Question

Which one of the following is not a derived unit?

A. Joule
B. Watt
C. Kilogram
D. Newton

Answer 1

Hint: Firstly, we will derive the units given in the options using the fundamental units (mass, length and time). We will be using the basic formulae to derive the units. Joule is the unit of work, Watt is the unit of power and Newton is the unit of force and kilogram is the unit of mass.

Formula used:
\[\begin{align}
  & W=F\times s \\
 & P=\dfrac{W}{t} \\
 & F=ma \\
\end{align}\]

Complete step by step answer:
Let us derive the units first.
Now consider, Joule. The Joule is a unit of work. The work is the product of force and displacement.
\[W=F\times s\]
The force is the product of mass and acceleration.
\[W=(ma)\times s\]
The acceleration is the velocity by time.
\[W=m\times \dfrac{v}{t}\times s\]
The velocity is the displacement by time.
\[\begin{align}
  & W=m\times \dfrac{s}{t}\times \dfrac{1}{t}\times s \\
 & \Rightarrow W=\dfrac{m{{s}^{2}}}{{{t}^{2}}} \\
\end{align}\]
Now, we will substitute the fundamental units: mass, length and time.
\[\begin{align}
  & W=m{{s}^{2}}{{t}^{-2}} \\
 & \Rightarrow W=[M{{L}^{2}}{{T}^{-2}}] \\
\end{align}\]
Now consider, Watt. The Watt is a unit of power. The power is the work done by time.
\[P=\dfrac{W}{t}\]

The work is the product of force and displacement.
\[P=\dfrac{F\times s}{t}\]
The force is the product of mass and acceleration.
\[P=\dfrac{(ma)\times s}{t}\]
The acceleration is the velocity by time.
\[P=\dfrac{ms}{t}\times \dfrac{v}{t}\]
The velocity is the displacement by time.
\[\begin{align}
  & P=\dfrac{ms}{t}\times \dfrac{s}{t}\times \dfrac{1}{t} \\
 & \Rightarrow P=\dfrac{m{{s}^{2}}}{{{t}^{3}}} \\
\end{align}\]
Now, we will substitute the fundamental units: mass, length and time.
\[\begin{align}
  & P=m{{s}^{2}}{{t}^{-3}} \\
 & \Rightarrow P=[M{{L}^{2}}{{T}^{-3}}] \\
\end{align}\]
Now consider Newton. The Newton is a unit of force. The force is the product of mass and acceleration.
\[F=ma\]
The acceleration is the velocity by time.
\[F=m\times \dfrac{v}{t}\]
The velocity is the displacement by time.
\[\begin{align}
  & F=m\times \dfrac{s}{t}\times \dfrac{1}{t} \\
 & \Rightarrow F=\dfrac{ms}{{{t}^{2}}} \\
\end{align}\]
Now, we will substitute the fundamental units: mass, length and time.
\[\begin{align}
  & F=ms{{t}^{-2}} \\
 & \Rightarrow F=[ML{{T}^{-2}}] \\
\end{align}\]
Now consider, kilogram.
The dimensional formula of the kilogram is \[kg=[M]\]
Therefore, kilogram is a fundamental unit, whereas, the joule, watt and Newton are the derived units.

As the kilogram is fundamental, thus, the option (C) is correct.

Note:
The dimensional formulae are derived from the units of the quantities. As, in this case, we have used the formula to find the dimensional units. The force is a product of mass and acceleration, the power is the work done by time and the work done is the product of force and displacement.