Fundamental and Derived Units form the basis of measurement in Physics and are essential for understanding all physical quantities. Fundamental units such as mass, length, and time are independent and form the foundation of the SI system, while derived units are obtained from combinations of these basic units. In this topic, we define the units of mass, length, and time and explore how they are used to express various physical quantities accurately.
- What are Fundamental and Derived Units ?
- What are the characteristics of a Unit ?
- How is the unit of Mass defined?
- How is the unit of Length defined?
- How is the unit of Time defined?
- Short Answer Types Question and Answers
- Multiple Choice Questions (MCQs) with Answers & Explanations
- Frequently Asked Questions (FAQs)
- Worksheet on Fundamental and Derived Quantities
- Buy Complete Study Material
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What are Fundamental and Derived Units ?
Fundamental units are those units, which can neither be derived from one another, nor can they be further resolved into any other units.
The quantities mass, length and the time are called fundamental physical quantities and their units are known as fundamental units.
For measuring mass, length and time, there are independent units such as kilogram, metre and second.
For measuring other physical quantities, if a separate unit is defined for each of them, then it will become very difficult to remember all of them as they will be quite unrelated to each other. It is found that the units of the various physical quantities can be expressed in terms of the fundamental units of mass, length and time and such units are called derived units.
The units of all such physical quantities, which can be expressed in terms of the fundamental units of mass, length and time, are called derived units.
For example, the unit of area is a derived unit. The unit of area is “area of a square” having sides each of unit length. The unit of volume is “volume of a cube” having sides each of unit length. In fact, the unit of any physical quantity can be derived from its defining equation.
To explain, let us consider the defining equation of speed. The fundamental relationship between speed, distance, and time is expressed as:
$$Speed = \frac{Distance\ Covered}{Time\ Taken}$$
As you noted, we use the standard SI units to derive the unit for speed:
- Unit of Distance (Length): metre (m)
- Unit of Time: second (s)
Therefore, the unit of speed is calculated as:
$$\text{Unit of Speed} = \frac{\text{metre}}{\text{second}} = \text{m/s} \text{ (or ms}^{-1})$$
The units of the physical quantities such as acceleration, momentum, force, work, etc are all derived units and they can be obtained by writing their defining equations in terms of fundamental physical quantities.
| 🧠 Brain Teaser |
| The derived unit of a physical quantity should not be expressed by using the slash (/). For example, speed of 10 metres per second should not be expressed as 10 m/s. It is preferred to write it in index of notation i.e. as 10 m s-1 |
What are the characteristics of a Unit ?
In early days, people used to measure a distance or length of an object in terms of cubit (length of the human arm from elbow to the tip of the middle finger), foot (length of the human foot), etc. Such units for length measurement cannot be regarded as standards and hence these were discarded with the passage of time. A unit selected for measuring a physical quantity must fulfil the following
requirements :
- It should be well defined.
- It should be of suitable size i.e. neither too large nor too small in comparison
to the quantity to be measured. - It should be easily reproducible at all places.
- It should not change with time and from place to place.
- It should not change with change in its physical conditions, such as
temperature, pressure, etc. - It should be easily accessible.
Keeping in view the requirements of a standard unit, the fundamental units of mass, length and time, namely kilogram, metre and second were defined and redefined as discussed in the following sections.
How is the unit of Mass defined?
Mass of a body is the quantity of matter it contains. It is an essential property of a material body and therefore for a material body, it can never be zero. It is not affected by the presence of other bodies. Further, it does not vary with change in temperature, pressure or location of the body in space.
The internationally accepted unit of mass is kilogram.
Definition of Unit of Mass: In the International System of Units (SI), the unit of mass is the kilogram (kg).
Originally, kilogram was defined as the mass of one cubic decimetre of water at 4°C (the temperature at which density of water is maximum).
Define kilogram
The General Conference of Weights and Measures defined kilogram as the mass of a platinum-iridium cylinder kept at the International Bureau of Weights and Measures at Sevres, near Paris, France.
In the near future, kilogram may be defined in terms of the mass of some fundamental particle, like proton.
| 🔑 Key Point |
| In atomic and nuclear physics, mass is measured in terms of atomic mass unit (a.m.u.) |
Define One a.m.u.
One a.m.u. (atomic mass unit) is defined as $\dfrac{1}{12}$th of the mass of one $^{12}\text{C}$ atom.
It can be calculated that:
$$1\ \text{a.m.u.} = 1.66 \times 10^{-27}\ \text{kg}$$
How is the unit of Length defined?
In 1960, the Eleventh General Conference of Weights and Measures decided to define metre by adopting atomic standards.
Define metre
On atomic standards, one metre is defined as to be equal to 1,650,763·3 wavelengths in vacuum of the orange-red coloured radiation emitted by krypton having mass number 86.
