Physics notes on Physical quantities, Units, dimensions and Measurements. This notes of physics is helpful for academic learning, competitive and government exams like SSC CGL, UPSC CSE, CDS, NDA, and others.

## Table of Contents

# Physical Quantity, Units and Dimensions

In the vast and intricate realm of science, understanding the properties and behaviors of the physical world requires a structured framework. This framework is built upon the fundamental concepts of physical quantities, units, and dimensions – the cornerstones of measurement and mathematical description. Whether you’re a curious science enthusiast or a dedicated student embarking on a journey through the intricacies of physics, delving into the world of physical quantities and their associated units and dimensions is an essential first step.

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## Definition of Physical Quantity:

A physical quantity is a property or attribute that can be measured and expressed quantitatively. It consists of a numerical value and a unit. For example, length, mass, time, temperature, and electric current are all physical quantities.

## Scalar and Vector Quantities:

Physical quantities can be classified into two main types: **scalar** and **vector** quantities.

** a. Scalar Quantity:** A scalar quantity has only magnitude and no direction. Examples include mass, temperature, volume, and speed.

** b. Vector Quantity:** A vector quantity has both magnitude and direction. Examples include **displacement, velocity, acceleration, force**, and **electric field.**

## Fundamental and Derived Quantities:

Physical quantities are further categorized as **fundamental** and **derived** quantities.

a. **Fundamental Quantities:** Fundamental quantities are independent and cannot be expressed in terms of other physical quantities. Examples include length, mass, time, and electric current. Fundamental quantities are typically chosen as base units for measurements.

** b. Derived Quantities:** Derived quantities are obtained by combining fundamental quantities. They are expressed in terms of fundamental quantities using mathematical equations. Examples include **speed** (derived from distance and time), **density** (derived from mass and volume), and **force** (derived from mass, length, and time).

## Units and Measurements:

To quantify physical quantities, units of measurement are used. Units provide a standardized and consistent way of expressing quantities. The **International System of Units (SI)** is the globally accepted system for measurements. It comprises **seven base units**, from which all other units are derived:

### a. Base Units:

**Length**: meter (m)**Mass**: kilogram (kg)**Time**: second (s)**Electric Current**: ampere (A)**Temperature**: kelvin (K)**Amount of Substance**: mole (mol)**Luminous Intensity**: candela (cd)

### b. Derived Units:

Numerous derived units are derived from the base units. For example, area (m²), volume (m³), velocity (m/s), acceleration (m/s²), and so on.

### c. Prefixes:

SI units can be modified using prefixes to represent values that are too large or too small. Common prefixes include kilo-, mega-, micro-, nano-, etc.

## Conversion of Units:

Converting units is a common requirement in physics. Conversion involves changing the numerical value of a quantity while keeping its physical meaning intact. Conversion factors are used to convert between units. The key steps involved are:

a. Identify the conversion factor relating the two units.

b. Multiply the given quantity by the conversion factor to obtain the desired unit.

**Length:**

- 1 meter (m) = 3.28084 feet (ft)
- 1 kilometer (km) = 0.621371 miles (mi)
- 1 inch (in) = 2.54 centimeters (cm)

**Energy:**

- 1 joule (J) = 10^7 ergs (erg)
- 1 calorie (cal) = 4.184 joules (J)
- 1 kilowatt-hour (kWh) = 3.6 × 10^6 joules (J)

**Force:**

- 1 newton (N) = 10^5 dynes (dyn)
- 1 pound-force (lbf) = 4.44822 newtons (N)

**Area:**

- 1 square meter (m²) = 0.0001 hectares (ha)
- 1 acre = 4046.86 square meters (m²)
- 1 square kilometer (km²) = 100 hectares (ha)

**Mass:**

- 1 kilogram (kg) = 2.20462 pounds (lb)
- 1 gram (g) = 0.03527396 ounces (oz)
- 1 metric ton (t) = 1000 kilograms (kg)

**Volume:**

- 1 liter (L) = 1000 cubic centimeters (cm³)
- 1 gallon (gal) = 3.78541 liters (L)
- 1 cubic meter (m³) = 1000 liters (L)

**Temperature:**

- °C to °F: (°C × 9/5) + 32 = °F
- °F to °C: (°F – 32) × 5/9 = °C
- K (Kelvin) = °C + 273.15

**Time:**

- 1 second (s) = 1000 milliseconds (ms)
- 1 minute (min) = 60 seconds (s)
- 1 hour (hr) = 60 minutes (min)

**Pressure:**

- 1 pascal (Pa) = 0.00001 bar
- 1 atmosphere (atm) = 101325 pascals (Pa)

**Speed:**

- 1 meter per second (m/s) = 2.23694 miles per hour (mph)
- 1 kilometer per hour (km/h) = 0.621371 miles per hour (mph)

**Power:**

- 1 watt (W) = 0.001 kilowatts (kW)
- 1 horsepower (hp) = 745.7 watts (W)

**Volume (continued):**

- 1 cubic foot (ft³) = 28.3168 liters (L)
- 1 cubic inch (in³) = 16.3871 cubic centimeters (cm³)
- 1 gallon (US) = 128 fluid ounces (fl oz)

