Math 129 Calculus II for Engineers and Scientists Engineering and Physics Problems T. Sakai with D. Crombecque and N. Emerson Fall, 2017
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Math 129
Calculus II for Engineers and Scientists
Engineering and Physics Problems
T. Sakaiwith D.Crombecque and N.Emerson
Fall, 2017
Chapter 5
1. (§ 5.6) A 10 meter long crane is used to move a rock horizontally at a construction site. When liftingthe crane, the cable is continually extended to maintain the rock at the same height with the height ofthe crane pivot so that the rock moves only horizontally. If the crane is lifted with an angular speeddθ/dt = 1/4 rad/s where θ is an angle between the crane and the ground and t represents time, how fastthe rock is moving toward the crane base when the cable is 8 meter?
θ
10 m 8 m
2. (§ 5.6) A simple pendulum consists of a small bob of mass m suspended by a massless cord of length Lis released (without a push) at t = 0 where the cord makes an angle θ = θ0. Knowing that at the lowestpoint, the velocity v of the mass is
v =√
2gL(1− cos θ0),
where g is a gravitational constant. If the velocity v is measured as√gL, find the initial angle θ0.
θ
10 m 8 m
L
m
v
θ0
3. (§ 5.7) The propagating speed of water waves with length L moving across a body of water with depthd is given as
v =
√gL
2πtanh
(2πd
L
),
where g is a gravitational acceleration.
θ
10 m 8 m
L
m
v
θ0
L
d
(a) If the water depth d is very shallow relative to the wave length L (i.e., d/L 1), approximate thewave speed v by linearly approximating tanh(2πd/L) for small d/L.
(b) If the water depth d is very deep relative to the wave length L (i.e., d/L 1), approximate v bytaking the limit d/L→∞.
1
4. (§ 5.7) Using principles of physics it can be shown that when a long cable hung between two poles, ittakes a shape of a curve y = f(x) that satisfies the differential equation
d2y
dx2=ρg
T
√1 +
(dy
dx
)2
,
where ρ, g, T are the linear density of the cable, the gravitational constant and the cable tension,respectively. By direct substitution verify that the function
y = f(x) =T
ρgcosh
(ρgx
T
)is a solution to this differential equation (same as Problem 47).
5. (§ 5.8) Current across an resonant electrical circuit, having a capacitance, an inductance and an alter-nating electric power, is given as a function of time t as
I(t) =
[2A
ω20 − ω2
sin(ω0 − ω)t
2
]sin
(ω0 + ω)t
2,
where A is some constant, ω0 is a characteristic frequency of the circuit and ω is a frequency of inputvoltage.
(a) Show that I(t) is bounded, that is there are constants m and M such that m ≤ I(t) ≤M for every t.
(b) Let I0 be the current if the input frequency ω approaches to the characteristic frequency ω0. Findan expression for I0 at arbitrary time t, that is find I0(t) = lim
ω→ω0
I(t).
(c) Show that I0(t) is unbounded. That is, given a constant M > 0, find a time T such that I0(T ) ≥M .
6. (§ 5.8) A metal cable has radius r and is covered by insulation, so that the distance from the center ofthe cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable isgiven by
v = −c( rR
)2ln( rR
),
where c is a positive constant. Find the limits a) limR→r+
v and b) limr→0+
v. (same as Problem 48)
7. (§ 5.8) If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment Pper unit volume is given by
P (E) =eE + e−E
eE − e−E− 1
E.
Show that limE→0+
P (E) = 0. (same as Problem 47)
2
Chapter 6 Techniques of Integration
1. (§ 6.1) A rocket accelerates by burning ins onboard fuel, so its mass decreases with time. Suppose thetotal mass of the rocket at liftoff (including fuel) is m, the fuel is consumed at rate r, and the exhaustgases are discharged with constant velocity vg relative to the rocket. Then the velocity of the rocket isgiven by
v(t) = −gt− vg lnm− rtm
,
where g is the gravitational constant. The altitude of the rocket after an arbitrary time is calculated byintegrating v with respect to time, as given by
h(t) =
∫ t
0v(x) dx.
Find h(t). (same as Problem 42)
2. (§ 6.1) A drug response curve describes the level of medication in the bloodstream after a drug is admin-istered. A surge function S(t) = Atpe−kt is often used to model the response curve, reflecting an initialincrease (surge) in the drug level and then a more gradual decrease. If, for a particular drug, A = 0.1,p = 1, k = 0.1, and t is measured in minutes, calculate the average value of the surge function duringthe first ten minutes, that is, you calculate the integral
1
10
∫ 10
0S(t) dt.
