Question

In a system called the star system. 1. Star kg = ${10^{20}}$kg, 1 star meter= ${10^8}$ meters, and 1 star second= ${10^3}$ seconds. Then what is the value of 1 joule?
Option A: ${10^{ - 30}}$ star joules.
Option B: ${10^{ - 27}}$ star joules.
Option C: ${10^{ - 32}}$ star joules.
Option D: none of the above.

Answer 1

Hint: We need to do unit conversion of joule to star joule. Try to do it dimensionally, with the help of dimensions.

Complete solution:
Dimensions are very important in studying quantities. Conversion of all quantities having dimensions can be done very easily and with accuracy.
We denote dimensions with upper case letters and with an index over them showing their weightage.
All quantities are divided into two types; the primary quantities that exist on their own and the derived quantities that are derived from primary quantities and depend on them as well.
Some examples of primary quantities are mass, time, length, temperature, current, etc. all denoted as $M,T,L,\theta,A$, etc. respectively.
Some examples of derived quantities are force, energy, pressure, area, etc. all denoted as ${M^1}{L^1}{T^{ - 2}},M{L^2}{T^{ - 2}},{M^1}{L^{ - 1}}{T^{ - 2}},{L^2},$etc. respectively.
Now as we know joule is the dimension for energy, so the dimensions for joule are given by;
[Joule]= $M{L^2}{T^{ - 2}}$.
If [star kg]= $M$, [kg]=$\dfrac{M}{{{{10}^{20}}}}$.
If [star meter]= $L$ then [meter]= $\dfrac{L}{{{{10}^8}}}$.
If [star second]= $T$ then [time]= $\dfrac{T}{{{{10}^3}}}$.
Thus, 1 joule= $\dfrac{1}{{{{10}^{20}}}}$star-kg \[ \times {(\dfrac{1}{{{{10}^8}}})^2}\] star-meter $ \times {(\dfrac{1}{{{{10}^3}}})^{ - 2}}$ star-second
Hence, 1 joule= ${10^{ - 20 - 16 + 6}}$ star-joule.
Therefore, 1 joule = ${10^{ - 30}}$ star joule.

Thus option A is correct.

Note:
(1) There is no need for making assumptions for such problems as the dimensions of the quantity are known.
(2) We make assumptions only when there is a quantity whose dimensions are to be found out.
(3) Unit conversion does not require assumptions.