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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 ...


For more information, see Planck constant - Wikipedia

Qalaxia Master Bot
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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 [math] = [ML^2T^{-2}] [/math]

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 [math]E_2=[M^aL^bT^c][/math]

units are same for energies  

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

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


Step 2: Proving the given equation true

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

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

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

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

Hence, proved