Re: [Phys-L] playing for keeps

[Bernard Cleyet <bernard@cleyet.org>]

I wrote the below and then finished reading the thread.  (OS 10.8 calls it a conversation) and see JD had anticipated me.



On 2013, Jun 29, , at 14:17, Bruce Sherwood <Bruce_Sherwood@NCSU.EDU> wrote:

> Good engineering and science students at good universities have memorized
> the formulas C = 2*pi*R and A = pi*R^2 but often do not know which applies
> when to what, nor that there is any connection to things they really do
> know. It would be awfully useful somewhere to start with the circumference
> being about 4*(2R) = 8R (the circumference of the bounding square) and then
> see that this is only a bit bigger than 2*pi*R, and to start with the area
> being about (2R)^2 = 4R^2 (the area of the bounding square) and then see
> that this is only a bit bigger than pi*R^2.
>
> An example of confusion is the frequently seen A = 2pi*R^2.
>
> Bruce
> _____________________________________________

let's see    arc length is radius times the angle  well 2PI is 360 so circle is 2pi r

area is add one to r's exp. and divide by the result.   now we have an area i.e. length times length

same with sphere surface is 4pi r*r  again add one and divide by the result.  4/3 pi r^3  the volume again. cheque  l* l*l  is volume.


bc's nemonic(s)


On 2013, Jun 29, , at 18:50, John Denker <jsd@av8n.com> wrote:

> On 06/29/2013 02:17 PM, Bruce Sherwood wrote:
>
> > An example of confusion is the frequently seen A = 2pi*R^2.
>
> This provides a delightful illustration of a couple of points 
> I've been trying to make.
>  -- Check your work.
>  -- Look for connections between things.
>
> 1a) Write down the formula for the circumference of a circle.
>    My guess is C = 2 π r
>
> 1b) Write down the formula for the area of a circle.
>    My guess is A = π r^2
>
> 1c) Check the work!
>    Look for connections!
>    The area of an annulus had better be 2 π r dr, and 
>    if you integrate that you get π r^2.
>    Conversely if you differentiate the area you'd better
>    get dA = 2 π r dr
>
> Similarly ....
>
> 2a) Write down the formula for the area of a sphere.
>    My guess is A = 4 π r^2.
>    In other words, there are 4 π steradians total.
>
> 2b) Write down the formula for the volume of a sphere.
>    My guess is V = 4/3 π r^3
>
> 2c) Check the work!
>    Look for connections!
>    There's a connection between the area and the volume,
>    and you can use that to check the work.