Algebraic Statements And Scaling

Scaling Often one is interested in how quantities change when an object or a system is enlarged or shortened Different quantities will change by different factors! Typical example: how does the circumference, surface, volume of a sphere change when its radius changes?

How does it scale? Properties of objects scale like the perimeter, the area or the volume –Mass scales like the volume (“more of the same stuff”) –A roof will collect rain water proportional to its surface area

Homework: Newton’s Law of Gravity Note that in order to compute a "factor of change" you can ask: by what factor do I have to multiply the original quantity in order to get the desired quantity? Example: Q: By what factor does the circumference of a circle change, if its diameter is halved? A: It changes by a factor 1/2 = 0.5, i.e. (new circumference) = 0.5 * (original circumference), regardless of the value of the original circumference. If the mass of the Sun was bigger by a factor 2.7, by what factor would the force of gravity change?  scales linear with mass  same factor If the mass of the Earth was bigger by a factor 2.2, by what factor would the force of gravity change?  scales linear with mass  same factor If the distance between the Earth and the Sun was bigger by a factor 1.2, by what factor would the force of gravity change?  falls off like the area  factor 1/ f 2 = 1/1.44 = 0.694

Reminder: Quantitative Reasoning Amazingly powerful tool to understand the world around us Fundamentals: –Area &Volume –Scaling –Arithmetical statements –Ratios

From Phrase to Equation Important skill: translate a relation into an equation, and vice versa Most people have problems with this arithmetical reasoning

Ratios Different types of ratios –Fractions: 45/7 = 6.42… Can subtract 7 from 45 six times, rest 3 –With units: 10 ft / 100ft Could be a (constant) slope, e.g. for every 10ft in horizontal direction have to go up 1 ft in vertical direction –Inhomogeneous ratios: $2.97/3.8 liters