3. (§ 6.2) Household electricity is supplied in the form of alternating current that varies from 155 V to −155V with a frequency of 60 cycles per second. The voltage is thus given by
E(t) = 155 sin(120πt),
where t is the time in seconds. Voltmeters read the RMS (Root-Mean-Square) voltage, which is thesquare root of the average value of E2(t) over one cycle (1/60 second), as given by
RMS =
√1
1/60
∫ 1/60
0E2(t) dt.
(a) Calculate the RMS voltage of household current.
(b) Many electric stove require an RMS voltage of 220 V. Find the corresponding amplitude A neededfor the voltage E(t) = A sin(120πt).
(same as Problem 66)
3
4. (§ 6.2) A charged rod of length L produces an electric field at point P (a, b) as given by
E(P ) =
∫ L−a
−a
λb
4πε0(x2 + b2)3/2dx,
where λ is the charge density per unit length on the rod and ε0 is the free space permittivity (see figure).Evaluate the integral to determine an expression for the electric field E(P ). (same as Problem 70)
θ
10 m 8 m
L
m
v
θ0
L
d
L 0
P(a, b)
x
y
5. (§ 6.3) A virus spreads to a population of N people from a single infected person. A simple mathematicalmodel describing the spreading assumes that the virus spreads out at a rate that is proportional to theproduct of the total infected population, say y, and the uninfected population (N − y), as given by theequation
dy
dt= ky(N − y),
where t represents time and k is a constant. Dividing the equation by y(N−y) and integrate with respectto t, ∫
1
y(N − y)
dy
dtdt =
∫k dt.
This gives that ∫dy
y(N − y)= kt+ C,
where C is an arbitrary constant. Evaluate the integral and find the infected population y as a functionof time t. Determine the arbitrary constant assuming initially infected population is one, i.e., y(0) = 1.
6. (§ 6.3) A calculator company has set up a production line to manufacture a new calculator. An industrialengineer mathematically modeled the time rate of production of these calculator after t weeks as
dx
dt= 5000
(1− 24
t2 + 10t+ 24
)calculators/week.
(Notice that production approaches to 5000 per week as time goes on, but the initial production is lowerbecause of the workers’ unfamiliarity with new techniques.) Find the number of calculators producedfrom the beginning of the third week to the end of the fourth week.
7. (§ 6.6) According to statistical mechanics, the average speed of molecules in an ideal gas is given by
v =4√π
(M
2RT
) 32∫ ∞0
v3e−Mv2/(2RT ) dv,
where M is the molecular weight of the gas, R is the gas constant, T is the gas temperature, and v isthe molecular speed. By evaluating the improper integral, show that
v =
√8RT
πM.
To make algebra simpler, it is suggested to let M/(2RT ) ≡ a and carry out the integral. (same asProblem 54)
4
8. (§ 6.6) The method of Laplace Transforms is often used to solve differential equations. You will learn thispowerful mathematical technique later you take Math 245. The Laplace Transform of a function f(t) isa function F (s) defined by the following
F (s) =
∫ ∞0
f(t)e−st dt,
where s is a constant parameter that is to be chosen so that the improper integral exists. Find theLaplace transform of the following functions
(a) f(t) = c, where c is a constant. Assume s > 0.
(b) f(t) = eat, where a is a constant. Assume s > a.
(c) f(t) = t. Assume s > 0.
5
Chapter 7 Applications of Integration
1. (§ 7.1) NACA (formerly NASA) standardized wing section profiles mathematically. The y-coordinates ofthe upper camber (yu) and the lower camber (yl) of NACA2412 airfoil with unit cord length (0 ≤ x ≤ 1)are defined as functions of the coordinate x (see figure below)
θ
10 m 8 m
L
m
v
θ0
L
d
L 0
P(a, b)
x
y yu(x)
yl(x)
ym(x)
x=0 x=1
100 m
100 m
yu(x) = ym(x) + θ(x),
yl(x) = ym(x)− θ(x),
where ym(x) is the y-coordinate of the mean camber and θ(x) is a half-thickness at the coordinate x, asprescribed by
ym(x) =2
9(−7 + 8x− x2),
θ(x) = 0.9√x− 0.08x− 0.2x2 + 0.2x3 − 0.06x4.
Find an area of the airfoil section.
2. (§ 7.2) A doctor needs to know the volume of a patient’s kidney in order to determine amount of dose ofa medicine being prescribed to the patient. A MRI scan produces equally spaced cross-sectional viewsof a human organ that provide information about the organ. Suppose that a MRI scan of a patient’skidney show cross-sectional sections spaced 1 cm apart. The kidney is 10 cm long and the cross-sectionalareas are 0, 13, 23, 30, 35, 36, 37, 34, 26, 25, 0 square meters. Use the Trapezoidal rule to estimate thevolume of the kidney.
3. (§ 7.2) A reservoir has a plane shape of an isosceles triangle with a length of 100 m and a width of 100 mas shown. The vertical cross section has a shape of a trapezoid whose bottom side and depth are both ahalf the length of the top side. Find the volume capacity of the reservoir.
θ
10 m 8 m
L
m
v
θ0
L
d
L 0
P(a, b)
x
y yt(x)
yb(x)
θ(x)
x=0 x=1
100 m
100 m
4. (§ 7.2) A fluid tank in the chemical plant, has the shape of a circular cylinder with the radius R and thelength L, is mounted so that its axis is parallel to the ground. To design a fluid quantity gauge of thetank, an engineer needs to establish a map of the volume of the fluid relative to the fluid level. Find thefluid volume V as a function of the fluid level h.
6
5. (§ 7.3) The external fuel tank for an military airplane has the shape obtained by rotating y = 0.3√x for
0 ≤ x ≤ 1 and y = 0.3 for 1 ≤ x ≤ 3 and y = 0.9 − x/5 for 3 ≤ x ≤ 4.5 about x-axis as shown below.Ignoring volume of the structure, find the fuel capacity of the tank. Use the shell method.
θ
10 m 8 m
L
m
v
θ0
L
d
L 0
P(a, b)
x
y yu(x)
yl(x)
ym(x)
x=0 x=1
100 m
100 m
1 3 4.5
x
y 𝑦𝑦 = 0.3 𝑥𝑥 𝑦𝑦 = 0.3 𝑦𝑦 = 0.9 −𝑥𝑥5
6. (§ 7.4) A custom beam is made by welding together two identical shells which are made from sheet metalsas shown. The sheet metal of width w is formed into a parabolic section with a height of a and a widthof 2b as prescribed by the function y = ax2/b2. Find the width of the sheet metal in terms of a and b.
b
a
w
2b
2a
welded
x
y
7. (§ 7.5) A group of engineers is building a parabolic satellite dish whose shape will be formed by rotatingthe curve y = ax2 about y-axis. If the dish is to have a 10-ft diameter and a maximum depth of 2 ft,find the value a and the surface area of the dish. (same as Problem 20)
8. (§ 7.6) When studying the formation of mountain ranges, geologists estimate the amount of energy (work)required to lift a mountain from sea level. Consider a mountain having a shape of a solid cone with thebase radius R and the height H. Assuming the sea level is at the base surface of the mountain andall the material is lifted from the sea level, find an expression of the total work required in forming themountain. Assume the mass density of the mountain is uniform, denoted as ρ, and use the gravitationalconstant denoted as g.
7
9. (§ 7.6) A satellite with the mass m is launched from the surface of the earth with the mass M and theradius R. Newton’s law of Gravitation states that the satellite and the earth attract each other with aforce
F = GMm
x2,
where x is the distance between the earth center and the satellite and G is the gravitational constant(mass of the launch vehicle is ignored).
b
a
w
2b
2a
welded
x
y
R M
O m
v0
r
(a) Find the work needed to propel the satellite from the earth surface x = R to the arbitrary positionx = r.
(b) Use an improper integral to find the work needed to propel the satellite out of the earth’s gravita-tional field.
(c) By using the fact that the initial kinetic energy of the satellite is 12mv
20, find an expression of the
escape velocity v0 that is needed to propel the rocket out of the gravitational field. Calculate v0, ifthe mass of the earth is 6× 1024 kg, the gravitational constant is G = 6.7× 10−11 Nm2/kg2 and theearth radius is R = 6.4× 106 m.
(similar to Problems 21, 22, 23)
10. (§ 7.6) A vertical dam has a shape of isosceles trapezoid with the length 2b on the top side and the lengthb on the bottom side and the hight h. Suppose water is filled to the top. Denote the water density as ρand the gravitational constant as g.
b
a
w
2b
2a
welded
x
y
R M
O m
v0
r
x l
wa
wg
R M
O m v=v0
at t=0
r (maximum distance)
v=0
F v
x(t)
2b
b
h F
Center of Pressure
yc
O
y
(a) Find the hydrostatic force on the dam.
