Question

Express 10N force in terms of $gcm{s^{ - 2}}$

Answer 1

$gcm{s^{ - 2}}$ is in CGS unit Newton is in SI unit. To convert Newton to $gcm{s^{ - 2}}$ , we have to use the dimensional equation. The dimensional equation of force is $[F] = [{M^1}{L^1}{T^{ - 2}}]$ . while solving the problem the dimensions cancel out and the remaining term that we are going to get is a numerical value.

Complete answer:
We can take ${n_1}{u_1} = {n_2}{u_2}$
Where ${n_1}$=10, ${u_1}$= N, ${n_2}$= ? , ${u_2}$ = $gcm{s^{ - 2}}$
${n_2}$ is the quantity that we have to find out, that is the converted value.
${n_2} = \dfrac{{{n_1}{u_1}}}{{{u_2}}}$
Replacing each term by its value we get ${n_2} = \dfrac{{10N}}{{gcm{s^{ - 2}}}}$
But we know that 1N= $kgm{s^{ - 2}}$ (SI unit)
Therefore, ${n_2} = \dfrac{{10*kgm{s^{ - 2}}}}{{gcm{s^{ - 2}}}}$
To solve this we have to convert kg into g and m into cm and the second cancels each other. Here we are using the dimensional equation to make sure that the dimension of force that is $[{M^1}{L^1}{T^{ - 2}}]$ remains the same and the unit is being converted.
$\Rightarrow n{}_2 = \dfrac{{10*1000g*100cm}}{{gcm}}$ , the g and cm on the numerator gets cancelled by the g and cm in the denominator. Hence the units are removed and a numerical value is obtained.
$\Rightarrow {n_2} = 10*1000*100$
$\Rightarrow {n_2} = 10,00,000gcm{s^{ - 2}}$
Therefore 1N is equivalent to ${10^6}gcm{s^{ - 2}}$

Note:
$gcm{s^{ - 2}}$ is known as 1 dyne. 1 dyne is equal to ${10^{ - 5}}N$ . Newton is the unit of force , so is dyne. Newton is used more because it is the SI unit of force. By using the dimensional equation we can convert any value in one unit to any other. The dimensional equation for one quantity will remain the same no matter how many times its unit is being converted.