1.10 A mass is measured to be m = 4.4 ± 0.2 kg and its speed v is measured to be 18 ± 2 m s−1. Find the kinetic energy of the mass. The kinetic energy is E = 12mv2, so the mean value of the kinetic energy, E0, is: E0 = 12 × 4.4 × 182 = 712.8 J Using: E m = + E0 m0 we find:
2× because of the square
v v0
E 0.2 2 = 2 × = 0.267 = 712.8 4.4 18 So: E = 712.8 × 0.2677 = 190.8 J To one significant figure, the uncertainty is E = 200 = 2 × 102 J; that is E = (7 ± 2) × 102 J.
Exam tip The final absolute uncertainty must be expressed to one significant figure. This limits the precision of the quoted value for energy.
1.11 The length of a simple pendulum is increased by 4%. What is the fractional increase in the pendulum’s period?
The period T is related to the length L through T = 2
L . g
Because this relationship has a square root, the fractional uncertainties are related by: T = T0
L L0
1 ×
2
because of the square root
We are told that
L = 4%. This means we have : L0
T 1 = × 4% = 2% T0 2
1 MEASUREMENT AND UNCERTAINTIES
15