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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Physics
Chapter
Units and Dimensions

NTA Abhyas 2020 · Physics · Units and Dimensions

If C (velocity of light), h (Planck's constant) and G & (Universal...

If CC (velocity of light), hh (Planck's constant)andGand G& (Universal gravitational constant) are taken as fundamental quantities, then the dimensional formula of mass is

Options

  1. Ah−12G12Coh^{- \dfrac{1}{2}}G^{\dfrac{1}{2}}C^{o}
  2. Bh12C12G−12h^{\dfrac{1}{2}}C^{\dfrac{1}{2}}G^{- \dfrac{1}{2}}
  3. Ch−12C12G−12h^{- \dfrac{1}{2}}C^{\dfrac{1}{2}}G^{- \dfrac{1}{2}}
  4. Dh−12C−12G−12h^{- \dfrac{1}{2}}C^{- \dfrac{1}{2}}G^{- \dfrac{1}{2}}

Correct answer

B. h12C12G−12h^{\dfrac{1}{2}}C^{\dfrac{1}{2}}G^{- \dfrac{1}{2}}

Step-by-step solution

Let, M=CahbGcM = {C}^{a}{h}^{b}{G}^{c}

ML0T0=[LT−1]M{L}^{0}{T}^{0} = {\left[L{T}^{- 1}\right]}^{a}{\left[M{L}^{2}{T}^{- 1}\right]}^{b}{\left[{M}^{- 1}{L}^{3}{T}^{- 2}\right]}^{c} ......(i)

Where, h=EnergyFreqyencyh = \dfrac{Energy}{Freqyency}

=[ML2T−2][T−1]=[ML2T−1]= \dfrac{\left[M{L}^{2}{T}^{- 2}\right]}{\left[{T}^{- 1}\right]} = \left[M{L}^{2}{T}^{- 1}\right]

C=MetreSecond=[LT−1]C = \dfrac{Metre}{Second} = \left[L{T}^{- 1}\right]

G=Force×(distance)2(mass)2G = \dfrac{Force \times {\left(distance\right)}^{2}}{{\left(mass\right)}^{2}}

=[MLT−2][L2][M2]=[M−1L3T−2]= \dfrac{\left[ML{T}^{- 2}\right]\left[{L}^{2}\right]}{\left[{M}^{2}\right]} = \left[{M}^{- 1}{L}^{3}{T}^{- 2}\right]

Comparing the coefficients M, L, T, of both sides we get

b−c=1b - c = 1 ......(ii)

a+2b+3c=0a + 2b + 3c = 0 ......(iii)

−(a+b+2c)=0- \left(a + b + 2c\right) = 0 .....(iv)

Solve the equations (ii), (iii) and (iv), we get

a=12,b=12,c=−12a = \dfrac{1}{2}, b = \dfrac{1}{2}, c = - \dfrac{1}{2}

So, M=h12C12G−12M = {h}^{\dfrac{1}{2}}{C}^{\dfrac{1}{2}}{G}^{- \dfrac{1}{2}}
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