Question

The period of a body under Simple Hormonic Motion is represented by T = P^a D^b S^c. Where P is pressure, D is density, and S is surface tension. Then the value of a, b, c...???

Please explain me clearly....

Answer 1

`T = S sqrt((D)/(P^3))`

Explanation:

Your claim is that: `T = P^a D^b S^c`

By dimensional analysis:

`s = ((kg \ m \ s^(-2))/m^2)^a ((kg)/m^3)^b ((kg \ m \ s^(-2))/m)^c`

Surface tension is accounted for here as force per unit length. If your understanding is different, adjust the calculation.

So:

`s = ((kg )/(m \ s^2))^a ((kg)/m^3)^b ((kg) /( s^2))^c`

Working mechanically through the exponentiation, this gives 3 equations in 3 unknowns:

• kg: `\qquad a + b + c = 0`

• m: `\qquad -a - 3b = 0`

• s: `\qquad -2a - 2 c = 1`

It solves as:

`a, b, c = -3/2, 1/2, 1`

Dimensionally at least, the following should hold:

`T = S sqrt((D)/(P^3))`