PhysicsChapter 1: Physical Quantities and Measurement
1.3.2 System International (SI) Derived Units
SI Derived Units
Explanation level
While there are only seven base units in the SI system, there are countless physical quantities that need to be measured. Units for all these other quantities are created by combining the seven base units. These are called SI derived units. This system ensures that every possible measurement is connected back to the seven fundamental building blocks.
Definition
SI Derived Units are units of measurement for derived physical quantities. They are obtained by multiplying and/or dividing the seven SI base units. Some derived units are given special names and symbols for convenience.
Main Idea
The main idea is that the SI system is comprehensive. It doesn't need a new, unrelated unit for every single quantity. Instead, it builds all necessary units from the seven base units. This creates a coherent and interconnected system where the unit of any quantity can be broken down into a combination of base units. This simplifies calculations and ensures consistency across all fields of science.
Core Concept
The core concept is that derived units are combinations of base units. You can create the unit for any derived quantity by analyzing its formula and replacing each base quantity with its corresponding base unit.
- Example 1: Area. The formula for the area of a rectangle is length × width. The base unit for length is the metre (m). Therefore, the derived unit for area is m × m = m2 (square metre).
- Example 2: Velocity (Speed). The formula for velocity is displacement / time. The base unit for displacement is the metre (m) and for time is the second (s). Therefore, the derived unit for velocity is m/s (metre per second).
- The derived unit for force is kg·m/s2. This is given the special name newton (N).
- The derived unit for pressure is kg/(m·s2). This is given the special name pascal (Pa).
- The derived unit for energy is kg·m2/s2. This is given the special name joule (J).
Even with a special name, a derived unit is always fundamentally a combination of base units.
Detailed Explanation
The table below shows some common derived quantities and how their units are built from the base units. This table is based on Table 1.2 from the provided textbook.
| Derived Quantity | Symbol | SI Derived Unit Name | Symbol | Derived from Base Units |
|---|---|---|---|---|
| area | A | square metre | m2 | m × m |
| volume | V | cubic metre | m3 | m × m × m |
| speed, velocity | v | metre per second | m/s\) or \(m·s-1 | m / s |
| acceleration | a | metre per second squared | m/s2\) or \(m·s-2 | m / s2 |
| density | ρ | kilogram per cubic metre | kg/m3\) or \(kg·m-3 | kg / m3 |
| force | F | newton (N) | N | kg·m/s2 |
| pressure | P | pascal (Pa) | Pa | kg/(m·s2)\) or \(N/m2 |
| energy | E, U | joule (J) | J | kg·m2/s2\) or \(N·m |
This table demonstrates the power and elegance of the SI system. Every complex unit can be systematically broken down into the seven fundamental base units.
Intermediate Level
The concept of derived units is crucial for checking the consistency of physical equations. This process is called dimensional analysis. Every physical equation must have the same units on both sides. For example, in the equation F = ma (force = mass × acceleration), the unit on the right side is kg · m/s2. This must be equal to the unit on the left side, which we call the newton (N). Therefore, 1 N = 1 kg·m/s2. This shows that the newton is not a fundamental unit but a derived one. This system ensures that all of physics is interconnected through the base units.
Important Points
- - Derived units are units for derived quantities.
- - They are formed by multiplying or dividing base units.
- - Some derived units are given special names (like newton, pascal, joule) to make them simpler to use.
- - Every derived unit can ultimately be expressed in terms of the seven SI base units.
Remember
- - Derived Units = Combinations of Base Units.
- - Example: Velocity is derived from length and time (m/s).
- - Special Names: Force is a derived unit called a newton (N).
- - Foundation: All derived units are built from the seven SI base units.
Common Mistakes
- - Thinking a unit with a special name is a base unit: The newton is a derived unit, not a base unit, even though it has a special name.
- - Forgetting to use base units to derive other units: When you need the unit of a complex quantity, always break it down to the base units to check your work.
- - Incorrectly combining units: Remember that the units follow the same mathematical operations as the quantities. For example, if velocity = distance/time, then the unit is metre/second, not metre × second.
Quick Revision
SI derived units are units for derived physical quantities and are created by multiplying or dividing the seven SI base units. For example, the unit for area is the square metre (m2), derived from the base unit of length. Many derived units, such as the newton for force and the joule for energy, are given special names for convenience, but they are all fundamentally combinations of the seven base units.
60-Second Revision
Derived units are made by combining the seven base units. Examples include speed (m/s) and area (m2). While some have special names like the newton (N) for force, they are all fundamentally combinations of base units like metre, kilogram, and second.
