Question

A physical quantity x is calculated from $x = a{b^2}{c^{( - 0.5)}}$. Calculate the percentage error in measuring x when the percentage errors in measuring a, b, and c are 4,2 and 3% respectively.
A. 7%
B. 9%
C. 11%
D. 9.5%

Answer 1

Uncertainty in measurement is known as error. It is the difference between the measured value and the true value. Here, we shall use the formula of the propagation error which shows the error due to variables which are in multiplicative or additive form and hence obtain the percentage error of the physical quantity.

Complete step by step answer:
Here, we have the relationship of the physical quantities as$x = a{b^2}{c^{( - 0.5)}}$
Here, x is the physical quantity, while a,b and c are the variables. The formula for propagation error for a quantity x for any formula $x = {a^p}{b^q}{c^r}$ where a,b and c are variables and p,q and r are any integers, then;
$\dfrac{{\Delta x}}{x} = p\dfrac{{\Delta a}}{a} + q\dfrac{{\Delta b}}{b} + r\dfrac{{\Delta c}}{c}$
Here, $\Delta x$ is the change in the value of x from its initial value, $\Delta a$is the change in the value of a from its initial value, $\Delta b$ is the change in the value of b from its initial value and $\Delta c$ is the change in the value of c from its initial value. The value of the change in a physical quantity over the actual value of the physical quantity is called relative error. Thus, in our case, the values of p are 1, q is 2 and r is 0.5. The relative error or the percentage error of the quantities a, b and c are given as 4%, 2% and 3% respectively. Thus, the percentage error in x is given by:

$$\dfrac{{\Delta x}}{x} = 1(4) + 2(2) + 0.5(3) \\\ \therefore \dfrac{{\Delta x}}{x} = 9.5\% \\\$$

Hence option D is the correct answer.

Note: There are different kinds of errors as follows:
Constant errors: The errors which repeat time to time are called constant errors.
Systematic errors: Errors occurring in a certain pattern or in a certain sequence or system are called system errors.
Gross errors: Errors which are produced due to improper setting of an instrument or device or due to miscalculations are called gross errors.