Question

In a system unit, the unit of force is 100N unit of length is 10m and the unit of time is 100s what is the unit of mass in the system?

Answer 1

Hint: Newton is the S.I unit of force, which is given by $1N=1kgm{{s}^{-2}}$. The S.I unit of mass is kilogram, denoted by (kg) and the S.I unit of length is meter, denoted by (m). Similarly, the S.I unit of time is second, denoted by (s).

Complete step by step solution:
Let’s start by understanding what is the information given in the problem. We are given a system unit where the unit of force is 100N. The unit of length is 10m and the unit of time 100s. We are asked to find the unit of mass in the system.
As per the formula of force (F), we know that: F=ma. Where, (m) is the mass of a body and (a) is the acceleration of the body, for (F) is the force acting on the body.
However, in the given problem, there isn’t any mention of acceleration. Hence, we will use the dimensional analysis.
We know that the S.I unit of force is Newton (N), which is given by: $1N=1kgm{{s}^{-2}}$. Therefore; $F=100N\Rightarrow \left[ F \right]=100\left[ ML{{T}^{-2}} \right]$.
This value of force is equal to the length, time and mass given in the problem. Hence, by putting in the values as per the dimension, we get: $\left[ F \right]=100\left[ ML{{T}^{-2}} \right]=x\left[ M \right]10\left[ L \right]{{\left( 100\left[ T \right] \right)}^{-2}}$
Since, these values are given in their S.I units, the above equation becomes: \[100N=\dfrac{\left( x\ kg \right)(10\ m)}{{{(100\ s)}^{2}}}\Rightarrow 100\ kgm{{s}^{-2}}=\dfrac{\left( x\ \right)(10\ )}{{{(100\ )}^{2}}}kgm{{s}^{-2}}\Rightarrow 10\ =\dfrac{\left( x\ \right)(1\ )}{{{(100\ )}^{2}}}\].
Therefore the value of mass becomes: $x={{10}^{5}}$. Hence the unit mass of the system is${{10}^{5}}kg=100000kg$.

Note: The above problem was an example of an unconventional method of finding a solution. Since, the normal formula of force is F=ma, however no value of acceleration was given in the solution. Hence, we had to use the dimensional analysis to solve this problem.