Class 11 NCERT PHYSICS • Chapter 1: Units and Measurement • Question 3

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We need to calculate the calorie magnitude in new units given its SI value and the new unit definitions.

Concept: Principle of homogeneity

The principle of homogeneity states that the physical quantity $Q$ remains constant regardless of the unit system used: $Q = n_1 u_1 = n_2 u_2$ . For a quantity with dimensions $[M^a L^b T^c]$ , the numerical value in a new system is found using:

n_2 = n_1 \left( \frac{M_1}{M_2} \right)^a \left( \frac{L_1}{L_2} \right)^b \left( \frac{T_1}{T_2} \right)^c

Given:

Magnitude in SI: $n_1 = 4.2$
SI base units: $M_1 = 1\,\text{kg}$ , $L_1 = 1\,\text{m}$ , $T_1 = 1\,\text{s}$
New base units: $M_2 = \alpha\,\text{kg}$ , $L_2 = \beta\,\text{m}$ , $T_2 = \gamma\,\text{s}$
Dimensions of Energy (Joule): $[M^1 L^2 T^{-2}]$

Solving:

Identify the dimensional exponents for energy:
- From the unit $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}$ , we have $a = 1$ , $b = 2$ , and $c = -2$ .
Substitute the ratios of the base units into the conversion formula:
- Ratio of mass: $\frac{M_1}{M_2} = \frac{1\,\text{kg}}{\alpha\,\text{kg}} = \frac{1}{\alpha}$
- Ratio of length: $\frac{L_1}{L_2} = \frac{1\,\text{m}}{\beta\,\text{m}} = \frac{1}{\beta}$
- Ratio of time: $\frac{T_1}{T_2} = \frac{1\,\text{s}}{\gamma\,\text{s}} = \frac{1}{\gamma}$
Calculate the new magnitude $n_2$ :

\begin{aligned} n_2 &= 4.2 \left( \frac{1}{\alpha} \right)^1 \left( \frac{1}{\beta} \right)^2 \left( \frac{1}{\gamma} \right)^{-2} \\[8pt] &= 4.2 \cdot \alpha^{-1} \cdot \beta^{-2} \cdot (\gamma^{-1})^{-2} \\[8pt] &= 4.2 \cdot \alpha^{-1} \cdot \beta^{-2} \cdot \gamma^2 \\[8pt] &= 4.2 \alpha^{-1} \beta^{-2} \gamma^2 \end{aligned}

Answer

4.2 \alpha^{-1} \beta^{-2} \gamma^2

NCERT Solutions

Units and Measurement

SOLUTION

We need to calculate the calorie magnitude in new units given its SI value and the new unit definitions.

Concept: Principle of homogeneity

Given:

Solving:

Answer