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Physical World and Measurement - Units of Measurement

Grade 11ICSEPhysics

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

🔑Concepts

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Fundamental Units: These are independent units that cannot be derived from others. The SI system has seven: Mass (kgkg), Length (mm), Time (ss), Electric Current (AA), Thermodynamic Temperature (KK), Amount of Substance (molmol), and Luminous Intensity (cdcd).

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Supplementary Units: Includes Radian (radrad) for plane angles and Steradian (srsr) for solid angles.

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Derived Units: Units expressed as combinations of fundamental units, such as Force (kg⋅m⋅s−2kg \cdot m \cdot s^{-2} or Newton) and Pressure (N⋅m−2N \cdot m^{-2} or Pascal).

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Dimensions: The powers to which fundamental units are raised to represent a physical quantity, denoted as [MaLbTc][M^a L^b T^c].

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Principle of Homogeneity: States that the dimensions of all terms in a physical equation must be the same. For example, in v=u+atv = u + at, the dimensions of vv, uu, and atat are all [LT−1][LT^{-1}].

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Significant Figures: The digits in a measurement that are known with certainty plus one digit that is uncertain. Rules include: all non-zero digits are significant, and trailing zeros in a decimal are significant.

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Errors in Measurement: Absolute error is the magnitude of the difference between the true value and measured value. Relative error is the ratio of absolute error to the mean value.

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Least Count: The smallest value that can be measured by a measuring instrument. For Vernier Calipers, LC=1 MSD−1 VSDLC = 1 \text{ MSD} - 1 \text{ VSD}.

📐Formulae

[Q]=[MaLbTcIdΘeNfJg][Q] = [M^a L^b T^c I^d \Theta^e N^f J^g]

Absolute Error: Δai=∣amean−ai∣\text{Absolute Error: } \Delta a_i = |a_{mean} - a_i|

Relative Error: δa=Δameanamean\text{Relative Error: } \delta a = \frac{\Delta a_{mean}}{a_{mean}}

Percentage Error: Δameanamean×100%\text{Percentage Error: } \frac{\Delta a_{mean}}{a_{mean}} \times 100\%

Error in Sum/Difference: ΔZ=ΔA+ΔB\text{Error in Sum/Difference: } \Delta Z = \Delta A + \Delta B

Error in Product/Quotient: ΔZZ=ΔAA+ΔBB\text{Error in Product/Quotient: } \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}

Error in Power (if Z=ApBq/Cr): ΔZZ=pΔAA+qΔBB+rΔCC\text{Error in Power (if } Z = A^p B^q / C^r \text{): } \frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}

Least Count (Vernier Caliper): LC=Value of 1 Main Scale DivisionTotal Number of Vernier Scale Divisions\text{Least Count (Vernier Caliper): } LC = \frac{\text{Value of 1 Main Scale Division}}{\text{Total Number of Vernier Scale Divisions}}

💡Examples

Problem 1:

Check the dimensional consistency of the equation s=ut+12at2s = ut + \frac{1}{2}at^2, where ss is displacement, uu is initial velocity, aa is acceleration, and tt is time.

Solution:

Dimensions of LHS: [s]=[L][s] = [L]. Dimensions of RHS: [ut]=[LT−1][T]=[L][ut] = [LT^{-1}][T] = [L] and [12at2]=[LT−2][T2]=[L][\frac{1}{2}at^2] = [LT^{-2}][T^2] = [L].

Explanation:

Since the dimensions of all terms on both sides of the equation are the same ([L][L]), the equation is dimensionally consistent according to the Principle of Homogeneity.

Problem 2:

The radius of a sphere is measured as (5.3±0.1)(5.3 \pm 0.1) cm. Calculate the percentage error in its volume.

Solution:

Volume of a sphere V=43πr3V = \frac{4}{3}\pi r^3. The relative error is ΔVV=3Δrr\frac{\Delta V}{V} = 3 \frac{\Delta r}{r}. Percentage error =3×0.15.3×100≈5.66%= 3 \times \frac{0.1}{5.3} \times 100 \approx 5.66\%.

Explanation:

In calculations involving powers, the relative error is multiplied by the power of the quantity. Here, the radius is cubed, so we multiply its relative error by 3.

Problem 3:

State the number of significant figures in 0.0070.007 m and 2.64×10242.64 \times 10^{24} kg.

Solution:

0.0070.007 has 1 significant figure; 2.64×10242.64 \times 10^{24} has 3 significant figures.

Explanation:

For 0.0070.007, leading zeros are never significant. For 2.64×10242.64 \times 10^{24}, all digits in the decimal part of scientific notation are significant, while the power of 10 does not affect significant figures.

Units of Measurement - Revision Notes & Key Formulas | ICSE Class 11 Physics