Frequently Asked Questions (FAQs)
Q1. What is an SI derived unit? Answer: An SI derived unit is a unit of measurement for a derived physical quantity. It is formed by multiplying and/or dividing the seven SI base units. Examples include the metre per second (m/s) for speed and the newton (N) for force.
Q2. How are derived units formed? Answer: Derived units are formed based on the mathematical formula of the physical quantity. You replace each base quantity in the formula with its corresponding base unit. For example, since speed = distance/time, the unit for speed is the unit of distance (metre) divided by the unit of time (second), resulting in m/s.
Q3. Why do some derived units have special names? Answer: Some derived units, like the newton (N) for force and the joule (J) for energy, are used so frequently in physics that they are given special names. This is purely for convenience and simplicity in writing and communication. The special names are just shorthand for the combination of base units.
Q4. Is the newton a base unit or a derived unit? Answer: The newton is a derived unit. It is the unit of force and is derived from the base units of mass (kg), length (m), and time (s) as 1 N = 1 kg·m/s2.
Q5. What are the base units for the derived unit "joule"? Answer: The joule (J) is the unit of energy. It can be derived from the base units of mass, length, and time: 1 J = 1 kg·m2/s2.
Q6. Can a derived unit be formed by dividing a base unit by another base unit? Answer: Yes. A simple example is velocity, where the metre (base unit of length) is divided by the second (base unit of time) to give the derived unit m/s. This is a very common way of forming derived units.
Multiple Choice Questions (MCQs)
- Q1. Units of derived quantities are obtained by:
- A. Adding base quantities
- B. Subtracting base quantities
- C. Multiplying and/or dividing base quantities
- D. Replacing base quantities
- Correct Answer: C
- Explanation: The text states that units of derived quantities are obtained by multiplying and/or dividing base units.
- Q2. The unit for area is m2. This is an example of a:
- A. Base unit
- B. Derived unit
- C. Fundamental unit
- D. Simple unit
- Correct Answer: B
- Explanation: The unit for area is a derived unit because it is formed by combining base units (m × m = m2).
- Q3. Which of the following is the SI derived unit for force?
- A. Joule (J)
- B. Watt (W)
- C. Pascal (Pa)
- D. Newton (N)
- Correct Answer: D
- Explanation: The text specifically mentions that the derived unit of force, kg·m/s2, is given the special name 'newton' (N).
- Q4. The SI unit for energy is the:
- A. Newton
- B. Joule
- C. Pascal
- D. Kilogram
- Correct Answer: B
- Explanation: The table in the text lists the joule (J) as the derived unit for energy.
- Q5. The unit for velocity (m/s) is derived from which two base units?
- A. Mass and time
- B. Length and mass
- C. Length and time
- D. Time and temperature
- Correct Answer: C
- Explanation: Velocity is defined as distance (length) divided by time, so its unit is derived from the base units of length (metre) and time (second).
- Q6. The unit for density is derived from which base units?
- A. Mass and length
- B. Length and time
- C. Mass, length, and time
- D. Mass and time
Correct Answer: A Explanation: Density is mass per unit volume. Volume is derived from length. So, the unit for density is kg/m3, derived from the base units of mass (kg) and length (m).
- Q7. Which of the following is a derived unit?
- A. Kilogram (kg)
- B. Second (s)
- C. Pascal (Pa)
- D. Ampere (A)
- Correct Answer: C
- Explanation: The pascal (Pa) is a derived unit for pressure. The other three are base units.
- Q8. The derived unit for pressure, the pascal (Pa), is equivalent to:
- A. kg·m/s2
- B. kg/(m·s2)
- C. kg·m2/s2
- D. kg/m3
- Correct Answer: B
- Explanation: The table shows that the pascal (Pa) is derived from kg/(m·s2), which is the same as N/m2.
- Q9. The newton (N) is the special name for the derived unit of force. What is its equivalent in base units?
- A. kg/s2
- B. m/s2
- C. kg·m/s2
- D. kg·m2/s2
- Correct Answer: C
- Explanation: The text explicitly states that force has a derived unit of kg·m/s2, which is given the special name newton (N).
- Q10. The joule (J), the derived unit for energy, is equivalent to:
- A. kg·m/s2
- B. kg/(m·s2)
- C. kg·m2/s2
- D. kg/m3
- Correct Answer: C
- Explanation: The table shows that the joule (J) is derived from kg·m2/s2.
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Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.1.3.2 System International (SI) Derived Units
Sticky Notes · Quick Revision
Key Point 1
- Derived Unit = Build from Base Units.
Key Point 2
- Area = m × m = m2
Key Point 3
- Velocity = m / s
Key Point 4
- Force = kg·m/s2 = newton (N)
Key Point 5
- Energy = kg·m2/s2 = joule (J)
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