**Pressure (continued):**

- 1 bar = 100000 pascals (Pa)
- 1 psi (pound per square inch) = 6894.76 pascals (Pa)

**Temperature (continued):**

- °C to K: Kelvin = °C + 273.15
- K to °C: Celsius (°C) = Kelvin – 273.15

**Energy (continued):**

- 1 kilocalorie (kcal) = 4184 joules (J)
- 1 electronvolt (eV) = 1.60218 x 10^-19 joules (J)

**Mass (continued):**

- 1 ton (US ton) = 2000 pounds (lb)
- 1 ounce (oz) = 28.3495 grams (g)

**Frequency:**

- 1 hertz (Hz) = 1 cycle per second
- 1 kilohertz (kHz) = 1000 hertz (Hz)
- 1 megahertz (MHz) = 10^6 hertz (Hz)

**Data Storage:**

- 1 byte (B) = 8 bits
- 1 kilobyte (KB) = 1024 bytes
- 1 megabyte (MB) = 1024 kilobytes (KB)
- 1 gigabyte (GB) = 1024 megabytes (MB)

**Electricity:**

- 1 ampere (A) = 1 coulomb per second (C/s)
- 1 volt (V) = 1 joule per coulomb (J/C)
- 1 ohm (Ω) = 1 volt per ampere (V/A)

**Angles:**

- 1 degree (°) = π/180 radians (rad)
- 1 radian (rad) = 180/π degrees (°)

**Fuel Efficiency:**

- 1 mile per gallon (mpg) = 0.4251 kilometers per liter (km/L)
- 1 liter per 100 kilometers (L/100 km) = 235.215 miles per gallon (mpg)

**Density:**

- 1 kilogram per cubic meter (kg/m³) = 0.0624 pounds per cubic foot (lb/ft³)

## Dimensional Analysis:

Dimensional analysis is a powerful tool for **understanding the relationships between physical quantities**. It involves checking the dimensions of different quantities to determine their compatibility in equations. Each physical quantity has a dimension that represents its nature. For example, length has the dimension [L], mass has [M], time has [T], and so on. By comparing the dimensions on both sides of an equation, you can verify its correctness.

### Important physical quantities along with their SI units and dimensions

No. | Measurement | Symbol | Measurement Unit | Dimensions |

1 | Length | m | Meter | [L] |

2 | Mass | kg | Kilogram | [M] |

3 | Time | s | Second | [T] |

4 | Electric Current | A | Ampere | [I] |

5 | Temperature | K | Kelvin | [Θ] |

6 | Amount of Substance | mol | Mole | [N] |

7 | Luminous Intensity | cd | Candela | [J] |

8 | Speed | m/s | Meter per second | [L]/[T] |

9 | Acceleration | m/s² | Meter per second squared | [L]/[T]² |

10 | Force | N | Newton | [M][L]/[T]² |

11 | Work, Energy, and Quantity of Heat | J | Joule | [M][L]²/[T]² |

12 | Power | W | Watt | [M][L]²/[T]³ |

13 | Pressure | Pa | Pascal | [M]/[L][T]² |

14 | Electric Charge | C | Coulomb | [I][T] |

15 | Electric Potential | V | Volt | [M][L]²/[T]²[I] |

16 | Electric Resistance | Ω | Ohm | [M][L]²/[T]³[I] |

17 | Electric Current Density | A/m² | Ampere per square meter | [I]/[L]² |

18 | Electric Field Strength | V/m | Volt per meter | [M][L]/[T]³[I] |

19 | Magnetic Flux | Wb | Weber | [M][L]²/[T]²[I] |

20 | Magnetic Field Strength | T | Tesla | [M][I]/[T]² |

21 | Frequency | Hz | Hertz | 1/[T] |

22 | Voltage | V | Volt | [M][L]²/[T]²[I] |

23 | Electric Capacitance | F | Farad | [M]⁻¹[L]⁻²[T⁴][I²] |

24 | Electric Inductance | H | Henry | [M][L]²[T]⁻²[I]⁻² |

25 | Electric Conductance | S | Siemens | [M]⁻¹[L]⁻²[T³][I²] |

26 | Electric Potential Energy | J | Joule | [M][L]²/[T]² |

27 | Density | kg/m³ | Kilogram per cubic meter | [M]/[L]³ |

28 | Volume | m³ | Cubic meter | [L]³ |

29 | Area | m² | Square meter | [L]² |

30 | Angular Velocity | rad/s | Radian per second | 1/[T] |

31 | Torque | N·m | Newton-meter | [M][L]²/[T]² |

32 | Electric Power | W | Watt | [M][L]²/[T]³ |

33 | Impulse | N·s | Newton-second | [M][L]/[T] |

34 | Momentum | kg·m/s | Kilogram-meter per second | [M][L]/[T] |

35 | Electric Field Intensity | N/C | Newton per coulomb | [M]/[T]²[I] |

36 | Magnetic Field Intensity | A/m | Ampere per meter | [I]/[L] |

37 | Sound Intensity | W/m² | Watt per square meter | [M]/[L][T]³ |

38 | Gravitational Constant | N·m²/kg² | Newton meter squared per kilogram squared | [M]⁻¹[L]³[T]⁻² |

