Question

In a new system, the unit of mass is $10kg$, the unit of length is $5m$ and the unit of time is $10s$. Then $10Pa = $
a. $500$ new units
b. $1000$ new units
c. $1500$ new units
d. $2000$ new units

Answer 1

Hint: Dimensional analysis is the very easy method to convert one system of units to another system of units. So, when the charge unit is varying, then the numerical values also may change. We can write the conversion of units like length, mass and time.

Complete step by step answer:
This can be solved by dimensional analysis method:
Given, SI unit of pressure is $Pascal$ $(Pa)$ or $N{m^{ - 2}}$
Let dimensional formula for pressure is $\left[ {M{L^{ - 1}}{T^{ - 2}}} \right]$
In SI system of units ${M_1} = 1kg,{L_1} = 1m,{T_1} = 1s$
In SI system of units ${M_2} = 10kg,{L_2} = 5m,{T_2} = 10s$
To find $10Pa$
We have, ${n_1}{u_1} = {n_2}{u_2}$
$ \Rightarrow 10Pa = {n_2}$…………………….(1)
$ \Rightarrow {n_2} = \dfrac{{{u_1}}}{{{u_2}}}{n_1}$
$ \Rightarrow {n_2} = \dfrac{{SI}}{{{\text{new system}}}} \times 10$
Now apply dimensional formula we get,
$ \Rightarrow {n_2} = \dfrac{{\left[ {{M_1}{L_1}^{ - 1}{T_1}^{ - 2}} \right]}}{{\left[ {{M_2}{L_2}^{ - 1}{T_2}^{ - 2}} \right]}} \times 10$
${M_1},{L_1},{T_1}$ in terms of SI system
${M_2},{L_2},{T_2}$ in New system of units.
Then, ${n_2} = 10 \times \left[ {\dfrac{{1kg}}{{10kg}}} \right]{\left[ {\dfrac{{1m}}{{5m}}} \right]^{ - 1}}\left[ {\dfrac{{1{s^{ - 2}}}}{{10{s^{ - 2}}}}} \right]$
Seconds’ term, a kg and m term cancels out. Then we have,
$ \Rightarrow {n_2} = 10 \times \left[ {\dfrac{1}{{10}}} \right]{\left[ {\dfrac{1}{5}} \right]^{ - 1}}{\left[ {\dfrac{1}{{10}}} \right]^{^{ - 2}}}$
$ \Rightarrow {n_2} = {\left[ {0.2} \right]^{ - 1}}{\left[ {0.1} \right]^{^{ - 2}}}$
 $ \Rightarrow {n_2} = 5 \times 100 = 500$
Now equation (1) becomes,
$10Pa = 500{\text{ }}n{\text{ew units}}$

Hence, the correct answer is option (A).

Additional information:
Dimensional formula is the expression of showing the powers to which the physical quantity must be raised to represent fundamental quantities.
Dimensions for the fundamental physical quantities

1) Mass	[ M ]
Length	[ L ]
Time	[ T ]
Temperature	[ K ]
Electric current	[ A ]
Luminous intensity	[ cd ]
Amount of substance	[ mol ]

Constants having dimensional formulas are called Dimensional constants.
Eg : Planck’s constant, speed of light, Universal gravitational constant..
 Physical quantities having no dimensional formula are called Dimensionless quantities.
Eg : Angle, Strain, Relative Density.
Limitations of Dimensional analysis
Proportionality constants cannot be determined by dimensional analysis.
Formulae containing non- algebraic functions like sin, cos, log, exponential etc., cannot be derived.

Note: Square bracket [ ] is used to represent dimension of the physical quantity Dimensional formulae can be used to convert one system of unit to another system of unit. And also it can be used to derive the relationship among different physical quantities.