Question

Conversion of $1MW$ power in a system of units having basic units of mass, length and time as $10kg,1decimeter$and $1min$ respectively is:
$$\begin{array}{rl}& A.2.16\times {10}^{10}units\\ & B.2\times {10}^{4}units\\ & C.2.16\times {10}^{12}units\\ & D.1.26\times {10}^{12}units\end{array}$$

Answer 1

Hint: Here we need to convert the basic unit which is $1kg$,$1m$ and $1s$ to $10kg,1decimeter$and $1min$ respectively .We can define the terms and use their formula to the get the relationship between the $1MW$ with respect to basic unit $1kg$,$1m$ and $1s$, and then substitute the conversion of $10kg,1decimeter$and $1min$. Here we are using the units for conversion from basic to new unit systems.

Formula used:
$1J/s={\displaystyle \frac{1kg\times 1m/s^{-2}\times 1m}{1s}}$ and $1dm=0.1m$ also $1s={\displaystyle \frac{1}{60}}min$

Complete step by step answer:
Here, we have $1MW={10}^{6}W$
We know that one watt is the energy consumed if $1J$ of energy flows for $1s$.
Then we can write $1MW={10}^{6}W={10}^{6}J/s$
Also energy is the ability of an object to do work, then we can say $1J$ is the work done in moving an object through a distance of $1m$ when $1N$ force acts on it.
Then we get, $1J=1N\times 1m$
Where $1N$ is the force that can accelerate $1kg$ mass at $1m/s$ or $1N=1kg\times 1m/s^{-2}$
Then we have $1J/s={\displaystyle \frac{1kg\times 1m/s^{-2}\times 1m}{1s}}$
Then, $1MW={10}^{6}J/s={\displaystyle \frac{{10}^{6}kg\times 1m/s^{-2}\times 1m}{1s}}={\displaystyle \frac{{10}^{6}kg\times 1m^2}{1s^3}}$
Clearly the basic unit here is $1kg$,$1m$ and $1s$
Here, we have to convert $1kg$ to $10kg$, $1m$ to $1dm$ and$1sec$ to $1min$
We know that $1dm=0.1m$ and $1s={\displaystyle \frac{1}{60}}min$
Then when substituting the values in $MW$, we get,
$1MW={\displaystyle \frac{{10}^6\times 0.1kg\times (10dm{)}^2}{{\left({\displaystyle \frac{1}{60}}min\right)}^3}}=216\times {10}^{10}{\displaystyle \frac{kg.d{m}^2}{mi{n}^3}}=2.16\times {10}^{12}unit$
Thus we get $1MW=2.16\times {10}^{12}units$

Hence the answer is option $$C.2.16\times {10}^{12}units$$

Note:
Another alternative and simple way would be using dimensional analysis. We know that $1W$ has dimensions, $[M{L}^2{T}^{-3}]$ we can use the conversion $1kg$ to $10kg$, $1m$ to $1dm$ and$1sec$ to $1min$ in the form of ratio to where $M$ is the mass and will have the ratio of $1kg$ to $10kg$, $L$ is the length will have the ratio of $1m$ to $1dm$ and $T$ is the time and will have the ratio of $1sec$ to $1min$ respectively. This will also give the same answer as above.