From Fraction to Fart…

Ah, fractions.  The cornerstone of 4th grade math.  Why is it so hard to get our little heads around fractions?  In math a fraction is simply a way of describing a part of a whole.  The Latin origin is helpful:  the word fraction entered English in the 14th century from the Anglo-French word fraccioun, which came from the Old French fraccion.  The Late Latin word fractionem meant a breaking and came from fractus, the past participle of frangere meaning to break something into parts.  Here is where it gets interesting.  Yes, the English words fragment and fracture share the same roots, entering English around the same time.  The Proto-Italic root *frang- came from the Proto-Indo-European *bhreg- meaning to break (and root for the word break) but also Old English brecan meaning to break, Lithuanian brasketi meaning to crash or crack and Old Irish braigim meaning to break wind.  Yes, all things 4th grade eventually make their way back to breaking wind. A fraction, then is just a part broken off from a whole.  That wasn’t so hard, was it?  Now, who farted? 

Mathematics

Kinematics help

Cont’d from “Kinematics example”.

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Different problems involving kinematics often share common properites which can be used (or not used) depending on the scenario. Once you become more adept at solving these problems individually, you’ll begin to easily recognise situations where some conditions apply and others may not and start to adapt the conditions to fit the scenario. A bit of thought is required to decide logically whether one condition may be physicaly viable in the situation or not.

In this post, I will look at topics that are not strictly kinematics but have applications within the field and in wider mechanics.

Boundary conditions

There are some common boundary conditions (i.e. what happens at the extreme-most points of the motion) which come from logical reasoning.

For motion of an object with its weight perpendicular to flight motion it can be assumed that the x-component of velocity stays constant throughout the flight. This means that the final xvelocity, vx, is equal to the initial x velocity, v0, x.

Themaximum height of flight, ymax,is always the point where the y velocity, vy = 0. This is a direct result of calculus, wherein dy/dt = 0 at a turning point (a maximum or minimum of a function). Now, if the time to reach the maximum height is equal to the time to reach the ground again (i.e. for perfect parabolic flight where the launch height = landing height) then we can assume that the time to reach ymax is equal to half of the full flight time.

Additionally, if we set the original position of the object at the origin of our system of co-ordinates, the point at which we find it at the same height again (the point where it lands, assuming it lands at the same height) by setting y = 0.

From there we can select an appropriate SUVAT equation.

Forces and vector resolution

See “Resolving vectors.

Sometimes you may have to combine energy conservation, momentum or force superposition (mostly, for equibrium) into your answer.

Now, if something is accelerating then its vectorial sum of all forces is equal to ma, where a is its resultant acceleration. If it is in equilibrium (“stationary” or “moving at constant velocity”) then the vector sum of its forces is equal to zero. This is a very important consequence and can be used extensively to solve problems.

You should always remember to resolve all forces/vectors in a free body diagram when working on the x and y components separately. Unless a vector is at 90° from the direction you’re resolving it to it will contribute that component. For example, the vector W acts only in the y-direction, meaning we can leave it out of our x-component resolution.

Remember, if a vector points in the opposite direction to the direction you’re working in, give its magnitude a negative sign. This is what’s intended by the vectorial (or algebraic) sum as opposed to a simply scalar summation.

Resistive forces and their applications

See a question about friction.

Generally, in kinematics we ignore the effects of resistive forces – especially air resistance – since they can make problems more complicated to solve. However, resistive forces such as the friction force Ff or the reaction force R can be used to our advantage.

The magnitude of the frictional is given by

Ff = μR,

where μ is the friction coefficient. Vectorially, we define Ff to be perpendicular to the reaction force R (along the surface of the material) and opposite to the direction of motion with unitary vector v. Hence we can express this as,

Ff = − μR v,

although the magnitude will usually suffice.

Now, we shall further examine what is meant by the reaction force. The reaction force is defined in a direction perpendicular to the surface the object rests upon and will often have its y-component Rx equal in magnitude and opposite in direction to the weight W of the object, assuming the weight acts purely in the y-direction. See Newton’s third law of motion for more information. Thus we have,

Rx = W,

assuming the object is fully supported by the surface it sits upon.

These results can be useful when applied simultaneously. If we know the frictional force and friction coefficient, we can find the reaction force and, in turn, the object’s weight. Since we know that

W = mg

from Newton’s second law of motion, we can hence find the object’s mass and its overall acceleration.

4

I should probably stop now. It’s 11:20 pm and I’ve checked 10 out of 18 of the things I had to do this weekend. Today I studied: • History (Middle Ages) • Chemistry (Periodic Table properties) • Algebra • a bit of the SAT’s writing section I guess it was a pretty productive day! oh and I downloaded that tree app and it’s so good that I actually felt bad for killing the tree. /February 28th 2015/

I think that mathematically interested sixth formers, wanting to study the subject at University, should have some background knowledge.  Many would have heard about the Riemann Hypothesis and a link to prime numbers; this subject is beyond the scope of this blog but school pupils, with knowledge of prime numbers and geometric progressions, can see how the Zeta Function relates to prime numbers in the above bit of mathematics.  I like it anyway.