Question

The period of a body under S.H.M. is represented by: $T = P^{a}D^{b}S^{c}$ where P is pressure, D is density and S is surface tension, then values of a, b and c are(a) $\frac{-3}{2},\frac{1}{2}, 1$(b) -1, -2, 3(c) $\frac{1}{2},\frac{3}{2},\frac{1}{2}$(d) 1,2,\frac{1}{3}[/tex]

Answer 1

The answer is option (a)

Answer 2

Given that,

The period of a body under S.H.M. is represented by :

$T=P^aD^bS^c$

Here,

where P is pressure, D is density and S is surface tension

To find,

The values of a,b and c.

Solution,

It can be calculated using dimensional analysis such that,

Dimensional formula of time period, [T]=[T]

Dimensional formula of pressure, [P]=[ML⁻¹T⁻²]

Dimensional formula of density, [D]=[ML⁻³]

Dimensional formula of surface tension, [D]=[ML⁻²]

Writing dimensional formula as :

$[T]=[ML^{-1}T^{-2}]^a[ML^{-3}]^b[ML^{-2}]^c\\\\\ [T]=[M]^{a+b+c}[L]^{-a-3b}[T]^{-2a-2c}$

On comparing powers on both sides we get :

$-2a-2c=1\\\\a + b + c = 0\\\\- a - 3 b = 0$

On solving the above three equations, we get:

$a=\dfrac{-3}{2}\\\\b=\dfrac{1}{2}\\\\c=1$

Hence, the correct option is (a).

Learn more,

Dimensional analysis

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