Question

The ratio of SI units to CGS units of acceleration due to gravity is
A. ${10^2}$
B. $10$
C. ${10^{ - 1}}$
D. ${10^{ - 2}}$

Answer 1

Hint: Acceleration due to gravity is the acceleration which acts on a body as a consequence of the force acting because of Earth’s gravitation. It has the same dimensions as that of acceleration. In SI, its fundamental units are meter and seconds, while in CGS these are centimeters and seconds. Take the ratio of these in their respective numeric equivalent to get the required answer.
Formula Used:
Dimensional Formula of acceleration due to gravity: $\left[ g \right] = \left[ {{L^1}{T^{ - 2}}} \right]$
where L is length,
and T is time.

Complete step by step answer:
The acceleration due to gravity is represented using the symbol ‘g’. It is directly proportional to the force F exerted by the earth on an object and inversely proportionally to the mass of the object m.
The acceleration due to gravity is given mathematically as:
$g = \dfrac{F}{m}$
where F is the force exerted by the earth on an object of mass m.
The dimensional formula for the acceleration due to gravity is
$\left[ g \right] = \left[ {{L^1}{T^{ - 2}}} \right]$
where L is length,
and T is time.
In the SI unit system, meter and second are used to represent the physical quantity of length and time respectively. Whereas, in CGS unit systems, centimeters and second are used to represent length and time respectively.
We know that, $1m = 100cm$
So, taking ratio of acceleration due to gravity in SI unit system and CGS unit system we have:
$\eqalign{
  & \dfrac{{g\left( {{\text{SI}}} \right)}}{{g\left( {{\text{CGS}}} \right)}} = \dfrac{{{m^1}{s^{ - 2}}}}{{c{m^1}{s^{ - 2}}}} \cr
  & \Rightarrow \dfrac{{g\left( {{\text{SI}}} \right)}}{{g\left( {{\text{CGS}}} \right)}} = \dfrac{{{m^1}}}{{c{m^1}}} \cr
  & \Rightarrow \dfrac{{g\left( {{\text{SI}}} \right)}}{{g\left( {{\text{CGS}}} \right)}} = \dfrac{{100cm}}{{1cm}}{\text{ }}\left[ {\because 100cm = 1m} \right] \cr
  & \Rightarrow \dfrac{{g\left( {{\text{SI}}} \right)}}{{g\left( {{\text{CGS}}} \right)}} = {10^2} \cr} $
Therefore, the correct answer is A. i.e., the ratio of SI units to CGS units of acceleration due to gravity is ${10^2}$.

Note: For the conversion of any and all physical quantities in one unit system to another, one needs to just multiply them to the factor values of their derived quantities in the respective system they are being converted to. Additionally, for conversion of a SI formula into CGS replace ${\varepsilon _ \circ }{\text{ to }}\dfrac{1}{{4\pi \times {{10}^{ - 3}}{c^2}}}$ and ${\mu _ \circ }{\text{ to }}4\pi \times {10^{ - 7}}$.