Question

If the units of force were kilonewton, that of time millisecond and that of power kilowatt, what would be the units of mass in kg?

Answer 1

Hint: First we will calculate the value of length L using the power formula Power is the usage or transfer of quick energy power is the sum of energy separated by the time it took to use the energy. Power = $\dfrac{Energy}{Time}$Two factors depend on the amount of work done when a force works on a body:
The size of the force which acts on the object
The distance by which the force causes the body to travel in the force's direction
Work done = force × distance An object's acceleration, as caused by a net force, is directly proportional to the magnitude of the net force, in the same direction as the net force, and is inversely proportional to the mass of the object.
Force = mass× acceleration

Complete step by step solution:
Force unit is represented as = F = kilo Newton = kN = ${{10}^{3}}$ N
Time unit is represented as = t = millisecond = ms = ${{10}^{-3}}$ s
Power unit is represented as = P= kilo watt =Kw = ${{10}^{3}}$w
In order to find unit of mass
Power = $\dfrac{Energy}{Time}$
[Energy = Work]
Power = Force × $\dfrac{Dis\tan ce}{Time}$------ (1)
Substituting the values in equation (1)
${{10}^{3}}$w = ${{10}^{3}}$N× Length/${{10}^{-3}}$ s
Length = ${{10}^{-3}}$ m = 1mm
Force = mass × acceleration
${{10}^{3}}$N = mass × $\dfrac{Length}{tim{{e}^{2}}}$
${{10}^{3}}$N = mass × $\dfrac{{{10}^{-3}}}{{{({{10}^{-3}})}^{2}}}$
${{10}^{-3}}$ = mass × ${{10}^{-3}}$
Mass = 1kg
Hence, mass = 1kg

Hence unit of mass is in kilogram (kg)

Note:
We can solve it using dimensional analysis
Dimension of power P= $M{{L}^{2}}{{T}^{-1}}$ -------- (a)
Dimension of force F= $ML{{T}^{-2}}$
Dimension of ${{F}^{2}}$ = ${{M}^{2}}{{L}^{2}}{{T}^{-4}}$ --------- (b)
Taking equation (a) and (b)
= $\dfrac{P}{{{F}^{2}}}$ ---------------- (2)
Putting the value in equation (2)
= $\dfrac{M{{L}^{2}}{{T}^{-1}}}{~{{M}^{2}}{{L}^{2}}{{T}^{-4}}~~}$
=$\dfrac{P}{{{F}^{2}}}$= $\dfrac{{{T}^{3}}}{M}$
= M= $\dfrac{{{F}^{2}}{{T}^{3}}}{P}$---------- (3)
Putting the value in equation (3)
M= $\dfrac{{{({{10}^{3}})}^{2}}{{({{10}^{-3}})}^{3}}}{{{10}^{-3}}}$
M= 1kg
Hence unit of mass is in kilogram (kg)