Re: [Phys-L] calc probs for physics

[John Denker <jsd@av8n.com>]

On 12/21/2013 01:31 PM, Paul Lulai wrote:

> Does anyone have a favorite resource from which they select
> calc-based problems for physics students?

In addition to the previous answer, here is a stratagem that allows
you to /construct/ (rather than select) calculus problems.

Executive summary:  Turn a cut-and-dried calculation into an optimization.

Here's an example.  I flipped open a book, picked a calculation
almost at random, and promoted it to an optimization.  There must
be hundreds of calculations that could be promoted in this way.

a) Start by considering the following cut-and-dried calculation.
A version of this appears in more-or-less every physics text known 
to man:  We have brick sliding on a table, being pulled by a string.
The coefficient of friction is 0.5 and the weight of the brick is 
255 N and the string makes a 40° angle to the horizontal, what is 
the tension in the string?

                         /
                        /
                       /
               _______/
               |XXXXXX|
   =============================================
            A                                  B


That's fine as far as it goes.  Tangential point, FWIW:  This can 
be made more abstract as follows:

b) The coefficient of friction is μ, the weight is mg, and the angle
(relative to horizontal) is θ.  Find a general expression for the 
tension.  As a final step, plug in the specific values for μ, mg, 
and θ given above.

My main non-tangential point is that this can be turned into a calculus 
exercise:

c) The weight and the coefficient of friction are still considered 
"knowns" with specified values ... but the angle θ is now variable.
The goal is to optimize the angle so as to pull the brick using the 
minimum amount of tension in the string.  What is the optimal angle,
and what is the required tension?

Remark:  The calculation in part (c) is only slightly messier than 
in part (b), if you know anything about calculus.  The result is 
nontrivial without being unduly complicated.  It has an obvious 
connection to practical applications.

Important remark:  This is just the flea on the tail of the penguin
on the tip of the iceberg.  There are *lots* of elementary physics
calculations that can be turned into optimizations.  We can discuss
additional specific examples later.

d) For extra credit:  Continuing the brick-dragging exercise: We are 
no longer concerned with minimizing the tension.  Instead we want to 
minimize the /work/ required to drag the nose of the brick from point 
A to point B.  What is the optimal angle of pull, and what is the 
required tension?

Remark:  Again this has an obvious connection to practical applications.