Question

The percentage error in measuring ${\text{ }}M,L,{\text{ }}$ and ${\text{ }}T{\text{ }}$ are ${\text{ }}1\% ,{\text{ }}1.5\% {\text{ }}$ and ${\text{ 3% }}$ respectively. Then the percentage error in measuring the physical quantity with dimensions ${\text{ }}\left[ {M{L^{ - 1}}{T^{ - 1}}} \right]$ is:
A) ${\text{ }}1\% $
B) ${\text{ }}3.5\% $
C)${\text{ }}3\% $
D) ${\text{ }}4.5\% $
E) ${\text{ }}5.5\% {\text{ }}$

Answer 1

Hint: A quantity that can be measured in terms of numerical values and can be expressed in terms of numerical values is called a physical quantity. Dimensions of a physical quantity are the powers to which the base quantities are to be raised to represent that quantity. Physical quantities like mass, length, time, etc. can be expressed in terms of dimensions. The total error of a quantity can be calculated by summing up the errors in the base quantities.

Formula used:
$\dfrac{{\Delta x}}{x} \times 100 = \dfrac{{\Delta M}}{M} \times 100 + \dfrac{{\Delta L}}{L} \times 100 + \dfrac{{\Delta T}}{T} \times 100$
Where ${\text{ }}\dfrac{{\Delta x}}{x} \times 100{\text{ }}$is the percentage error in the physical quantity
${\text{ }}\dfrac{{\Delta M}}{M} \times 100{\text{ }}$ stands for the percentage error in the mass (${\text{ }}M{\text{ }}$)
${\text{ }}\dfrac{{\Delta L}}{L} \times 100{\text{ }}$ stands for the percentage error in the length (${\text{ }}L{\text{ }}$)
and, ${\text{ }}\dfrac{{\Delta T}}{T} \times 100{\text{ }}$ stands for the percentage error in the time (${\text{ }}T{\text{ }}$)

Complete step by step solution:
All physical quantities can be expressed as a combination of fundamental or base quantities. The seven fundamental quantities as Mass (${\text{ }}M{\text{ }}$), Length (${\text{ }}L{\text{ }}$), Time (${\text{ }}T{\text{ }}$), Electric current${\text{ }}\left( A \right){\text{ }}$, Thermodynamic temperature${\text{ }}\left( K \right){\text{ }}$, Luminous intensity, ${\text{ }}(cd){\text{ }}$ and Amount of substance${\text{ }}\left( {mol} \right){\text{ }}$
We can write the dimension of any physical quantity by putting the suitable powers to the base quantities that represent that quantity.
When we use an instrument to measure a quantity, the measured value will be different from its true value. The difference between the measured value and the true value is called the error.
The percentage error in measuring${\text{ }}M{\text{ }}$,${\text{ }}\dfrac{{\Delta M}}{M} \times 100{\text{ }}$${\text{ }} = 1\% {\text{ }}$
The percentage error in measuring${\text{ }}L{\text{ }}$,${\text{ }}\dfrac{{\Delta L}}{L} \times 100{\text{ }}$${\text{ }} = 1.5\% $
The percentage error in measuring, ${\text{ }}T$${\text{ }}\dfrac{{\Delta T}}{T} \times 100$=${\text{ }}3\%.$
The percentage error in the physical quantity with dimension ${\text{ }}\left[ {M{L^{ - 1}}{T^{ - 1}}} \right]{\text{ }}$can be obtained by substituting these values in the equation,
$\dfrac{{\Delta x}}{x} \times 100 = \dfrac{{\Delta M}}{M} \times 100 + \dfrac{{\Delta L}}{L} \times 100 + \dfrac{{\Delta T}}{T} \times 100$

Substituting the values in the equation,
$\dfrac{{\Delta x}}{x} \times 100$$ = 1 + 1.5 + 3$
$\dfrac{{\Delta x}}{x} \times 100 = 5.5\% $

The answer is Option (E):${\text{ }}5.5\%.$

Note: Percentage error is the relative error expressed in percentage. The dimensional formula of a given physical quantity is an expression showing the dimensions of the fundamental quantities. An equation connecting the physical quantity with its dimensional formula is called the dimensional equation of that physical quantity.