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Measurement – Foundation of Science - Need for a Common System of Units-advanced

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Measurement is the process of comparing an unknown physical quantity with a known, fixed standard quantity called a unit. A physical quantity QQ is expressed as Q=n×uQ = n \times u, where nn is the numerical value and uu is the unit.

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The need for a common system arises because local units (like cubit, handspan, or foot) vary from person to person and region to region, leading to confusion in trade and scientific collaboration.

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A standard unit must be well-defined, easily reproducible, invariable with time/place, and internationally accepted.

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The SI System (Système International d'Unités) is the modern metric system used globally. It consists of seven base units: Length (mm), Mass (kgkg), Time (ss), Electric Current (AA), Thermodynamic Temperature (KK), Amount of Substance (molmol), and Luminous Intensity (cdcd).

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Supplementary units in SI include the Radian (radrad) for plane angles and the Steradian (srsr) for solid angles.

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The SI system is a Coherent System, meaning derived units are obtained by simple multiplication or division of base units without introducing numerical factors other than 11.

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It is also a Rational System, as it assigns only one unit to a particular physical quantity (e.g., Joule for all forms of energy like mechanical, heat, and electrical).

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Prefixes are used to express very large or small magnitudes in powers of 1010, such as micro (10−610^{-6}), milli (10−310^{-3}), kilo (10310^{3}), and mega (10610^{6}).

📐Formulae

Q=n×uQ = n \times u

n1u1=n2u2n_1 u_1 = n_2 u_2

Density=MassVolume=ML3=kg⋅m−3\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{M}{L^3} = kg \cdot m^{-3}

Force=Mass×Acceleration=kg⋅m⋅s−2=Newton (N)\text{Force} = \text{Mass} \times \text{Acceleration} = kg \cdot m \cdot s^{-2} = \text{Newton (N)}

Pressure=ForceArea=Nm2=Pascal (Pa)\text{Pressure} = \frac{\text{Force}}{\text{Area}} = \frac{N}{m^2} = \text{Pascal (Pa)}

💡Examples

Problem 1:

A piece of metal has a density of 7.8 g/cm37.8 \text{ g/cm}^3. Convert this value into the SI unit of density (kg/m3kg/m^3).

Solution:

Given n1=7.8n_1 = 7.8, u1=g/cm3u_1 = \text{g/cm}^3, and we need to find n2n_2 for u2=kg/m3u_2 = kg/m^3. We know: 1 g=10−3 kg1 \text{ g} = 10^{-3} \text{ kg} 1 cm=10−2 m⇒1 cm3=(10−2)3 m3=10−6 m31 \text{ cm} = 10^{-2} \text{ m} \Rightarrow 1 \text{ cm}^3 = (10^{-2})^3 \text{ m}^3 = 10^{-6} \text{ m}^3 Using the relation n2=n1×u1u2n_2 = n_1 \times \frac{u_1}{u_2}: n2=7.8×10−3 kg10−6 m3n_2 = 7.8 \times \frac{10^{-3} \text{ kg}}{10^{-6} \text{ m}^3} n2=7.8×103=7800n_2 = 7.8 \times 10^3 = 7800 So, the density is 7800 kg/m37800 \text{ kg/m}^3.

Explanation:

To convert units, we substitute the equivalent SI values for grams and cubic centimeters and simplify the power of ten.

Problem 2:

Show that the unit of Work is a coherent derived unit in the SI system.

Solution:

Work is defined as Force ×\times Displacement. Work=F×s\text{Work} = F \times s In SI base units: Force (N)=kg⋅m⋅s−2\text{Force (N)} = kg \cdot m \cdot s^{-2} Displacement (s)=m\text{Displacement (s)} = m Work=(kg⋅m⋅s−2)×m=kg⋅m2⋅s−2\text{Work} = (kg \cdot m \cdot s^{-2}) \times m = kg \cdot m^2 \cdot s^{-2} In SI, this combination is defined as 1 Joule (J)1 \text{ Joule (J)}.

Explanation:

Since the unit of Work (Joule) is derived by multiplying base units without any extra numerical constant, it proves the SI system is coherent.