportation, and medicine. For example, understanding electromagnetism has led to electric power, and Newton's laws enabled the design of cars and airplanes — all contributing to better comfort, productivity, and health.

Q3. What is meant by significant figures in measurement?

Ans: Significant figures in a measurement are the digits that carry meaningful information about the precision of the quantity. They include all the certain digits plus the first uncertain digit. They help reflect the accuracy of measurement.

Q4. Find the number of significant figures in the following quantities, quoting the relevant laws:

(i) 426.69 Ans: 5 significant figures.

(ii) 4200304.002 Ans: 10 significant figures.

(iii) 0.3040 Ans: 4 significant figures.

(iv) 4050 m Ans: 3 significant figures.

(v) 5000 Ans: 1 significant figure.

Q5. The length of an object is 3.486 m. If it is expressed in centimetre (i.e., 348.6 cm), will there be any change in number of significant figures in the two cases?

Ans: No, there will be no change in the number of significant figures. Both 3.486 m and 348.6 cm have four significant figures. Changing the unit does not affect the number of significant digits.

Q6. What are the four applications of the principles of dimensions? On what principle are the above based?

Ans: The four applications of dimensional analysis are:

1. To check the correctness of physical equations.
2. To convert units from one system to another.
3. To derive relations between physical quantities.
4. To identify the dimension of unknown quantities.

These are all based on the principle of dimensional homogeneity, which states that all terms in a physical equation must have the same dimensional formula.

Q7. The mass of the sun is 2 × 10³⁰ kg. The mass of a proton is 2 × 10⁻²⁷ kg. If the sun was made only of protons, calculate the number of protons in the sun.

Ans: $\text{Number of protons}=\frac{\text{Mass of sun}}{\text{Mass of one proton}}=\frac{2\times 10^{30}}{2\times 10^{-27}}=10^{57}$

Answer: $10^{57}$ protons.

Q8. Earlier the wavelength of light was expressed in angstroms. One angstrom equals 10⁻⁸ cm. Now the wavelength is expressed in nanometers. How many angstroms make one nanometre?

Ans:

• 1 angstrom = 10⁻⁸ cm
• 1 nanometre = 10⁻⁷ cm

$\text{Number of angstroms in 1 nm}=\frac{10^{-7}\text{ cm}}{10^{-8}\text{ cm}}=10$

Answer: 10 angstroms make one nanometre.

Q9. A radio station operates at a frequency of 1370 kHz. Express this frequency in GHz.

Ans:

• 1 GHz = 10⁹ Hz
• 1370 kHz = 1370 × 10³ Hz = 1.37 × 10⁶ Hz

$\text{Frequency in GHz}=\frac{1.37\times 10^6}{10^9}=1.37\times 10^{-3}\text{ GHz}$

Answer: $1.37\times 10^{-3}$ GHz.

Q10. How many decimetres are there in a decametre? How many MW are there in one GW?

Ans:

Decimetres in one decametre:

• 1 decametre = 10 metres
• 1 metre = 10 decimetres
• 1 decametre = 10 × 10 = 100 decimetres

Megawatts in one gigawatt:

• 1 gigawatt = 10⁹ watts
• 1 megawatt = 10⁶ watts

$\text{Number of MW in 1 GW}=\frac{10^9}{10^6}=1000$

Answer: 100 decimetres in a decametre; 1000 megawatts in one gigawatt.

Terminal Exercise

1. Define physical quantity. Distinguish between fundamental and derived quantities with examples.

2. State the principle of homogeneity of dimensions. Check the correctness of the equation $v^2=u^2+2as$ using dimensional analysis.

3. Convert 1 newton into dyne using dimensional analysis.

4. What are the limitations of dimensional analysis?

5. Find the dimensional formula of: (a) Planck's constant, (b) Gravitational constant G.

6. Distinguish between scalar and vector quantities. Give four examples of each.

7. State and explain the triangle law of vector addition.

8. Two vectors $\vec{A}=3\hat{i}+4\hat{j}$ and $\vec{B}=\hat{i}-2\hat{j}+2\hat{k}$. Find $\vec{A}+\vec{B}$, $|\vec{A}+\vec{B}|$, $\vec{A}\cdot \vec{B}$, and $\vec{A}\times \vec{B}$.

9. Show that the scalar product is commutative ($\vec{A}\cdot \vec{B}=\vec{B}\cdot \vec{A}$) but the vector product is anti-commutative ($\vec{A}\times \vec{B}=-\vec{B}\times \vec{A}$).

10. The percentage error in measurement of length and time period of a simple pendulum are 1% and 2% respectively. Find the maximum percentage error in the measurement of acceleration due to gravity using the formula $g=4{\pi }^{2}L/T^2$.

Worked Examples

Example 1: Significant Figures

Problem: A student measures the length of a rod as 2.50 cm using a vernier calliper. How many significant figures are in this measurement?

Solution:

• The digits 2, 5, and 0 are all meaningful
• The trailing zero after the decimal IS significant
• Number of significant figures = 3

Example 2: Dimensional Check

Problem: Check if the equation $s=ut+\frac{1}{2}a{t}^2$ is dimensionally correct.

Solution:

• LHS: $[s]=[L]$
• RHS: $[ut]=[L{T}^{-1}][T]=[L]$
• $[\frac{1}{2}a{t}^2]=[L{T}^{-2}][T^2]=[L]$

Since all terms have dimension $[L]$, the equation is dimensionally correct. ✅

Example 3: Dot Product

Problem: Find the angle between $\vec{A}=2\hat{i}+2\hat{j}-\hat{k}$ and $\vec{B}=6\hat{i}-3\hat{j}+2\hat{k}$.

Solution: $\vec{A}\cdot \vec{B}=(2)(6)+(2)(-3)+(-1)(2)=12-6-2=4$

$|\vec{A}|=\sqrt{4+4+1}=\sqrt{9}=3$ $|\vec{B}|=\sqrt{36+9+4}=\sqrt{49}=7$

$\cos \theta =\frac{\vec{A}\cdot \vec{B}}{|\vec{A}||\vec{B}|}=\frac{4}{3\times 7}=\frac{4}{21}$

$\theta ={\cos }^{-1}\left(\frac{4}{21}\right)$

Common Mistakes

1. Confusing precision with accuracy: Precision relates to the closeness of repeated measurements; accuracy relates to how close a measurement is to the true value.
2. Miscounting significant figures in numbers without decimal points: 5000 has 1 significant figure, but 5000. (with decimal) has 4.
3. Forgetting that dimensional analysis cannot give dimensionless constants: It cannot determine values like $\frac{1}{2}$, $\pi$, etc.
4. Treating vectors as scalars: Direction matters for vectors; forgetting direction leads to wrong results in addition/subtraction.
5. Confusing dot product and cross product results: Dot product gives a scalar; cross product gives a vector.

Quick Revision