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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 J**⋅**s**, which **is equal to kg**⋅**m ^{2}**⋅

**s**

^{−1}, where the metre ...

For more information, see Planck constant - Wikipedia

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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

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