I Love Math

Anyone who knows the Pennerfive at all will know immediately from the title of this post that it must have been written by Mike.

I love math.  Always have.  My mom says I was counting by two’s well before kindergarten.  In elementary school, I recall doing extra challenge work by myself in the library.  I have vivid memories of learning about Pi and being completely pumped about it.  My yearbook caption in Grade 7 was “The Wiz” – based on the fact that I aced just about every math test we had that year.  My mark in Grade 12 calculus was 98%. 

Okay, so I LOVE math.  But what I love more is the practical applications of math.  I suppose that’s why I became a mechanical engineer instead of a mathematics professor.  Math is useful every single day of our lives.  We use math all the time.  For example,

Q:  The Canucks & Predators Game 5 starts at 5:00 p.m. today and it’s currently 9:30 a.m.  How much time do I have left to help the girls get their Mother's Day shopping done before I need to be on the couch?

A:  It doesn’t matter because the PVR is set to record the series, so I can start watching it whenever I want.

Okay, that’s not a good example.  How about this problem, one that I’m sure everyone has encountered and wondered how to solve? 

When making a peanut butter and banana sandwich, how big a knife do I need to use (that is, how wide does the blade need to be), to ensure that the banana slices stick to the blade long enough so that I can accurately place them on my peanut-butter-covered piece of bread, instead of falling off prematurely and ending up on the kitchen counter, or worse, rolling along the counter-top, over the edge, and onto the floor?

Now, this is a complex problem with many, many variables.  As an engineer, I’ve been taught to reduce the complexity of a problem by making reasonable, simplifying assumptions in order to minimize the number of variables.  With that in mind, here is my simplified analysis of this perplexing problem:

First, let’s define all the relevant variables and make those simplifying assumptions:

Average Knife Blade-Width, K – This affects the area in contact with the banana slice.  This is the variable that we’re trying to solve for.  We are trying to find the minimum blade-width required to ensure the banana will stick to the blade for at least 5 seconds – enough time to get the banana slice into its proper place.  K is in units of metres [m].  Note:  SI (metric) units will be used throughout.  We are in Canada after all.

Diameter of the Banana, D.  There is some variability here of course, as bananas come in various sizes and the diameter changes a bit along its length, but we will assume a maximum banana diameter of 30 mm or 0.03 m.  Caution: this is an ASSUMED maximum banana diameter.  Bananas with larger diameters do exist!  If you’re handling a banana larger than 0.03 m in diameter, the results may NOT be valid.  Apply the results of this analysis at your own risk.

Thickness of the Banana Slice, T.  Everyone has their own preference here perhaps, but there is probably a fairly narrow range of acceptable values to 95% of the peanut butter and banana sandwich eating population.  Similar to the banana diameter, a maximum thickness can be assumed.  I think the peanut butter should never be overpowered by banana slices that are too thick.  I will assume that T = 3 mm or 0.003 m.

Gravity, g.  This is the force exerted by the earth’s gravitational pull on the slice of banana.  Although this is technically a variable depending on the altitude and latitude of the banana, I'll use the standard average as an approximation, which is 9.81 m/s^2.

Density of the Banana, d.  This will be assumed to be a constant, no matter what type, size or ripeness of banana.  1140 kg/m^2 is the value I obtained from Answers.com with a quick Google search.  Of course, if I was stamping and sealing my analysis as a P.Eng., I would need to be a little more thorough in my research than that.  But for my purposes here, Answers.com is good enough.

Surface Roughness of the Blade, u.  Assumed to be constant – all the knives are pretty similar in our kitchen – and this parameter will be incorporated into the next variable anyway, to simplify the calculation.

Stickiness Coefficient of a Banana Slice, SC.  We’re not talking about friction here, because the banana sticks to the knife, even though the knife blade is perfectly vertical.  There’s an inherent “stickiness” of the banana slice when in contact with the knife blade surface.  You can even turn the knife beyond the vertical, and in fact, you do this at the last second before laying the banana slice into the preferred location. 

This is where it gets really tough because the “Stickiness Coefficient” likely depends on the type of banana (Cavendish, Plantain, Burro, etc), surface roughness of the knife, and what I call the "Ripeness Factor".

To simplify the analysis, we will consider only the Cavendish (a.k.a. Chiquita) banana, as it is the most popular banana used in the manufacturing of peanut butter and banana sandwiches today (at least North American made sandwiches).  As mentioned above, the surface roughness of the knife is assumed to be constant.  That leaves only one variable affecting the "Stickiness Coefficient" then, the "Ripeness Factor".