(b) The center of pressure is a point on which the all hydrostatic force virtually acts. Since the damis symmetric, horizontal location of the center of pressure is on the center line. Vertical position ofthe center of pressure measured from the top of the dam can be calculated as the moment aboutthe top surface divided by the hydrostatic force, namely,
yc =(Moment about the top side)
(Total Hydrostatic force)=
1
FlimN→∞
N∑i=1
yi ∆Fi =1
FlimN→∞
N∑i=1
yi (ρgwi∆y) =
=1
F
∫y ρgw(y)dy,
where y is a distance from the top, yc is a vertical position of the center of pressure and F is a totalhydrostatic force on the dam and w(y) is a width of the dam measured at y. Find yc.
8
11. (§ 7.6) Resultant force acting on an airplane wing is a difference between aerodynamic force (lift) andweight of the wing as shown. Suppose the distribution of the aerodynamic force wa is given by
wa = w0
√1−
(xl
)2(per unit wing span)
and the weight distribution wg is given by
wg =1
3w0
(1− x
l
)(per unit wing span),
where l is the length of the wing span, x is a spanwise distance from the wind root and w0 is a constant.Find (a) the total resultant force F acting on the wing and (b) resultant moment M at the wing root,that is, an integral of force times distance from the wing root (M =
∫xf(x) dx, where f(x) is a force at
measured at x), and (c) center of force xc which is the resultant moment divided by the resultant force,namely, xc = M/F .
b
a
w
2b
2a
welded
x
y
R M
O m
v0
r
2b
b
h F
Center of Pressure
yc
x l
wa
wg
12. (§ 7.7) We will look at the escape velocity of a satellite in Problem 8 from dynamical point of view.Consider the satellite of mass m is projected with the initial velocity v0 at time t = 0 and x = R fromthe earth surface as shown. The satellite feels the gravitational drag force F and, therefore, according
b
a
w
2b
2a
welded
x
y
R M
O m
v0
r
2b
b
h F
Center of Pressure
yc
x l
wa
wg
R M
O m v=v0
at t=0
r (maximum distance)
v=0
F v
x(t)
to Newton’s 2nd law we have an equation of motion as
mdv
dt= −GMm
x2,
where v is a velocity of the satellite at an arbitrary time, dv/dt is an acceleration, x = x(t) is a distancefrom the center of the earth, G is the gravitational constant and M is the mass of the earth. Noting thefact that the velocity v depends on the position x and the position x depends on time t, i.e., v = v(x(t)),from the Chain rule we have the relation
dv
dt=
dv
dx
dx
dt=
dv
dxv,
where we used the fact that v = dx/dt. Substituting this relation into the equation of motion, we havethat
vdv
dx= −GM
x2.
(a) Solve the differential equation for v as a function of x. Note that the satellite is at x = R initially.
(b) When the satellite reaches to the maximum distance x = r, the velocity vanishes to zero. Find theinitial velocity v0 such that the satellite can achieve the maximum distance r. The escape velocityof the satellite can be obtained by taking the limit r → ∞, namely, limr→∞ v0. Find the escapevelocity as well. You should get the same result as in Problem 8.
9
Chapter 8 Series
1. (§ 8.2) After injection of a dose D of insulin, the concentration of insulin in a patient’s system decaysexponentially in time and it can be modeled as De−at, where t represents time in hours and a is apositive constant. Suppose a dose D is injected every T hours as shown. Then referring at the figure,
D
T 2T 3T O
s1
s2
s3 D D
D
t
Concentration of Insulin
the concentration just before the second dosage, denoted as s1, is given by
s1 = De−aT (just before the 2nd injection).
When the second insulin is injected, the concentration of the insulin increases byD to s1+D = De−aT+D,and then after T hours the concentration decays by a factor of e−aT to s2 = (s1 +D)e−aT = (De−aT +D)e−aT which is written as
s2 = D(e−aT + e−2aT ) (just before the 3rd injection).
Continuing for the third injection, the insulin increase to s2 +D = D(e−aT + e−2aT ) +D and it decaysafter T hours to s3 by a factor e−aT as given by
s3 = D(e−aT + e−2aT + e−3aT ) (just before the 4th injection).
