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Dimensional Analysis

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Question

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Q(1). If speed of light (e) universal gravitations constant (G), plank constant (h) are the fundamental unit. Express :-

(i) Mass in terms of (GH)

(ii) Length in terms of (GH)

(iii) Times in term of (GH)

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Solution

Here, $c equals open square brackets L T to the power of negative 1 end exponent close square brackets equals 3 cross times 10 to the power of 8 space end exponent m divided by s space equals 3 cross times 10 to the power of 10 space c m divided by s$

$G equals open square brackets M to the power of negative 1 end exponent L cubed T to the power of negative 2 end exponent close square brackets equals 6.67 cross times 10 to the power of negative 8 end exponent space d y n e space c m squared g to the power of negative 2 end exponent$

$h equals open square brackets M L squared T to the power of negative 1 end exponent close square brackets equals 6.6 cross times 10 to the power of negative 27 end exponent space e r g. s.$

Now, $fraction numerator h c over denominator G end fraction equals fraction numerator 6.6 cross times 10 to the power of negative 27 end exponent cross times 3 cross times 10 to the power of 10 over denominator 6.67 cross times 10 to the power of negative 8 end exponent end fraction$

or, $fraction numerator M L squared T to the power of negative 1 end exponent cross times L T to the power of negative 1 end exponent over denominator M to the power of negative 1 end exponent L cubed T to the power of negative 2 end exponent end fraction equals 2.9685 cross times 10 to the power of negative 9 end exponent$

or, $M squared equals 2.9685 cross times 10 to the power of negative 9 end exponent$

or, $M equals square root of 2.9685 cross times 10 to the power of negative 9 end exponent end root equals 0.5448 cross times 10 to the power of negative 4 end exponent space g$

So, $1 space g equals fraction numerator 1 over denominator 0.5448 cross times 10 to the power of negative 4 end exponent end fraction equals 1.8355 cross times 10 to the power of 4$ new unit of mass.

Again, $fraction numerator h over denominator M c squared end fraction equals fraction numerator 6.67 cross times 10 to the power of negative 27 end exponent over denominator 0.5448 cross times 10 to the power of negative 4 end exponent cross times open parentheses 3 cross times 10 to the power of 10 close parentheses squared end fraction$

or, $fraction numerator M L squared T to the power of negative 1 end exponent over denominator M L squared T to the power of negative 2 end exponent end fraction equals 1.3603 cross times 10 to the power of negative 43 end exponent space s$

or, $T equals 1.3603 cross times 10 to the power of negative 43 space end exponent s$

So, $1 space s equals fraction numerator 1 over denominator 1.3603 cross times 10 to the power of negative 43 end exponent end fraction equals 0.735 cross times 10 to the power of 43 space end exponent$ new unit of time.

Again, $c cross times T equals 3 cross times 10 to the power of 10 cross times 1.3603 cross times 10 to the power of negative 43 end exponent$

or, $left parenthesis L T to the power of negative 1 end exponent right parenthesis cross times T equals L equals 4.0809 cross times 10 to the power of negative 33 space end exponent c m$

So, $1 space c m equals fraction numerator 1 over denominator 4.0809 cross times 10 to the power of negative 33 end exponent end fraction equals 0.2450 cross times 10 to the power of 33$ new unit of length.

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