Phys 326 Sample Final Exam II SOLUTION 8-11 am 18 December 2012 144 Loomis
Closed book, no calculators, but you are permitted one two-sided 8 1/2 x 11" set of notes.
1] The three degree of freedom system
Has M and K matrices
0
1 0 0
2 1
0 1 0 ; [K ] = 1
Phys 326 HW 8A
due 30 October 2012
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name
1. ( 40 points) Use Runge-Kutta (use a Runge-Kutta step size t near 0.2 ) to study the
driven damped pendulum
v = 0.10v sin x + cos( t)cos(x)
x=v
( This is different from the DPP discussed in class, as the force i
Phys 326 HW 6B due October 16
2012
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.71 .612 .355
1. 20 points A certain rotation matrix is [Q] = .71 .612 .355
0 .500 .866
a) Show that it is orthogonal
b) Recalling that the axis of rotation must obey [Q]cfw_v = cfw_v, find the direction (i.e
Phys 326 HW 6A
due 16 October 2012
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name
1. 10 points A (axisymmetric) football ( Ia = 0.015 kg m2 It= 0.025 kg m2 ) is thrown
with a nominal spin rate of 3 radians per second ) If not thrown perfectly, it may
wobble. With what frequency can it wobble?
2
Phys 326 HW 5B
due 9 October 2012
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name
1. 20 points A thin square a x a solid object of uniform density is skewered on an axis
perpendicularly through point O which is a distance a/4 x a/4 away from the square's
center at C. The axis is supported at poi
Hw 3B Phys 326 due September 18, 2012
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name
HW 3B.1] This 2dof system of carts and springs has, using coordinates x and y
representing displacement of the carts, the following Lagrangian L, and M and K matrices,
and modes and natural frequencies:
L = 5 x
Phys 326 HW 5A due October 9 2012
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name
1. 20 points A spool rolls without slipping on its inner radius r (point P) and is pulled
by a rope wrapped around its outer radius R (at point O). The rope is pulled by a force F.
The spool has mass m and moment o
HW 4B Phys 326 Due 27 September 2012
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name
1. 20 points This 3dof semidefinite system is initially at rest, and is then excited by a unit impulse
applied to the first cart. What is the subsequent motion of the third cart z(t) ?
The M and K matrices are
1
HW 4A Phys 326 due Thursday September 27!, 2012 _
name
1. ( 20 points) Consider a particle of mass
m on a platform rotating with a constant
speed = k. (This is taken from
HW10B1 in Phys 325) The particle is
described by its Cartesian coordinates x(t)
and
Hw 3A Phys 326 due September 18, 2012
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name
(you may need a calculator for these problems)
Hw3A.1 A certain system has
2 1
4 0
[M ] =
[K ] =
;
1 2
0 8
It has (un-normalized) modes
6.31
1.69
cfw_u (1) =
cfw_u (2) =
;
8.62
0.62
Show that th
Phys 326 HW 2B
Due 11 am Tuesday September 11 2012
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name
1] Find the matrices [M] and [K], and the mode shapes and natural
frequencies for small motions of the pictured double pendulum. Without
loss of generality, assume g = l = m = 1 and obtain the natu
Phys 326
HW 2A
due 11am 11 September 2012
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name
1. Find the three natural frequencies and mode shapes cfw_u for the symmetric 3dof
system of identical masses and springs
( You will get a cubic, but it is not hard to solve )
Draw a picture of the relative
Phys 326 Discussion 7, Thursday Oct 11, 2012
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A top that is not spinning will fall over after being given an infinitesimal perturbation away from
being fully upright, because that is an unstable equilibrium. If it spins fast, though, we know that
it
Phys 326 HW 8B
due 30 October 2012
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name
1) (20 points) An infinitely long string has tension T and mass density . It is supported by a
stretchy membrane with stiffness per length . Its transverse displacement is (x,t). Its
Lagrangian density is therefor
Phys 326 HW 7A
due 23 October 2012
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name
1-d and 2-d maps
1. (10 points) Consider the Tent map
x j +1 = x j if x j < 0.5
= [1 x j ] if x j 0.5
Consider two cases for the parameter , = 0.8 and = 1.4 For each case, plot the
dynamics for 20 generations star
Phys 326 Sample Final Exam II 8-11 am 18 December 2012 144 Loomis
Closed book, no calculators, but you are permitted one two-sided 8 1/2 x 11" set of notes.