Here you notice that an extra term (De−anT ) is added after each injection.
(a) Find an expression of the concentration sn after n-th injection (just before n + 1-th injection). Ifthe insulin is injected continually, find the asymptotic concentration, namely limn→∞ sn.
(b) If the concentration of insulin must always remain at or above a critical value C, determine aminimal dosage D in terms of C, a and T .
(Ref. Problem 42)
2. (§ 8.6) The force due to gravity on an object with mass m at the surface of the earth with radius R isexpressed by Newton’s gravitational law as GMm/R2. This force is also equal to the experimental lawof gravity mg:
mg =GMm
R2.
Now suppose the object is at the height h from the earth surface. Then the gravitational force acting onthe object is given by
F =GMm
(R+ h)2=GMm
R2
R2
(R+ h)2= mg
R2
(R+ h)2.
(a) Express F as a series in powers of h/R.
(b) Observe that if we approximate F by only the first term in the series, we get the expression F ≈ mgthat is usually used when h is much smaller than R. Use the Alternating Series Error EstimationTheorem, estimate the range of values of h for which the approximation F ≈ mg is accurate withinone percent. (Use R = 6400 km.)
10
3. (§ 8.6) The Bessel function of order zero is defined by the power series
J0(x) =∞∑n=0
(−1)nx2n
22n(n!)2.
The Bessel functions are known as the solutions of the Bessel’s differential equation, and there arenumerous applications in physics and engineering, such as propagation of electromagnetic waves, heatconduction, vibrations of a membrane, quantum mechanical waves (and many more!), that are all set upin a cylindrical domain. You will learn this function (or hear at least) later year.
In[1]:= T2[x_] = 1 - x^2 2^2;
In[2]:= T4[x_] = T2[x] + x^4 2^4 Factorial[2]^2;
In[3]:= T6[x_] = T4[x] - x^6 2^6 Factorial[3]^2;
In[4]:= T8[x_] = T6[x] + x^8 2^8 Factorial[4]^2;
In[28]:= Plot[BesselJ[0, x], T2[x], T4[x], T6[x], T8[x],
x, 0, 5, PlotLegends → "J0", "T2", "T4", "T6", "T8",
PlotLabel → "J0(x) and Truncated Polynomials",
PlotRange → 0, 5, -0.5, 1.1, AxesLabel → "x"]
Out[28]=
1 2 3 4 5x
-0.5
0.5
1.0
J0(x) and Truncated Polynomials
J0
T2
T4
T6
T8
In[29]:= Plot[BesselJ[0, x], x, 0, 15,
AxesLabel → "x", "J0(x)", PlotLabel → "J0(x) Bessel Function"]
Out[29]=
2 4 6 8 10 12 14x
-0.4
-0.2
0.2
0.4
0.6
0.8
1.0
J0(x)J0(x) Bessel Function
In[1]:= T2[x_] = 1 - x^2 2^2;
In[2]:= T4[x_] = T2[x] + x^4 2^4 Factorial[2]^2;
In[3]:= T6[x_] = T4[x] - x^6 2^6 Factorial[3]^2;
In[4]:= T8[x_] = T6[x] + x^8 2^8 Factorial[4]^2;
In[28]:= Plot[BesselJ[0, x], T2[x], T4[x], T6[x], T8[x],
x, 0, 5, PlotLegends → "J0", "T2", "T4", "T6", "T8",
PlotLabel → "J0(x) and Truncated Polynomials",
PlotRange → 0, 5, -0.5, 1.1, AxesLabel → "x"]
Out[28]=
1 2 3 4 5x
-0.5
0.5
1.0
J0(x) and Truncated Polynomials
J0
T2
T4
T6
T8
In[29]:= Plot[BesselJ[0, x], x, 0, 15,
AxesLabel → "x", "J0(x)", PlotLabel → "J0(x) Bessel Function"]
Out[29]=
2 4 6 8 10 12 14x
-0.4
-0.2
0.2
0.4
0.6
0.8
1.0
J0(x)J0(x) Bessel Function
(a) Write the first four terms of J0(x).
(b) Show that J0(x) has a local maximum at x = 0.
(c) Approximate J0(1) and J0(2) within an error of 0.001.
(d) If
∫ 1
0J0(x)dx is approximated by the first three terms of the power series of J0(x), estimate the
accuracy of the approximation.
