Question

If power (P), surface tension (S) and Planck's constant (h) are arranged so that the dimensions of time in their dimensional formulae are in ascending order, then which of the following is correct?
A.)P, S, h
B.)P, h, S
C.)S, P, h
D.)S, h, P

Answer 1

Hint: Formula for power is work done divided by time, while for surface tension is Force divided by length and for Planck's constant is energy divided by frequency, we have to break each and every conventional formula into their dimensional form and then compare them to get the outcome.

Complete step by step answer:
For power(P), Formula for power is,
P= Work Done/ Time.
We know that, for work done dimensional form is

$[M{{L}^{2}}{{T}^{-2}}]$ ,

And for time is,
$[T]$ ,
Now putting these values in the equation we have,
P$=\dfrac{\text{work}}{\text{Time}}\therefore [P]=\dfrac{[M{{L}^{2}}{{T}^{-2}}]}{[T]}=[M{{L}^{2}}{{T}^{-3}}]$

For Surface tension,
Formula for surface tension is ,
S=Force/length.
We know that, for Force dimensional form is

$[M{{L}^{1}}{{T}^{-2}}]$ ,

And for Length is [L],
Now putting these values in the equation we have,
S=$\dfrac{\text{Force}}{\text{Length}}\therefore [S]=\dfrac{[ML{{T}^{-2}}]}{[L]}=[M{{L}^{0}}{{T}^{-2}}]$.
And for frequency is,
$[{{T}^{-1}}]$ ,
Now putting these values in the equation we have,

h$=\dfrac{\text{Energy}}{\text{Frequency}}\therefore

[h]=\dfrac{[M{{L}^{2}}{{T}^{-2}}]}{[{{T}^{-1}}]}=[M{{L}^{2}}{{T}^{-1}}]$.

Now from the above three derived dimensional formulas, we can say that in the ascending order of dimension of time, Power[P] comes first then, the Surface tension[S], and at the last comes planck's constant[h],
That is,

$[M{{L}^{2}}{{T}^{-3}}]<[M{{L}^{2}}{{T}^{-2}}]<[M{{L}^{2}}{{T}^{-1}}]$, is the correct order.
Hence option A is the correct option.

Note: The question wanted to know the answer in ascending order of time, we must note that all the time factors in the formulas are in negative power, so when negative power comes the highest number with the negative sign becomes the lowest. When replacing each and every conventional formula to a dimensional one never do step-jumping as it may lead to error.