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# Algebraic Statements And Scaling - PowerPoint PPT Presentation

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!

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Presentation Transcript

• 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?

• 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

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

• Amazingly powerful tool to understand the world around us

• Fundamentals:

• Area &Volume

• Scaling

• Arithmetical statements

• Ratios

• Important skill: translate a relation into an equation, and vice versa

• Most people have problems with this arithmetical reasoning

• 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