39 | Planck’s Constant | J·s | Joule-second | [M][L]²[T]⁻¹ |

40 | Universal Gas Constant | J/(mol·K) | Joule per mole-kelvin | [M][L]²/[T]²[N]⁻¹ |

## Practice Set on Physical Quantities

1. What is the SI unit of electric current?

- a) Kilogram
- b) Ampere
- c) Second
- d) Kelvin

2. Which of the following is a derived quantity?

- a) Length
- b) Mass
- c) Time
- d) Velocity

3. What is the SI unit of force?

- a) Pascal
- b) Newton
- c) Kilogram
- d) Watt

4. Which physical quantity is represented by the symbol “T”?

- a) Time
- b) Temperature
- c) Torque
- d) Electric current

5. What is the SI unit of power?

- a) Watt
- b) Joule
- c) Pascal
- d) Volt

6. The speed of light in a vacuum is approximately:

- a) 3 x 10^8 m/s
- b) 9.8 m/s²
- c) 1.6 x 10^-19 C
- d) 6.63 x 10^-34 J·s

7. Which physical quantity has the dimension [L^2T^-2]?

- a) Velocity
- b) Acceleration
- c) Force
- d) Energy

8. What is the SI unit of pressure?

- a) Pascal
- b) Joule
- c) Newton
- d) Volt

9. Which of the following is a scalar quantity?

- a) Velocity
- b) Force
- c) Acceleration
- d) Temperature

10. What is the SI unit of electric charge?

- a) Ampere
- b) Volt
- c) Coulomb
- d) Ohm

11. Which physical quantity is measured in units of Coulombs?

- a) Electric charge
- b) Electric resistance
- c) Electric potential
- d) Electric current

12. What is the SI unit of density?

- a) Kilogram per cubic meter
- b) Meter per second
- c) Joule per second
- d) Newton per meter

13. The quantity of matter in a substance is measured in:

- a) Moles
- b) Ampere
- c) Kelvin
- d) Candela

14. Which physical quantity is measured in units of Hertz?

- a) Frequency
- b) Electric potential
- c) Electric current
- d) Magnetic field

15. What is the SI unit of volume?

- a) Cubic meter
- b) Square meter
- c) Meter per second
- d) Pascal

16. Which of the following is a vector quantity?

- a) Speed
- b) Temperature
- c) Acceleration
- d) Distance

17. What is the SI unit of angular velocity?

- a) Radian per second
- b) Meter per second
- c) Ampere
- d) Newton

18. The rate of change of velocity is known as:

- a) Speed
- b) Acceleration
- c) Force
- d) Power

19. What is the SI unit of magnetic field strength?

- a) Tesla
- b) Ohm
- c) Watt
- d) Volt

20. Which physical quantity has the dimension [ML^2T^-3]?

- a) Pressure
- b) Electric current
- c) Force
- d) Velocity

21. What is the SI unit of electric power?

- a) Watt
- b) Joule
- c) Newton
- d) Volt

22. The SI unit of impulse is the same as the unit of:

- a) Momentum
- b) Force
- c) Power
- d) Voltage

23. What is the SI unit of momentum?

- a) Kilogram-meter per second
- b) Newton
- c) Pascal
- d) Joule

24. The SI unit of electric field intensity is the same as the unit of:

- a) Electric potential
- b) Electric charge
- c) Electric resistance
- d) Electric current

25. Which physical quantity is measured in units of Watt per square meter?

- a) Sound intensity
- b) Pressure
- c) Electric potential
- d) Magnetic field intensity

**Answers**:

1. b) Ampere

2. d) Velocity

3. b) Newton

4. c) Torque

5. a) Watt

6. a) 3 x 10^8 m/s

7. c) Force

8. a) Pascal

9. d) Temperature

10. c) Coulomb

11. a) Electric charge

12. a) Kilogram per cubic meter

13. a) Moles

14. a) Frequency

15. a) Cubic meter

16. c) Acceleration

17. a) Radian per second

18. b) Acceleration

19. a) Tesla

20. c) Force

21. a) Watt

22. a) Momentum

23. a) Kilogram-meter per second

24. a) Electric potential 25. a) Sound intensity

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## FAQs on Physical Quantities

**Q1: What is a physical quantity?**

A1: A physical quantity is a property or characteristic of a phenomenon, substance, or object that can be measured. Examples include length, mass, time, temperature, etc.

**Q2: What is a unit of measurement?**

A2: A unit of measurement is a specific standard quantity used to express a particular physical quantity. Units provide a standardized way to communicate measurements universally.

**Q3: What is the difference between a scalar and a vector quantity?**

A3: Scalar quantities have only magnitude (numerical value) and no direction, e.g., mass, or temperature. Vector quantities have both magnitude and direction, e.g., displacement, and velocity.

**Q4: What are the SI base units?**

A4: The SI (International System of Units) base units are: meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity.