How many significant figures are there in the following measured values, and what is the uncertainty in each measurement? (a) \(0.001 \mathrm{~kg}\) (b) \(0.00010 \mathrm{~m}\) (c) \(102 \mathrm{~L}\) (d) \(2.600 \times 10^{-3} \mathrm{~m}\) (e) \(1.1 \times 10^{6} \mathrm{~km}\)

Short Answer

Expert verified
(a) There is one significant figure in \(0.001 \mathrm{~kg}\), and the uncertainty is \(\pm 0.001 \mathrm{~kg}\). (b) There are two significant figures in \(0.00010 \mathrm{~m}\), and the uncertainty is \(\pm 0.00001 \mathrm{~m}\). (c) There are two significant figures in \(102 \mathrm{~L}\), and the uncertainty is \(\pm 1 \mathrm{~L}\). (d) There are three significant figures in \(2.600 \times 10^{-3} \mathrm{~m}\), and the uncertainty is \(\pm 0.001 \times 10^{-3} \mathrm{~m}\). (e) There are two significant figures in \(1.1 \times 10^{6} \mathrm{~km}\), and the uncertainty is \(\pm 0.1 \times 10^{6} \mathrm{~km}\).

Step by step solution

01

(a) Number of significant figures for \(0.001 \mathrm{~kg}\)

To find the significant figures, let's identify the non-zero digits and zeros according to the rules mentioned in the analysis. In this measurement, there is only one non-zero digit (1), so there is only one significant figure in \(0.001 \mathrm{~kg}\).
02

(a) Uncertainty for \(0.001 \mathrm{~kg}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.001 \mathrm{~kg}\). So the uncertainty for this measurement is \(\pm 0.001 \mathrm{~kg}\).
03

(b) Number of significant figures for \(0.00010 \mathrm{~m}\)

In this measurement, the significant figures include the 1 and the trailing zero after the decimal point (0). So there are two significant figures for \(0.00010 \mathrm{~m}\).
04

(b) Uncertainty for \(0.00010 \mathrm{~m}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.00001 \mathrm{~m}\). So the uncertainty for this measurement is \(\pm 0.00001 \mathrm{~m}\).
05

(c) Number of significant figures for \(102 \mathrm{~L}\)

In this measurement, both non-zero digits (1 and 2) are significant figures. There are no trailing zeros with a decimal point. So there are two significant figures for \(102 \mathrm{~L}\).
06

(c) Uncertainty for \(102 \mathrm{~L}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(1 \mathrm{~L}\). So the uncertainty for this measurement is \(\pm 1 \mathrm{~L}\).
07

(d) Number of significant figures for \(2.600 \times 10^{-3} \mathrm{~m}\)

In this measurement, all non-zero digits (2, 6) and the trailing zero after the decimal point (0) are significant figures. There are three significant figures for \(2.600 \times 10^{-3} \mathrm{~m}\).
08

(d) Uncertainty for \(2.600 \times 10^{-3} \mathrm{~m}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.001 \times 10^{-3} \mathrm{~m}\). So the uncertainty for this measurement is \(\pm 0.001 \times 10^{-3} \mathrm{~m}\).
09

(e) Number of significant figures for \(1.1 \times 10^{6} \mathrm{~km}\)

In this measurement, both non-zero digits (1 and 1) are significant figures. There are no zeros with a decimal point. So there are two significant figures for \(1.1 \times 10^{6} \mathrm{~km}\).
10

(e) Uncertainty for \(1.1 \times 10^{6} \mathrm{~km}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.1 \times 10^{6} \mathrm{~km}\). So the uncertainty for this measurement is \(\pm 0.1 \times 10^{6} \mathrm{~km}\).

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