Question

Ratio of S.I to C.G.S units of kinetic energy is
A. \[{{10}^{6}}\]
B. \[{{10}^{-7}}\]
C. \[{{10}^{7}}\]
D. \[{{10}^{8}}\]

Answer 1

Hint:To convert it into C.G.S units, you need to know the S.I units of energy and then apply certain conversion factors on it to get the corresponding value in C.G.S unit.The centimetre–gram–second system of units (abbreviated CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time.

Complete step by step answer:
Kinetic energy is the energy an object has due to its motion. For example, A car moving on the road, Shooting a rubber band etc. Kinetic energy depends on the mass and speed of the object. Mathematically, Kinetic Energy is given by:
\[K.E=\dfrac{1}{2}m{{v}^{2}}\]
where $m$ is the mass of the object and $v$ is the speed.

The S.I unit of kinetic energy can be found out by substituting the S.I units of mass and speed in the formula. We know the S.I unit of mass is ‘kg’ and the S.I unit of speed is ‘m/s’.Therefore, the S.I unit of kinetic energy is \[kg\dfrac{{{m}^{2}}}{{{s}^{2}}}\] which is defined as Joules. Therefore, \[kg\dfrac{{{m}^{2}}}{{{s}^{2}}}=J\].

Now to convert it into C.G.S units, we need to know the C.G.S units of mass and speed. So the C.G.S unit of mass is ‘g’ and the C.G.S unit of speed is ‘cm/s’
Conversion factor:
\[1\,kg=1000\,g \\
\Rightarrow 1\,m=100\,cm \]
Let’s convert the S.I unit of energy to C.G.S unit of energy
\[J=kg\dfrac{{{m}^{2}}}{{{s}^{2}}}=1kg\times 1\dfrac{{{m}^{2}}}{{{s}^{2}}} \\
\Rightarrow J=1000g\times \dfrac{{{\left( 100cm \right)}^{2}}}{{{s}^{2}}} \\
\Rightarrow J=1000g\times 10000\dfrac{c{{m}^{2}}}{{{s}^{2}}} \\
\Rightarrow J={{10}^{7}}g\dfrac{c{{m}^{2}}}{{{s}^{2}}} \\
\Rightarrow J={{10}^{7}}ergs \\
\Rightarrow 1J={{10}^{7}}ergs \]
Therefore, Ratio of S.I to C.G.S units of kinetic energy is
\[\dfrac{J}{ergs}=\dfrac{{{10}^{7}}ergs}{ergs} \\
\therefore \dfrac{J}{ergs}={{10}^{7}}\]

Hence, the correct answer is option C.

Note:You can also use dimensional analysis for interconversion. Similarly, you can convert other quantities like this, you just need to know the units and the conversion factors which are required to convert from S.I to C.G.S.