Question

A single sheet of aluminium foil is folded twice to produce a stack of four sheets. The total thickness of the stack of sheets is measured to be $\left( {0.80 \pm 0.02} \right)mm$ . This measurement is made using a digital calliper with a zero error of $\left( { - 0.20 \pm 0.02} \right)mm$ .
What is the percentage uncertainty in calculated thickness of a single sheet?
(A) $1.0\%$
(B) $2.0\%$
(C) $4.0\%$
(D) $6.7\%$

Answer 1

To solve this question, we have to determine the true value of the thickness by subtracting the zero error from the measured value. And the uncertainty will be obtained by the addition of the uncertainties of the measured value and the zero error. Dividing these values by the total number of sheets, we will get the thickness of a single sheet from which the percentage uncertainty can be calculated.

Complete step-by-step solution
We know that the true value of a quantity as measured by a measuring instrument is obtained by subtracting the zero error of the measuring instrument from the measured value of the quantity.
So we have
$TV = MV - e$ ............................(1)
According to the question, the measured value of the total thickness of the stack of sheets is equal to $0.80mm$ . Also, the zero error is equal to $- 0.20mm$ . Therefore substituting $MV = 0.80mm$ and $e = - 0.20mm$ in (1) we get the true value of the thickness as
$TV = 0.80 - \left( { - 0.20} \right)$
$\Rightarrow TV = 1.00mm$
Since there are a total of four sheets comprising the stack, so the true value of thickness of a single sheet becomes
$t = \dfrac{{1.00mm}}{4}$
$\Rightarrow t = 0.25mm$ ............................(2)
Now, since the errors are always added, the uncertainty in the true value of the total thickness of the stack of sheets will be equal to the sum of the uncertainties in the measured value and the zero error of the digital caliper. According to the question, the uncertainty in the measurement of the thickness is equal to $0.02mm$ and the uncertainty in the zero error of the digital caliper is equal to $0.02mm$ . So the uncertainty in the true value of the thickness becomes $0.04mm$ , that is,
$\Delta TV = 0.04mm$
So the uncertainty in the measurement of thickness of a single sheet becomes
$\Delta t = \dfrac{{0.04mm}}{4}$
$\Rightarrow \Delta t = 0.01mm$ ............................(3)
Now, the percentage error in the measurement of the thickness of a single sheet is given by
$E = \dfrac{{\Delta t}}{t} \times 100$
Putting (2) and (3) in the above equation, we get
$E = \dfrac{{0.01}}{{0.25}} \times 100$
$\Rightarrow E = 4\%$
Thus, percentage uncertainty in calculated thickness of a single sheet is equal to $4.0\%$ .
Hence, the correct answer is option C.

Note
We had subtracted the zero error from the measured value to obtain the true value of the thickness. But do not subtract the uncertainties too. This is because the uncertainty is nothing but the error in the measurement of a quantity, and we know that errors are always added.