1] The three degree of freedom system
Has M and K matrices
0
1 0 0
2 1
0 1 0 ; [K ] = 1 3 2
[M
Phys 326 Sample Midterm I 2 October 2012, 11:00 am 12:20 pm
7 short questions to be done in 80 minutes. 360 total points, 50 or 60 points per question, closed book
except for one 8 1/2 x 11" notes sheet of your devising. Calculators are not permitted, but
Phys 326 Midterm Exam II
6 November 2012 11:00-12:20
1. The 1-d map xn+1 = 3 sin xn has a fixed point at x = = 2.2789
Show that this fixed point is unstable. ( you may want a calculator for this)
2. A yo-yo (inner radius r, outer radius R,
mass M, moment
Phys 326 HW13B Due Tuesday Dec 11, 2012
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name
Poiseuille Flow in an annular pipe (30 points)
Consider steady uni-directional axi-symmetric laminar flow in a cylindrical pipe with an
inner radius R1 and an outer radius R2 . Pressure is fixed at the two en
Phys 326 HW11A Due Tuesday Nov 27, 2012
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1. a) (10 points) Show that the outgoing circular wave
(r,t) = f (r ct) / r
does NOT satisfy the 2-d wave equation.
1 1 2
(x, y,t) = c (x, y,t) = c [
r +
] (r, ,t)
r r r r 2 2
2
2
2
(But that the discrepanc
Phys 326 HW13A Due Tuesday Dec 11, 2012
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name
Free surface laminar flow.
(30 points) Honey runs down an inclined plane that is tilted an angle from the horizontal.
Gravity pulls the fluid down, viscosity resists the flow. The sheet of honey has thickness
Phys 326 HW12A Due Tuesday Dec 4, 2012
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name
1. (10 points) A stress tensor varies in space over some volume as described here:
xz z 2 0
[ ] = z 2
0 y
0 y 3
Assuming the material is in static equilibrium, what must the body force density f be?
f = f x
Phys 326 HW 9B
due 13 November 2012
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name
1. (15 points) A uniform bar of length L , bending stiffness B and lineal density is pinned
( = " = 0) at each end and has compressive force C in it. Find its modes and natural
frequencies. At what value of C doe
Phys 326 HW11B Due Tuesday Nov 27, 2012
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name
1. (20 points) A certain stress tensor is
0 1
[] = 1 0
0 0
0
0
3
a) Find its principle stresses and their directions.
compressive and which tensile?
Which of these directions are
b) Assuming the material is
Phys 326 HW12B Due Tuesday Dec 4, 2012
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name
1. Taylor 16.36 (20 points) The speed of sound in air is ( B/ )1/2 , but what is B? Newton,
who used the isothermal bulk modulus of air, got this wrong. Air expands and contracts when
an acoustic wave goes by,
Phys 326 HW 9A
due 13 November 2012
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name
1(20 points) a. What is the Lagrangian for a string of uniform density and tension T
fixed at x = 0, and free at x = L but with a point mass m attached there?
b. Take the variation of the action for this Lagrangi
Phys 326 HW 7B due October 23
2012
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name
Numerical solutions of differential equations
1. ( 25 points) Use Runge-Kutta and a computer to solve the Van-der Pol system
+ (x 2 1) x + x = 0
x
for the case = 0.3
Ie, solve the coupled 1st order equations
v =
Phys 326 HW 1B
Due 11 am Tuesday September 4 2012
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1. A symmetric two degree of freedom system consists of two identical rigid masses m.
The two outside springs k are identical; the interior spring has a tiny stiffness k, with
< 1.
Find the coupled
Phys 326 Lecture 15
Oct 18, 2012
Nonlinear Differential equations & chaos continued
Here is a more complicated system, also 2-d so with no chaos:
The Van-der-pol oscillator. If = 0, it is a linear single-degree of freedom system Simple
Harmonic Oscillator