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Planck constant - Wikipedia

is defined by taking the fixed numerical value of h to be 6.62607015×10−34 when expressed in the unit Js, which is equal to kgm2s1, where the metre ...

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Units and Dimensions - Dimensional Analysis, Formula, Applications

The units that are used to measure these derived quantities are called derived units. Fundamental and supplementary physical quantities in SI system: ...

For more information, see Units and Dimensions - Dimensional Analysis, Formula, Applications

Teja
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Given that

Energy E_1 = 4.2 joules

1 joule = 1kg m^2 s^{-2}

Energy E_1=4.2 1kg m^2 s^{-2}

Magnitude of the Energy with new units E_2 = ?

Show that E = 4.2 \alpha^{-1} \beta^{-2} \gamma^{2}

Step 1: Write the dimensional formula of energy

SI units: \text{ Mass } M_1 = 1 kg, \text{ length } L_1 = 1m^2, \text{ time }T_1 = 1 s^{-2}

Dimensional formula of energy $= [ML^2T^{-2}]$

New units: \text{ Mass } M_2 = \alpha kg, \text{ length } L_2 = \beta m^2, \text{ time }T_2 = \gamma s^{-2}

Dimensional formula of energy $E_2=[M^aL^bT^c]$

units are same for energies

$[ML^2T^{-2}] = [M^aL^bT^c]$

Equating the powers of both sides a = 1, b = 2 \text{ and } c = -2

Step 2: Proving the given equation true

$E_2 = E_1 [(\frac{M_1}{M_2})^a + (\frac{L_1}{L_2})^b + (\frac{T_1}{T_2})^c]$

$E_2 = 4.2 [(\frac{M_1}{M_2})^1 + (\frac{L_1}{L_2})^2 + (\frac{T_1}{T_2})^{-2}]$

$E_2 = 4.2 [(\frac{1kg}{\alpha kg})^1 + (\frac{m^2}{\beta m^2})^2 + (\frac{s^{-2}}{\gamma s^{-2}})^{-2}]$

E_2 = 4.2 \alpha^{-1} \beta^{-2} \gamma^{2}

Hence, proved