I suppose the overall “Stickiness Coefficient” could be assumed to be constant, no matter the ripeness of the banana.  The problem is that we really have to do some first-hand experimentation to come up with a numerical value for this physical property at all, since “Stickiness Coefficient” is not in the currently published data for Cavendish bananas.  If we’re going to do some experimentation anyway, we might as well develop an empirical formula to allow us to determine the Stickiness Coefficient depending on the Ripeness Factor of the banana, rated on a scale of, say, 1 to 10, 1 being as green as my Aunt Loreen on a Tilt-A-Whirl and 10 being as brown as . . . well, you know.

Assuming you’re not bored already (big assumption, I know), I won’t risk boring you with the details of the design and execution of the experiment.  But the results showed that the Stickiness Coefficient of a Cavendish banana slice on a standard Pennerfive kitchen knife is essentially a constant, especially across the range of Ripeness Factors that most manufacturers would consider acceptable for use in a peanut butter and banana sandwich.  The Ripeness Factor is fairly subjective anyway, so it’s a good thing the Stickiness Coefficient is constant, otherwise it could be a potential source of error in the calculation.

So, the results of the experiment showed that the Stickiness Coefficient, SC, can be taken as a constant equal to approximately 67.2 N/m^2, no matter how ripe the banana is.  This greatly simplifies the calculation. 

We now calculate as follows:

Weight of the banana = W = (pi/4) x D^2 x T x p x g = 0.0237 N

In order for the banana slice to stay stuck to the knife, the weight of the banana slice must be countered by the resistive force provided by the “stickiness” of the banana slice.  This is simply the area of the banana in contact with the knife, A, multiplied by the Stickiness Coefficient, SC, which was determined based on a “sticking time” of the desired 5 seconds.

So, the basic formula is simply:

W = SC x A

The area of the banana slice in contact with the knife is assumed to be as defined by the shaded portion of the circle in the sketch below:

From the above geometry we can calculate the contact area as follows:

A = ½ x (D/2)^2 x (Ө – Sin Ө)

where Ө is the central angle of the circular segment in units of Radians.  Sorry to disappoint, but no, I didn’t know this formula from memory.  I got it here: http://en.wikipedia.org/wiki/Circular_segment.

I suppose I could have derived the formula, but as an engineer, I’m always careful not to “reinvent the wheel”.  If someone else has derived the formula already, why bother doing it again?  So, with that philosophy in mind, the formula for the central angle in terms of D and K has also been derived previously (same link as above).  It is:

Ө = 2arccos(1 - 2K/D)

Substituting, we get the area in terms of D and K:

A = ½ x (D/2)^2 x [2arccos(1 – 2K/D) – Sin 2arccos(1 - 2K/D)]

Then, using

W = SC x A

and inserting our values of W, SC and D, we get:

0.0237 N = 67.2 N/m^2 x ½ x (0.03m/2)^2 x [2arccos(1 – 2K/0.03m) – Sin 2arccos(1 - 2K/0.03m)]

Only one variable left, so all we have to do is solve for K.  Easy, right?  Ummmm, no.  I don’t remember how to solve this multi-term, trigonometric equation by hand.  You see, although I spent 4 years learning advanced math and calculus in university (and loving it), as an engineer in the real world, I would estimate that I've used about 1% of the math I learned in school.  So, I haven’t solved an equation like this in at least 15 years.  What's an engineer to do?  Give up?  Go back to school?  No and no!  Cheat!

Not really.  I solved for K by trial and error and found that K = 0.015 m, which is 15 mm.

You may have noticed that that this is exactly one half of the diameter of the banana slice, which we took as 30 mm.  This provides a very useful rule-of-thumb for slicing bananas when manufacturing a peanut butter and banana sandwich.  Use a knife with a blade-width of at least one half the diameter of the banana.  This will ensure that the banana slice will stick to the knife long enough for you to position it exactly where you want it, instead of risking a rogue banana slice finding its way to your kitchen floor.

See?  We use math for useful things every day.  I love math.  This post illustrates why.

Incidentally, this post also illustrates why a lot of people, including my wife, suspect that engineers may be a different species from the rest of the population.  Actually, I have a confession to make: although tempted to do so, I did not in fact conduct an experiment to determine the "Stickiness Coefficient" of banana slices versus their "Ripeness Factor".  I made up the value of the Stickiness Coefficient and this entire post was written with my tongue planted firmly in my cheek.  That probably does nothing to dispell the notion, does it?