Question

A physical quantity X is represented by (\[X={{M}^{x}}{{L}^{-y}}{{T}^{-z}}\]). The maximum percentage errors in the measurement of M, L and T, respectively are \[a\%,\text{ }b\%\text{ }and\text{ }c\%\]. The maximum percentage error in the measurement of X will be:
\[\begin{align}
  & A)\text{ }(ax+by-cz)\% \\
 & B)\text{ }(ax-by-cz)\% \\
 & C)\text{ }(ax+by+cz)\% \\
 & D)\text{ }(ax-by+cz)\% \\
\end{align}\]

Answer 1

Hint: We can derive the errors involved in a physical relation from the equations itself. The errors are dependent on the operations we do in the formulae. The dimensional formulae is the indirect approach to the operations involved and thus we can derive the errors involved.

Complete answer:
The dimensional formula of a physical phenomenon gives the idea on the physical quantities involved or related to them. They also give the operations involved in the relation, thus giving us the idea to calculate the errors involved with the calculation of the physical phenomenon.
Let us recap the error analysis method for different mathematical operations.
For, \[z=x-y\], the error involved is
\[\pm \partial z=\pm \partial x\pm \partial y\]
For,\[d=\dfrac{ab}{c}\], the error involved is
\[\dfrac{\partial d}{d}=\dfrac{\partial a}{a}+\dfrac{\partial b}{b}+\dfrac{\partial c}{c}\]
For, \[d=\dfrac{{{a}^{p}}{{b}^{q}}}{{{c}^{r}}}\], the error involved is
\[\dfrac{\partial d}{d}=\dfrac{p\partial a}{a}+\dfrac{q\partial b}{b}+\dfrac{r\partial c}{c}\text{ --(1)}\]
Now, let us consider the given dimensional formula,
\[X={{M}^{x}}{{L}^{-y}}{{T}^{-z}}\], the percentage errors of each of the quantities are also given.
Using the method for error analysis given in (1), we can write that,
\[\dfrac{\partial X}{X}=\dfrac{x\partial M}{M}+\dfrac{-y\partial L}{T}+\dfrac{-z\partial T}{T}\],
But the errors are given in percentage form. It is just multiplying each error term by 100. i.e.,
\[\dfrac{\partial X}{X}\times 100=\dfrac{x\partial M}{M}\times 100+\dfrac{-y\partial L}{T}\times 100+\dfrac{-z\partial T}{T}\times 100\]
As we can understand that all the terms on the Left-hand side are already given as a, b and c respectively. So, we can calculate the percentage error involved in the physical formula on the Right-hand side by adding them together and multiplying with their powers as –
\[\begin{align}
  & \dfrac{\partial X}{X}\times 100=x\times a+(-y\times b)+(-z\times c) \\
 & \Rightarrow \text{ }\dfrac{\partial X}{X}\times 100=(ax-by-cz)\% \\
\end{align}\]
This gives the percentage error involved in the physical formula of X.

So, the correct answer is “Option B”.

Additional Information:
The use of percentage errors is highly recommended in any experiment.

Note:
The error analysis is not just a method to find the errors involved in the calculation or physical setup of the experiment, it also gives ideas in the different aspects where mathematical approach is limited to accuracy in the practical approach.