Adonis Diaries

Engineering Problems Transformed Mathematically 

Posted on: October 26, 2008

Article #51; (August 23, 2006)

“Basic Engineering and Physics Problems Transformed Mathematically” (draft)           

 This article exposes several practical exercises in engineering and physics that experimental data or observations had generated laws.  Consequently, the resolution of their mathematical equations provided the necessary incentives to open up new mathematical methods and fields of studies.

Conversely, many mathematical discoveries done on purely theoretical basis lead to practical applications later on.  We should keep in mind that mathematical equations are merely an abstraction of the reality, obtained by disregarding certain physical facts which seem to be of minor influence. In complicated physical situations, there may be no way of judging a priori the importance of various circumstances.

Let us state that rate of a quantity is the proportion of change in that quantity over a specific duration; the mathematical definition is written as the differential of that quantity (Q) over time (t) or dQ/dt.  For the following exercises the first step of the mathematical formulations of the laws will be provided without any attempt to solving them:

1)  Newton’s law of cooling:  Experiments show that the rate of change of the temperature T1 of a conductive material to heat is proportional to the difference between T and the temperature of the surrounding medium. The mathematical formulation is: dT/dt = -k*(T1-T2); where k is a positive constant.  Let us suppose that a copper ball is heated to a temperature T1 = 100 degrees C.  At time t=0 the ball is placed in water which is maintained at a temperature T2 = 30.  At the end of 3 minutes the temperature of the ball is reduced to T3 = 70.  Find the time at which the temperature of the ball is reduced to T4 = 31.

2) Newton’s law of gravitation:  Experiments show that the acceleration (a) of a body is proportional to inverse of the square of the distance (r) between the body and the center of the earth, or a(r) = k/r(2).  Find the minimum initial velocity of a body which is fired in radial direction from the earth and is supposed to escape from earth; neglect the air resistance and the gravitational pull of other celestial bodies. Formulation: a(r) = – g*R(2)/r(2)

3)  Torricelli’s law:  Experiments show that the velocity with which a liquid issues from an orifice is v = [0.6*square root (2*g*h)] where h is the instantaneous height of the liquid above the orifice and the constant 0.6 was introduced by Borda to account for the fact that the cross-section of the out flowing stream of liquid is somewhat smaller than that of the orifice.  A funnel whose angle at the outlet is 60 degrees and whose outlet has a cross-sectional area of 0.5 cm(2), contains water at a height of h = 10 cm.  At time t = 0 the outlet is opened and the water flows out.  Determine the time when the funnel will be empty.  Formulation: the change in volume of water lowing out during a short interval of time is: dV = 0.5* v*dt.

4)  Boyle-Mariotte’s law for ideal gases:  Experiments show that for a gas at low pressure (p) and constant temperature (T) the rate of change of the volume is: dV = -V/p

5)    The radiation of element radium law:  Experiment show that radium disintegrates at a rate proportional to the amount of radium (M) instantaneously present or: dM/dt = k*M.  What is its half-life or the time in which 50% of the amount M will disappear?  If the half-time is 1590 years, then what per cent will disappear in one year?

6)    The atmospheric pressure law:  Observations show that the rate of change of the atmospheric pressure (p) with altitude (h) is proportional to the pressure, or  dp/dh = – k*p.  If p at 18,000 ft is half its value at sea level, find the formula for the pressure at any height.

7)    The evaporation law:  Observations show that a wet porous substance in the open air loses its moisture at a rate proportional to the moisture content (Q) or:  dQ/dt = -k*Q.  If a sheet hung in the wind loses half its moisture during the first hour, when will it have lost 99% under the same weather condition?

8)    Sugar cane dilution law:  Experiments show that the rate of change of the inversion of cane sugar in diluted solution is proportional to the concentration (Y) of the unaltered sugar, or d(1/Y)/dt = k*Y.  If the concentration is 1/100 at t = 0 and is 1/250 at t = 5 hours, then find Y(t).

9)    The law of boiling liquids:  Observations show that the ratio of the quantities of two liquids of each passing off as vapor at any instant is proportional to the ratio of the quantities x and y still in the liquid state, or dy/dx = k*y/x.

10) Lambert’s law of absorption:  Observations show that the absorption of light in a very thin transparent layer is proportional to the thickness (x) of the layer and to the amount incident (A) on that layer, or dA/dx = -k*A.

11)  The law of mass action:  Experiments show that the velocity (v) of a chemical reaction, under a constant temperature, is proportional to the product of the concentrations (a) and (b) in moles per liter of the substances which are reacting.  If y is the number of moles per liter which have reacted after time (t), then the rate of reaction is: dy/dt = k*(a – y)*(b – y).

12)  Falling body law:  Experiments show that, if a body falls in vacuum due to the action of gravity and starting at time t = 0 with initial velocity v = 0, then the velocity of the body is proportional to the time or ds/dt = k*t, where s is the displacement or the distance of the body from its initial position.

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