Determine the absolute and percent relative uncertainty expressing the result in the correct number of significant digits
a.) 6.2 (+-0.2)-4.1(+-0.1)
b.)[9.23(+-0.03)][4.21(+-0.20}/21.1(+-0.2)
c.)log[3.1415(+-0.001)]
d) log[(0.104(+-0.0060)]^1/2 /0.0511(+-0.00090]
To determine the absolute and percent relative uncertainty for each expression, we need to use the given uncertainties and the rules of uncertainty propagation.
a) To determine the absolute uncertainty for the expression 6.2 (+-0.2) - 4.1 (+-0.1), we need to add the absolute uncertainties of each term.
Absolute uncertainty = absolute uncertainty of 6.2 + absolute uncertainty of 4.1
Absolute uncertainty = 0.2 + 0.1 = 0.3
The result of the expression is 6.2 - 4.1 = 2.1
To determine the percent relative uncertainty, we divide the absolute uncertainty by the result and multiply by 100.
Percent relative uncertainty = (absolute uncertainty / result) * 100
Percent relative uncertainty = (0.3 / 2.1) * 100 = 14.29%
Therefore, the result with the correct number of significant digits, considering the largest uncertainty, is 2.1 (+-0.3) with a percent relative uncertainty of 14.29%.
b) To determine the absolute uncertainty for the expression [9.23 (+-0.03)][4.21 (+-0.20)] / 21.1 (+-0.2), we need to calculate the absolute uncertainties of each term and then propagate them using the rules of uncertainty propagation.
Absolute uncertainty of the numerator = absolute uncertainty of 9.23 + absolute uncertainty of 4.21
Absolute uncertainty of the numerator = 0.03 + 0.20 = 0.23
Absolute uncertainty of the denominator = absolute uncertainty of 21.1
Absolute uncertainty of the denominator = 0.2
The absolute uncertainty for the expression is given by:
Absolute uncertainty = [(absolute uncertainty of the numerator / result) + (absolute uncertainty of the denominator / result)] * expression
Absolute uncertainty = [(0.23 / 36.38) + (0.2 / 21.1)] * [9.23 * 4.21 / 21.1]
Absolute uncertainty ≈ 0.0131551
The result of the expression is [9.23 * 4.21] / 21.1 ≈ 1.8427
Percent relative uncertainty = (absolute uncertainty / result) * 100
Percent relative uncertainty = (0.0131551 / 1.8427) * 100 ≈ 0.71%
Therefore, the result with the correct number of significant digits, considering the largest uncertainty, is 1.8427 (+-0.013) with a percent relative uncertainty of 0.71%.
c) To determine the absolute uncertainty for the logarithm expression log[3.1415 (+-0.001)], we need to calculate the absolute uncertainty of the argument.
Absolute uncertainty = absolute uncertainty of 3.1415
Absolute uncertainty = 0.001
The result of the expression is log(3.1415) ≈ 0.497
Percent relative uncertainty = (absolute uncertainty / result) * 100
Percent relative uncertainty = (0.001 / 0.497) * 100 ≈ 0.20%
Therefore, the result with the correct number of significant digits, considering the largest uncertainty, is 0.5 (+-0.001) with a percent relative uncertainty of 0.20%.
d) To determine the absolute uncertainty for the logarithmic expression log[(0.104 (+-0.0060))^1/2 / 0.0511 (+-0.00090)], we need to calculate the absolute uncertainties of each term and then propagate them using the rules of uncertainty propagation.
First, let's calculate the absolute uncertainty for (0.104 (+-0.0060))^1/2:
Absolute uncertainty = (absolute uncertainty of 0.104 / result) * expression
Absolute uncertainty = (0.0060 / 0.104) * (0.104)^1/2 ≈ 0.0060
Now, let's calculate the absolute uncertainty for the denominator 0.0511 (+-0.00090):
Absolute uncertainty = absolute uncertainty of 0.0511
Absolute uncertainty = 0.00090
The absolute uncertainty for the logarithmic expression is given by:
Absolute uncertainty = [(absolute uncertainty of the numerator / result) + (absolute uncertainty of the denominator / result)] * expression
Absolute uncertainty ≈ [(0.0060 / 0.0527) + (0.00090 / 0.0511)] * log(0.104)^1/2 / 0.0511
Absolute uncertainty ≈ 0.00543
The result of the expression is log[(0.104)^1/2 / 0.0511] ≈ -0.098
Percent relative uncertainty = (absolute uncertainty / result) * 100
Percent relative uncertainty = (0.00543 / 0.098) * 100 ≈ 5.54%
Therefore, the result with the correct number of significant digits, considering the largest uncertainty, is -0.098 (+-0.005) with a percent relative uncertainty of 5.54%.