Question

If the unit of force is \[1KN\], unit of length is \[1Km\] and the unit of time is \[100s\] in a new system, then the new unit of mass is:
\[\begin{align}
  & \text{A}\text{. 1000kg} \\
 & \text{B}\text{. 1kg} \\
 & \text{C}\text{.10,000kg} \\
 & \text{D}\text{. 100kg} \\
\end{align}\]

Answer 1

Hint: We know that the force causes a body to move. Also, the unit of motion is generally, \[N\] when the mass of the body is in \[kg\], distance is in $m$ and time is in $s$. Here, these quantities are changed in order to find a new force.

Formula used: \[F={{M}^{1}}{{L}^{1}}{{T}^{-2}}\],where $M$ is the mass, $L$ is the length and $T$ is the time.

Complete step-by-step solution:
Given that the new units are, $F=1KN=1000N$, $L=1km=1000m$ and $T=100s$
We know that $F=ma$, where $m$ is mass and $a$ is acceleration.
We also know that the dimension of force is given as \[F={{M}^{1}}{{L}^{1}}{{T}^{-2}}\], where $M$ is the mass, $L$ is the length and $T$ is the time.
We can substitute the given values to find the unknown dimension, here $M$.
Then we get, $1000=\dfrac{M\times 1000}{100^{2}}$
Or we get $M=100^{2}=10000kg$
Hence the answer is (C.) \[10,000kg\]
Additional information:
Dimensional analysis is the relationship between different physical quantities. The physical quantities are expressed in terms of the basic quantities like mass, length, and time. These basic quantities are raised to their specific powers to represent the physical quantity.
 This method is all called the factor-label method or unit-factor method. This method was found by Joseph Fourier in the year 1822. There are also a few constants called dimensionless quantities. These as the name suggests do not have any dimensions or units.

Note: Dimensions are different form units. There can be different units for the same dimension. For eg: say the dimension is length, then the different units are millimeters, centimeters, meters, kilometers, yards, inch, foot, a mile, and so on. There can also be dimensions without units, called the dimensionless quantity.