First slide

Accuracy; precision of instruments and errors in measurements

Question

A gas bubble from an expression under water, oscillates with a period T proportional to $p^{a} d^{b} E^{c}$ , where p is the static pressure, d is the density of water and E is the total energy of explosion. Find the value of a.

Moderate

A

$\frac{1}{2}$
B

$\frac{1}{3}$
C

$- \frac{1}{4}$
D

$- \frac{5}{6}$

Solution

$\begin{array}{l} T \propto p^{a} d^{b} E^{c} \\ [T] = {[M L^{- 1} T^{- 2}]}^{a} {[M L^{- 3}]}^{b} {[M L^{2} T^{- 2}]}^{c} \end{array} [T] = [M^{a + b + c} L^{- a - 3 b + 2 c} T^{- 2 a - 2 c}]$
$E q u a t i n g t h e e x p o n e r r t, a + b + c = 0$
$- a - 3 b + 2 c = 0 and - 2 a - 2 c = 1$
$B y s o l v i n g$
$a = - \frac{5}{6}, b = \frac{1}{2} a n d c = \frac{1}{3}$

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