Fechner's law in experimental psychology was first hypothesized in the 1860s. It describes the relationship between a change in sensory stimulus and the perceived change in the stimulus. Examples include • adding one tsp of sugar to a cup of tea (stimulus change) and sensing how much sweeter the tea tastes (perception change) • increasing the weight of an object you're holding (stimulus change) and sensing how much heavier the object feels (perception change) Let's look at this second situation. Let s be the weight of an object in grams, and let P(s) be the perceived weight of the object to someone holding it. Fechner's law says for some constant k > 0, k dP ds (a) Translate the differential equation to a sentence by filling in the blanks: The rate of change of with respect to a change in to the is Words you might use are time, weight, perceived weight, proportional, inversely proportional, equal. (b) Check that P(s) = k ln(s) + C is a solution to Fechner's law for any constant C. (c) Suppose that 0.5 grams is the threshhold stimulus, the highest actual weight at which a person cannot perceive any weight. In other words, holding 0.5 grams feels like holding zero weight. Find the particular solution to Fechner's law given this initial value. (d) What does your solution from part (b) predict for the perceived weight of an object of weight 0.25 grams should be? Does this make sense? If so, explain why, and if not, suggest a way to improve the model. (e) Does Fechner's law predict that it is easier to tell the difference between objects weighing 5 and 10 grams or between objects weighing 100 and 105 grams? Explain your answer in 1-2 sentences, referring to the differential equation.
