Question

What is the relationship between the C.G.S. and S.I. unit of density?
A) \[1gc{m^{ - 3}} = 1000kg{m^{ - 3}}\]
B) \[1gc{m^{ - 3}} = 1kg{m^{ - 3}}\]
C) \[1gc{m^{ - 3}} = 10kg{m^{ - 3}}\]
D) \[1gc{m^{ - 3}} = 10000kg{m^{ - 3}}\]

Answer 1

Hint: There are three main systems of measurement.
i) F.P.S. = foot pond second
ii) M.K.S. (S.I.) = meter kilogram second
iii) C.G.S. = centimetre gram second
We can get fundamental and derived units of physical quantities in these systems. And also find the relation between them. We can find the relation between them by many methods but here we only discuss dimensional and unit methods for it.

Complete step by step answer:
Here we discuss dimensional methods. We know that the unit of density in the M.K.S. and C.G.S. system is \[kg{m^{ - 3}}\] and \[gc{m^{ - 3}}\]respectively. So the dimension of density is given as \[\left[ {M{L^{ - 3}}} \right]\]
Now let the relation between S.I. and C.G.S. unit of density is given as
\[{n_1}gc{m^{ - 3}} = {n_2}kg{m^{ - 3}}\]
Now if put \[{n_1} = 1\]
\[1gc{m^{ - 3}} = {n_2}kg{m^{ - 3}}\]
Now, we can write the dimensions of density in both systems.
\[
   \Rightarrow 1 \times \left[ {{M_1}{L_1}^{ - 3}} \right] = {n_2} \times \left[ {{M_2}{L_2}^{ - 3}} \right] \\
   \Rightarrow {n_2} = 1 \times \left[ {{M_1}{L_1}^{ - 3}} \right]/\left[ {{M_2}{L_2}^{ - 3}} \right] \\
   \Rightarrow {n_2} = 1 \times \left[ {{M_1}/{M_2}} \right] \times \left[ {{L_1}^{ - 3}/{L_2}^{ - 3}} \right] \\
   \Rightarrow {n_2} = 1 \times \left[ {gm/kg} \right] \times \left[ {c{m^{ - 3}}/{m^{ - 3}}} \right] \\
   \Rightarrow {n_2} = 1 \times \left[ {gm/kg} \right] \times \left[ {{m^3}/c{m^3}} \right] \\
   \Rightarrow {n_2} = 1 \times \left[ {gm/1000gm} \right] \times \left[ {1000000c{m^3}/c{m^3}} \right] \\
   \Rightarrow {n_2} = 1 \times \left[ {1/1000} \right] \times \left[ {1000000} \right] \\
   \Rightarrow {n_2} = 1000 \\
\]
Hence the relation between C.G.S. and S.I. unit of density is given as
\[1gc{m^{ - 3}} = 1000kg{m^{ - 3}}\]

Therefore, option A is correct.

Additional information:
We can find the accuracy of physical formulas and different types of equations, and also establish the physical formulas on the basis of their dependence. The main reason for using dimensional method is, “the dimension of any physical quantity is a unique representation”.

Note:
Dimensional investigation is the examination of the connections between various physical amounts by distinguishing their base amounts, (for example, length, mass, time, and electric charge) and units of measure, (for example, miles versus kilometres, or pounds versus kilograms) and following these measurements as counts or examinations are performed. The change of units starting with one dimensional unit then onto the next is frequently simpler inside the measurement or SI framework than in others, because of the customary 10-base in all units.