Name: SI and Other Units: An Overview
Uploaded: 2019-07-19
Duration: 4 min 17 s
Description: In this video, let us discuss the fundamental and derived SI units used in physics and learn about some non-standard units used in ancient days. Learn more.

Question

The frequency of vibration of a string depends on the length

L

between the nodes, the tension

F

in the string and its mass per unit length

m

. Guess the expression for its frequency from the dimensional analysis?

Accepted Answer

The dimensional formula for frequency $f$ , length $L$ , Tension in string $F$ and mass per unit length $m$ are as follow,

$[f] = [M^{0} L^{0} T^{- 1}]$ , $[L] = [M^{0} L^{1} T^{0}]$ , $[F] = [M^{1} L^{1} T^{- 2}]$ and $[m] = [M^{1} L^{- 1} T^{0}]$

According to dimensional analysis,

Let $f = k L^{a} F^{b} m^{c}$

$[M^{0} L^{0} T^{- 1}] = {[L^{1}]}^{a} {[M^{1} L^{1} T^{- 2}]}^{b} {[M^{1} L^{- 1}]}^{c}$

$[M^{0} L^{0} T^{- 1}] = [M^{(b + c)} L^{(a + b - c)} T^{(- 2 b)}]$

$∴ b + c = 0 . . . . . . . (1)$

$a + b - c = 0 . . . . . . . (2)$

$- 2 b = - 1 \Rightarrow b = \frac{1}{2} . . . . . . . . (3)$

From $(1)$ and $(3)$

$c = - b = - \frac{1}{2}$

Put the value of $b$ and $c$ in equation $(2)$

$∴ a + \frac{1}{2} - (- \frac{1}{2}) = 0 \Rightarrow a = - 1$

$∴ f = k L^{- 1} F^{(\frac{1}{2})} m^{(- \frac{1}{2})}$

$∴ f = \frac{k}{L} \sqrt{\frac{F}{m}}$

The frequency of vibration of a string depends on the length $L$ between the nodes, the tension $F$ in the string and its mass per unit length $m$ . Guess the expression for its frequency from the dimensional analysis?

Important Questions on Introduction to Physics

Learn and Prepare for any exam you want !

The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from the dimensional analysis?

Important Questions on Introduction to Physics

Learn and Prepare for any exam you want !

The frequency of vibration of a string depends on the length $L$ between the nodes, the tension $F$ in the string and its mass per unit length $m$ . Guess the expression for its frequency from the dimensional analysis?