Algebraic statements and scaling
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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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Algebraic Statements And Scaling

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Algebraic statements and scaling

Algebraic Statements And Scaling


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

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

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

Reminder: Quantitative Reasoning

  • Amazingly powerful tool to understand the world around us

  • Fundamentals:

    • Area &Volume

    • Scaling

    • Arithmetical statements

    • Ratios


From phrase to equation

From Phrase to Equation

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

  • Most people have problems with this arithmetical reasoning


Ratios

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


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