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In all thesecases the decrease in the quantity of the substance introduced (or, conversely, therestoration of an insufficient quantity) is subject to the law N = N0 e−t/τ , where Nis the amount (in other words, the number of molecules) of the substance remainingin the body after time t has elapsed from the introduction of the amount N0 andτ is the so-called lifetime: the time elapsed when 1/e of the quantity originally introduced remains in the body. The lifetime, as one can easily verify, is 1.44 timeslarger than the half-life, which is the time elapsed when half of the original quantityof the substance remains.Suppose a radioactive substance leaves the body at a rate characterized by thelifetime τ0 , and at the same time decays spontaneously with lifetime τd .
Show thatin this case the lifetime τ characterizing the time the substance remains in the bodyis determined by the relation τ −1 = τ0−1 + τd−1 .d) A certain quantity of blood containing 201 mg of iron has been taken from adonor. To make up for this loss of iron, the donor was ordered to take iron sulfatetablets three times a day for a week, each tablet containing 67 mg of iron. Theamount of iron in the donor’s blood returns to normal according to an exponentiallaw with lifetime equal to approximately seven days.
Assuming that the iron fromthe tablets enters the bloodstream most rapidly immediately after the blood is taken,determine approximately the portion of the iron in the tablets that will enter theblood over the time needed to restore the normal iron content in the blood.e) A certain quantity of radioactive phosphorus P32 was administered to diagnose a patient with a malignant tumor, after which the radioactivity of the skinof the thigh was measured at regular time intervals. The decrease in radioactivity3045Differential Calculuswas subject to an exponential law. Since the half-life of phosphorus is known to be14.3 days, it was possible to use the data thus obtained to determine the lifetimefor the process of decreasing radioactivity as a result of biological causes. Find thisconstant if it has been established by observation that the lifetime for the overalldecrease in radioactivity was 9.4 days (see part c) above).4.
Absorption of radiation. The passage of radiation through a medium is accompanied by partial absorption of the radiation. In many cases (the linear theory) onecan assume that the absorption in passing through a layer two units thick is the sameas the absorption in successively passing through two layers, each one unit thick.a) Show that under this condition the absorption of radiation is subject to thelaw I = I0 e−kl , where I0 is the intensity of the radiation falling on the absorbingsubstance, I is the intensity after passing through a layer of thickness l, and k is acoefficient having the physical dimension inverse to length.b) In the case of absorption of light by water, the coefficient k depends on thewave length of the incident light, for example as follows: for ultraviolet k = 1.4 ×10−2 cm−1 ; for blue k = 4.6 × 10−4 cm−1 ; for green k = 4.4 × 10−4 cm−1 ; for redk = 2.9 × 10−3 cm−1 . Sunlight is falling vertically on the surface of a pure lake 10meters deep.
Compare the intensities of these components of sunlight listed abovethe surface of the lake and at the bottom.5. Show that if the law of motion of a point x = x(t) satisfies the equation mẍ +kx = 0 for harmonic oscillations, then2mẋ 2 (t)+ mx2 (t) is constant (E = K + U is the sum of the22mẋ 2 (t)of the point and its potential energy U = kx 2(t) at time t);2a) the quantity E =kinetic energy K =b) if x(0) = 0 and ẋ(0) = 0, then x(t) ≡ 0;c) there exists a unique motion x = x(t) with initial conditions x(0) = x0 andẋ(0) = v0 ;d) Verify that if the point moves in a medium with friction and x = x(t) satisfiesthe equation mẍ + α ẋ + kx = 0, α > 0, then the quantity E (see part a)) decreases.Find its rate of decrease and explain the physical meaning of the result, taking account of the physical meaning of E.6.
Motion under the action of a Hooke34 central force (the plane oscillator).To develop Eq. (5.156) for a linear oscillator in Sect. 5.6.6 and in Problem 5 let usconsider the equation mr̈(t) = −kr(t) satisfied by the radius-vector r(t) of a pointof mass m moving in space under the attraction of a centripetal force proportional tothe distance |r(t)| from the center with constant of proportionality (modulus) k > 0.Such a force arises if the point is joined to the center by a Hooke elastic connection,for example, a spring with constant k.34 R. Hooke (1635–1703) – British scientist, a versatile scholar and experimenter.