Krypton-86 emits light of several different wavelengths. The orange-red coloured light emitted by krypton-86 has wavelength $6057.8021\ \text{\AA}$ or $6.0578021 \times 10^{-7}\ \text{m}$. The number of these wavelengths in $1\ \text{m}$ can be counted by using an optical interferometer, which comes out to be $16,50,763.3$.
Formal Definition of metre
In 1983, one metre was defined as the length of the path travelled by light in vacuum during a time interval of $\frac{1}{299,792,458}$ of a second.
How is the unit of Time defined?
According to Einstein, time is simply what a clock reads. In 1964, the Twelfth General Conference of Weights and Measures held in Paris adopted atomic standard for measurement of time.
Define Second
One second was defined as to be equal to the duration of 9, 192, 631, 770 vibrations corresponding to the transition between two hyperfine levels of cesium-133 atom in the ground state.
Short Answer Types Question and Answers
Q: What are Fundamental Quantities?
A: Fundamental quantities are physical quantities that are independent and do not require other quantities for their definition. They are also known as absolute or base quantities. Examples include length, mass, and time.
Q: What are derived quantities?
A: Derived quantities are those that can be obtained through suitable multiplication or division of fundamental quantities. For example, area and volume are derived from length.
Q: Give Examples of Fundamental and Derived Quantities
A: The examples of Fundamental and Derived Quantities are as follows :
- Fundamental Quantities: Length, Mass, Time
- Derived Quantities: Area (Length²), Volume (Length³), Speed (Length/Time), Force (Mass × Acceleration)
Note: The choice of fundamental quantities in mechanics is arbitrary. If speed and time are chosen as fundamental, length becomes derived as Speed × Time. Similarly, if force and acceleration are fundamental, mass becomes derived as Force / Acceleration.
Q: What are fundamental units?
A: Fundamental units are the standard units for fundamental quantities. In mechanics, units of mass, length, and time are considered fundamental. Examples: Kilogram (kg), Meter (m), Second (s).
Q: What are derived units?
A: Derived units are expressed in terms of fundamental units. Examples include:
- Speed (m/s) → Derived from Length/Time
- Force (kg·m/s²) → Derived from Mass × Acceleration
- Density (kg/m³) → Derived from Mass/Volume
Q: Give Examples of Fundamental and Derived Units
A: The examples of Fundamental and Derived Units are as follows :
- Fundamental Units: Meter (m), Kilogram (kg), Second (s)
- Derived Units: m² (Area), m³ (Volume), m/s (Speed), kg·m/s² (Force)
Multiple Choice Questions (MCQs) with Answers & Explanations
Q1: Which of the following is a fundamental quantity?
A) Speed
B) Force
C) Mass
D) Acceleration
Answer: C) Mass
Explanation: Mass is independent and does not rely on other quantities, making it fundamental.
Q2: Which of the following is a derived quantity?
A) Length
B) Time
C) Force
D) Mass
Answer: C) Force
Explanation: Force is derived from the equation Force = Mass × Acceleration, making it a derived quantity.
Q3: The unit of force in the SI system is:
A) kg/m³
B) m/s
C) kg·m/s²
D) s⁻¹
Answer: C) kg·m/s²
Explanation: Force is given by F = Mass × Acceleration, and its SI unit is Newton (N) = kg·m/s².
Q4: If force and acceleration are chosen as fundamental, which quantity becomes derived?
A) Length
B) Mass
C) Speed
D) Time
Answer: B) Mass
Explanation: Mass is expressed as Force / Acceleration, making it a derived quantity in this case.
Q5: The SI unit of speed is:
A) m/s²
B) m/s
C) kg·m/s²
D) m³
Answer: B) m/s
Explanation: Speed is defined as Distance / Time, and its SI unit is meters per second (m/s).
Frequently Asked Questions (FAQs)
Q1: Why are fundamental quantities important?
A: Fundamental quantities are the basis for defining all other physical quantities. Without them, measurements and scientific equations would not be standardized.
Q2: Can fundamental quantities change?
A: The choice of fundamental quantities is arbitrary and can change based on the system of units. However, in mechanics, mass, length, and time are the standard fundamental quantities.
Q3: What is an example of a fundamental and derived unit?
A: The meter (m) is a fundamental unit (for length), whereas m² (square meter) is a derived unit (for area).
Q4: Is acceleration a fundamental or derived quantity?
A: Acceleration is a derived quantity, as it is expressed as Velocity / Time (m/s²).
Worksheet on Fundamental and Derived Quantities
Q1: Define fundamental and derived quantities with two examples each.
Q2: Explain why length, mass, and time are chosen as fundamental quantities in mechanics.
Q3: How can speed and time be chosen as fundamental quantities?
Q4: Classify the following as fundamental or derived quantities:
- Energy
- Temperature
- Work
- Density
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