Question

The surface tension of water in the CGS system is 72 dynes/cm. what is its value multiplied by 1000 in SI units?

Answer 1

Dimensional analysis is a very easy method to convert one system of the unit to another system of units. This is based on the fact that for a given physical quantity, the numerical value unit = constant. so, when units change, numerical value also may change. for conversion, we write its units in terms of mass, length, and time.

Complete step by step answer:
This can be solved by dimensional analysis method:
Let dimensional formula for surface tension is $\left[ {M{T^{ - 2}}} \right]$
Dyne/cm is the CGS system of unit and in SI units N/m
Given: 72dyne/cm = \\_\\_\\_N/m when it is multiplied by 1000.
We have, ${n_1}{u_1} = {n_2}{u_2}$
Where ${n_1}$, ${n_2}$ represents numerical values and ${u_1}$, ${u_2}$ and represents units.
$\Rightarrow {n_2} = \dfrac{{{u_1}}}{{{u_2}}}{n_1}$
$\Rightarrow {n_2} = \dfrac{{CGS}}{{SI}} \times 72$
Now apply the dimensional formula we get,
$\Rightarrow {n_2} = \dfrac{{\left[ {{M_1}{T_1}^{ - 2}} \right]}}{{\left[ {{M_2}{T_2}^{ - 2}} \right]}} \times 72$
${M_1}$ in terms of CGS system
${M_2}$ in SI system of units.
Then,
$\Rightarrow {n_2} = 72 \times \left[ {\dfrac{g}{{kg}}} \right]\left[ {\dfrac{{{s^{ - 2}}}}{{{s^{ - 2}}}}} \right]$
Seconds’ term cancels out. Then we have, $1kg = 1000g$
$\Rightarrow 1g = {10^{ - 3}}kg$
Substitute in the above equation we get,
$\Rightarrow {n_2} = 72 \times \left[ {\dfrac{{{{10}^{ - 3}}kg}}{{kg}}} \right]$
kg also cancels out, then
$\Rightarrow {n_2} = 72 \times {10^{ - 3}}$
Now, $72dyne/cm = 72 \times {10^{ - 3}}N/m$
Now multiply these values by 1000 which is given in the question then we get,
$\Rightarrow 72dyne/cm = 72 \times {10^{ - 3}}N/m \times 1000$

$\therefore 72dyne/cm = 72N/m$

Additional information:
The powers are the dimensions of a physical quantity to which the fundamental quantities. The formula that shows the powers to which the fundamental quantities must be raised and represent a physical quantity is called dimensional formula.
Dimensions for the fundamental physical quantities.

1. Mass| [ M ]
   ---|---
   Length| [ L ]
   Time| [ T ]
   Temperature| [ K ]
   Electric current| [ A ]
   Luminous intensity| [ cd ]
   Amount of substance| [ mol ]

Constants having dimensional formula are called Dimensional constants.
E.g.: Planck’s constant, speed of light, Universal gravitational constant.
Physical quantities having no dimensional formula are called Dimensionless quantities.
E.g.: Angle, Strain, Relative Density.
Limitations of Dimensional analysis.
Proportionality constants cannot be determined by dimensional analysis.
Formulae containing non- algebraic functions like sin, cos, log, exponential, etc., cannot be derived.
The dimensional analysis does not differentiate between a scalar and a vector quantity.

Note:
Square bracket [ ] is used to represent the dimension of the physical quantity
Applications of Dimensional analysis:
Dimensional formulae can be used to convert one system of the unit to another system of the unit.
Dimensional formulae can be used to check the correctness of a given equation.
Dimensional formulae can be used to derive the relationship among different physical quantities.