4. (§ 8.7) The electrostatic potential of an electric dipole, formed by two point charges, +q and−q Coulombs,
D
T 2T 3T O
s1
s2
s3 D D
D
t
Concentration of Insulin
π 2π 3π
1
–1
f(t)
t O
d d
y
x
r q
–q O x
y
separated by the distance 2d in y-direction, is given by
F (x, y) =1
4πε0
(q√
x2 + (y − d)2+
−q√x2 + (y + d)2
),
where ε0 is the permittivity in vacuum.
Assuming y d and using the relation x2 + y2 = r2, write the dipole potential F as a power series.Write down the first two terms. Then write the leading order approximation of the dipole potential, thatis, you truncate the second term and after in the series representation.
11
5. (§ 8.7) The Fourier series is used to represent a periodic function f(t) in terms of a sum of trigonometricfunctions. The Fourier series is frequently used in Engineering, and you will definitely learn that lateryear. Suppose the electronic signal has a form of piecewise periodic function with period 2π, as given by
f(t) =
1, 0 < t < π−1 π ≤ t < 2π,
where t represents time. This signal can be represented as a Fourier series as given by
D
T 2T 3T O
s1
s2
s3 D D
D
t
Concentration of Insulin
π 2π 3π
1
–1
f(t)
t O
f(t) =4
π
(sin t+
1
3sin 3t+
1
5sin 5t+
1
7sin 7t+ · · ·
).
By using the power series representation of tan−1 x, check that when t = π/2 this representation givesf(π/2) = 1 and that when t = 3π/2 this representation gives f(3π/2) = −1.
6. (§ 8.8) The resistivity ρ of a conducting wire is the reciprocal of the conductivity and is measured inunits of ohm-meters (Ω-m). The resistivity of a given metal depends on the temperature according tothe equation
ρ(t) = ρ20eα(t−20),
where t is the temperature in C and ρ20 and α are constants specific to the metal. Usually α is a smallvalue and, therefore, the resistivity does not change so much relative to change of temperature. Forthis reason it is common to approximate ρ(t) by its first (linear) or second (quadratic) degree Taylorpolynomial at t = 20.
(a) Find expressions for the linear and quadratic approximations.
(b) For copper α = 0.004 C−1 and ρ20 = 1.7 × 10−8 Ω-m. Find accuracy of the linear and quadraticapproximations of ρ of a copper wire for −20 C ≤ t ≤ 270 C?
a = 0.004
0.004
PlotExpa * t - 20, 1 + a * t - 20, 1 + a * t - 20 + a^2 2 * t - 20^2,
t, -200, 220, AxesLabel → "t", "ρ/ρ20",
PlotLegends → " ρ (t) ", "T1(t)", "T2(t)"
-200 -100 100 200t
0.5
1.0
1.5
2.0
ρ/ρ20
ρ(t)
T1(t)
T2(t)
12
7. (§ 8.8) A cantilever beam of length L is rigidly supported at one end (x = 0), and the vertical force P isapplied at another end (x = L) that bends down the beam. According to solid mechanics, the bendingmoment at an arbitrary section of the beam, denoted as M(x), is related to the vertical displacementw(x) of the beam through the differential equation
M
EI= − w′′
[1 + (w′)2]32
,
where E is a modulus of elasticity of the beam material and I is a polar moment of inertia of the beamcross section, and they are assumed to be constant.
D
T 2T 3T O
s1
s2
s3 D D
D
t
Concentration of Insulin
π 2π 3π
1
–1
f(t)
t O
d d
y
x
r q
–q O x
y P
w x O
L
M
(a) Assuming the slope of the bend w′(x) is very small, |w′| 1, write M/EI as a power series in termsof w′. Write down the first three terms. By truncating the second term and after in the series, showthat M/EI is approximated as a well-known relation
M
EI≈ −d2w
dx2.
This relation is practically used in engineering when calculating the displacement shape.
(b) If the displacement is given by
w(x) =P
6EIx2(3L− x),
estimate accuracy of the approximation (M/EI ≈ −w′′) relative to w′′, that is, estimate upper errorbound of M/EIw′′ in terms of P , L, E and I.
(c) Using the result of (b), express the error estimate in term of wmax/L where wmax is the maximumdisplacement that occurs at x = L, namely,
wmax = w(L) =PL3
3EIor
wmax
L=PL2
3EI.
Then find the maximum relative displacement wmax/L such that the approximation M/EI ≈ −w′′is accurate within one-percent.
13
Related Documents
