Homework 1 (due Feb 7) Introduction to Math. Modeling
Version: January 29, 2018
Linear Algebra
(1) Consider the following vectors in R3:
(a) Show that fv1;v2;v3g are linearly independent. (This implies that fv1;v2;v3g is a
basis of R3.)
(b) Express the vector [1;1;1] as a linear combination of the vectors vi.
(2) Find a vector v2R3 with the property:(3) The equation
3x 6y +z = 0 (1)
has in nitely many solutions b = [x;y;z]. The set of all solutions S is a 2-dimensional
vector space.
(a) Find two linearly independent solutions fb1;b2g of S. You need to demonstrate
their linear independence.
(b) Your choicefb1;b2gfrom (3)a forms a basis of S. Prove that any linear combination
of those two vectors yields a solution of (1).
(c) Show that any solution of (1) can be written as a linear combination of fb1;b2g.
Dimensional Analysis
(1) The equation for an elastic beam is
EI@
4u
@x4 +
@2u
@t2 = 0;
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Homework 1 (due Feb 7) Introduction to Math. Modeling
Figure 1: Shock wave produced by a nuclear explosion, at 6 ms, 16 ms, 25 ms and 90 ms. The
width of the white bar in each gure is 100 m. (ms = millisecond)
where u(t;x) is the transverse displacement, x2 [0;L] is the distance along the beam,
and L is the length of the beam. The boundary conditions are
u(t;0) = u0 sin(!t); @u@x(t;0) = 0
u(t;L) = @u@x(t;L) = 0:
The initial conditions are
u(0;x) = @u@t(0;x) = 0:
Here E is the elastic modulus, I is the moment of inertia, and is the mass per unit
length ([ ] = kg=m) of the beam. Nondimensionalize the problem in such a way that the
resulting boundary conditions contain no nondimensional parameters.
(2) Consider the dimensional problem for the motion of a projectile launched from close to
the surface of the Earth:
d2y
dt2 =
GME
(RE +y)2; y(0) = 2 m; y
0(0) = v0 m=s:
Assume the Earth to be spherical with a uniform. density E, so ME = 43 R3E E. Here,
y(t) is the height of the projectile t seconds after the launch, ME is the Earth’s mass, RE
is the Earth’s radius, v0 is the initial velocity, and G is Newton’s gravitational constant.
Let y(t) = Ly (t ); t = Tt , with a char. length L and a char. time T. Consider the
following cases:
(i) The fast projectile limit: RE; E xed, v0!1.
(ii) The dense Earth limit: RE;v0 xed, E !1.
(iii) The light Earth limit: RE;v0 xed, E !0.
(iv) The small Earth limit: v0;ME xed, RE !0.
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Homework 1 (due Feb 7) Introduction to Math. Modeling
For each case:
(a) Choose characteristic scalings L;T to normalize as many terms as possible.
(b) Choose the scalings so that the time it takes for the projectile to fall to the ground
should be nite for the given limit. Namely, the speed, acceleration and initial
height should not diverge, nor should falling take in nitely long in time, or happen
instantaneously.
(c) Write the scaled problem and identify all remaining dimensionless parameters.
(d) Identify the limiting small parameter, and for each case, write the problem when the
parameter is set to zero.
(3) The frequency ! of waves on a deep ocean is found to depend on the wavelength of the
wave, the surface tension of the water, the density of the water, and the gravitational
acceleration g.
(a) Use the Buckingham Pi Theorem to determine the functional dependence of ! on
; ; and g.
(b) In uid dynamics it is shown that
! =
p
gk + k3=
where k = 2 = is the wave number. How does this di er from your result?
(4) Consider a frictionless pendulum of mass m, hanging on a rod of length l, and released
from an initial angle (angles are dimensionless, given in radians). Its motion depends
on the grav. acceleration g = 9:81 m=s2. Use the Buckingham Pi Theorem to derive an
expression for the period t of the pendulum in terms of a constant (with units time) and
a function of a non-dimensional parameter.
(5) In a high energy explosion there is a very rapid release of energy E that produces an
approximately spherical shock wave that expands in time.
(a) Assuming that the radius R of the shock wave depends on E, the time t since the
explosion, and the density of the air. Use dimensional analysis to derive how R
depends on these quantities.
(b) It was shown that if E = 1 J and = 1 kg=m3, then R = t2=5 m=s2=5. Use this
information to obtain an exact formula for R.
(c) Assuming an air density = 1 kg=m3, use the photographs from Figure 1 to estimate
the energy released.