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Uncertainties Definitions Random uncertainties Uncertainties due to the inexperience of the experimenter and the difficulty of reading instruments. They can be reduced by repeated measurements and taking an average of the measurements. Systematic uncertainties Uncertainties mainly due to incorrectly calibrated instruments. They cannot be reduced by repeated measurements. Accurate measurements Measurements that have a small systematic error. Precise measurements Measurements that have a small random error. We measure the length of the side of a cube to be 25 ± 1 mm. The 25 mm represents the measured value of the length and the ±1 mm represents the absolute uncertainty in the measured value. The ratio 1 = 0.04 is 25 1 the fractional uncertainty in the length and 25 × 100% = 4% is the percentage uncertainty in the length. In general for a quantity Q we have Q = Q × 100% = percentage uncertainty. Q0
Q0
measured value
±
Q absolute uncertainty
,
Q = fractional uncertainty, Q0
Suppose quantities a, b and c have been measured with uncertainties, respectively, a, b and c. If we use these quantities to calculate another quantity Q, these uncertainties will cause uncertainties in Q. You need to know these (approximate) rules for calculating the uncertainty in Q. If Q = a ± b ± c, then Q = a + b + c I.e. for both addition and subtraction, add absolute uncertainties. Q ab c If Q = , then = a+ b+ Q c c b a I.e for multiplication and/or division, add fractional or percentage uncertainties to get the fractional or percentage uncertainty in the result. n Q If Q = am , then =n a +m b Q b a b I.e. for powers, add fractional or percentage uncertainties, multiplied by their individual power. Q 1 a 1 b + = If Q = ab or Q = a , then Q 2 a 2 b b I.e. for square roots and other roots, add fractional or percentage uncertainties and then divide by the power of the root. This is a special case of the rule for powers.
Test yourself 3
V . The uncertainty in the voltage is 4% and that in the current is I 6%. What is the absolute uncertainty in a calculated resistance value of 24 ? The resistance of a lamp is given by R =
Test yourself 4 Each side of a cube is measured with a fractional uncertainty of 0.02. Estimate the percentage uncertainty in the volume of the cube.
Test yourself 5
m k . What is the percentage uncertainty of the period if m is measured with a percentage uncertainty 4% and the k with a percentage uncertainty 6%?
The period of oscillation of a mass m at the end of a spring of spring constant k is given by T = 2
Physics for the IB Diploma