He discoveredthe cell structure of tissues and introduced the word cell. He was one of the founders of the mathematical theory of elasticity and the wave theory of light; he stated the hypothesis of gravitationand the inverse-square law for gravitational interaction.5.6 Examples of Differential Calculus in Natural Science305a) By differentiating the vector product r(t) × ṙ(t), show that the motion takesplace in the plane passing through the center and containing the initial positionvector r0 = r(t0 ) and the initial velocity vector ṙ0 = ṙ(t0 ) (a plane oscillator).
Ifthe vectors r0 = r(t0 ) and ṙ0 = ṙ(t0 ) are collinear, the motion takes place alongthe line containing the center and the vector r0 (the linear oscillator considered inSect. 5.6.6).b) Verify that the orbit of a plane oscillator is an ellipse and that the motion isperiodic. Find the period of revolution.c) Show that the quantity E = mṙ2 (t) + kr2 (t) is conserved (constant in time).d) Show that the initial data r0 = r(t0 ) and ṙ0 = ṙ(t0 ) completely determine thesubsequent motion of the point.7. Ellipticity of planetary orbits. The preceding problem makes it possible to regardthe motion of a point under the action of a central Hooke force as taking place ina plane.
Suppose this plane is the plane of the complex variable z = x + iy. Themotion is determined by two real-valued functions x = x(t), y = y(t) or, what is thesame, by one complex-valued function z = z(t) of time t. Assuming for simplicityin Problem 6 that m = 1 and k = 1, consider the simplest form of the equation ofsuch motion z̈(t) = −z(t).a) Knowing from Problem 6 that the solution of this equation corresponding tothe specific initial data z0 = z(t0 ), ż0 = ż(t0 ) is unique, find it in the form z(t) =c1 eit + c2 e−it and, using Euler’s formula, verify once again that the trajectory ofmotion is an ellipse with center at zero. (In certain cases it may become a circle ordegenerate into a line segment – determine when.)b) Taking account of the invariance of the quantity |ż(t)|2 + |z(t)|2 during themotion of a point z(t) subject to the equation z̈(t) = −z(t), verify that, in terms ofa new (time) parameter τ connected with t by a relation τ = τ (t) such that dτdt =2w|z(t)|2 , the point w(t) = z2 (t) moves subject to the equation ddτw2 = −c |w|3 , wherec is a constant and w = w(t (τ )).
Thus motion in a central Hooke force field andmotion in a Newtonian gravitational field turn out to be connected.c) Compare this with the result of Problem 8 of Sect. 5.5 and prove that planetaryorbits are ellipses.d) If you have access to a computer, looking again at Euler’s method, explainedin Sect. 5.6.5, first compute several values of ex using this method.
(Observe thatthis method uses nothing except the definition of the differential, more precisely theformula f (xn ) ≈ f (xn−1 ) + f (xn−1 )h, where h = xn − xn−1 .)Now let r(t) = (x(t), y(t)), r0 = r(0) = (1, 0), ṙ0 = ṙ(0) = (0, 1) and r̈(t) =r(t)− |r(t)|3 . Using the formulasr(tn ) ≈ r(tn−1 ) + v(tn−1 )h,v(tn ) ≈ v(tn−1 ) + a(tn−1 )h,where v(t) = ṙ(t), a(t) = v̇(t) = r̈(t), use Euler’s method to compute the trajectoryof the point. Observe its shape and how it is traversed by a point as time passes.3065Differential Calculus5.7 PrimitivesIn differential calculus, as we have verified on the examples of the previous section,in addition to knowing how to differentiate functions and write relations betweentheir derivatives, it is also very valuable to know how to find functions from relations satisfied by their derivatives. The simplest such problem, but, as will be seenbelow, a very important one, is the problem of finding a function F (x) knowing itsderivative F (x) = f (x).
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