Download PHYS4330 Theoretical Mechanics Fall 2011 Final Exam

Name:
PHYS4330
Theoretical Mechanics
Final Exam
Fall 2011
Wednesday, 14 Dec 2011
This exam has two parts. Part I has 20 multiple choice questions, worth two points each.
Part II consists of six relatively short problems, worth ten points each. The short problems
can be worked out on the front page of the sheet provided, but use the back if you need
more room. In any case please be neat!
Also, two extra pages are provided at the back. Use these if you need to, but be sure to
indicate which problem you are working on these pages.
You may use your textbook, course notes, or any other reference you may have other than
another human. You are welcome to use your calculator or computer, although the test is
designed so that these are not necessary.
Good luck!
Part I (Total):
Part II
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Problem 5:
Problem 6:
Total:
Part I: Multiple Choice Questions (Two points each)
1. A circular hoop of mass M and radius R rotates about an axis perpendicular to
the plane of the disk and passing through a point on its circumference. The rotational
inertia of the disk about this axis is
A. M R2 /2
B. M R2
C. 3M R2 /2
D. 2M R2
E. 5M R2 /2
2. Water flows at a constant rate of 400 gal/min into one end of a long, horizontal
pipe with circular cross section. The radius at the input of the pipe is 20 cm, and the
radius at the output is 10cm. At what rate does water emerge from the output?
A. 100 gal/min
B. 200 gal/min
C. 400 gal/min
D. 800 gal/min
E. 1600 gal/min
3. A bullet is fired horizontally in the direction of due North at the Equator. The
Coriolis force
A. deflects the bullet upward.
B. deflects the bullet downward.
C. deflects the bullet to the East.
D. deflects the bullet to the West.
E. does not affect the path of the bullet.
4. Which of the following is not a characteristic of a system acting “chaotically”?
A. Lack of any periodic response when driven periodically
B. State space orbits that cover the entire available phase space
C. Bifurcation into eight distinct periods when driven at one period
D. Exponential divergence of solutions with different initial conditions
E. Poincare sections that continuously cover bounded regions of state space
5. In the figure above, the pendulum swings in the same plane through which the
cart moves. How many degrees of freedom are represented by this system?
A. 0
B. 1
C. 2
D. 3
E. 4
6. An object moves in two dimensions with coordinates x and y, which obey the
coupled differential equations dx/dt = −αy and dy/dt = αx, where α is a constant.
At time t = 0, the position of the object is (x, y) = (A, 0). The path followed by the
object is best described as
A. oscillating along the x axis
B. oscillating along the y axis
C. oscillating along a straight line at 45 degrees relative to the x and y axes
D. a circle
E. an ellipse with eccentricity greater than zero
7. A student with mass M stands on a horizontal platform that rotates at a constant
rate, making one complete revolution in time T. If she stands a distance R from the
center of rotation, what is the magnitude of the force she feels that drives to drive her
radially outward?
A. 0
B. 4π 2 M T 2 /R
C. 4π 2 M R/T 2
D. M T 2 /4π 2 R
E. M R/4π 2 T 2
The following figure pertains to the following two questions.
Two masses hang by a fixed length string over a massless pulley as shown:
8. If p is the momentum conjugate to x, the Lagrangian function for this system is
A. (m1 + m2 )ẋ2 /2 + (m1 − m2 )gx
B. (m1 + m2 )ẋ2 /2 − (m1 − m2 )gx
C. (m1 + m2 )ẋ2 /2 + (m1 + m2 )gx
D. (m1 + m2 )ẋ2 /2 − (m1 + m2 )gx
E. p2 /2(m1 + m2 ) + (m1 + m2 )gx
9. If p is the momentum conjugate to x, the Hamiltonian function for this system is
A. p2 /2(m1 + m2 ) + (m1 − m2 )gx
B. p2 /2(m1 + m2 ) − (m1 − m2 )gx
C. p2 /2(m1 − m2 ) + (m1 + m2 )gx
D. p2 /2(m1 − m2 ) − (m1 + m2 )gx
E. p2 /2m1 + p2 /2m2 + (m1 + m2 )gx
10. An object with mass m moves vertically near the Earth’s surface. If +y measures
the upward direction, which of the following are Hamilton’s Equations of motion?
A. ÿ = −g
B. ẏ = p/m and ṗ = mg
C. ẏ = −p/m and ṗ = mg
D. ẏ = p/m and ṗ = −mg
E. ẏ = −p/m and ṗ = −mg
11. A 5 kg mass sits at the origin and a 1 kg mass sits on the x-axis. If the 1 kg
mass is at x = 6 cm, then the x position of the center of mass is
A. 1 cm
B. 1.2 cm
C. 3 cm
D. 4.8 cm
E. 5 cm
12. The eigenvectors of a symmetric real matrix are necessarily
A. zero.
B. normalized.
C. orthogonal.
D. either normalized or orthogonal.
E. both normalized and orthogonal.
13. A line drawn from any planet to the Sun will sweep out equal areas in equal times,
regardless of the orbital eccentricity. This fact follows directly from conservation of
A. time.
B. energy.
C. momentum.
D. angular momentum.
E. phase space volume.
14. A pendulum of length ` and bob mass m hangs inside a moving car that is
moving forward but slowing down. The angle the pendulum makes with the vertical is
A. zero.
B. nonzero, constant, with the bob behind the vertical
C. nonzero, constant, with the bob forward of the vertical
D. initially behind the vertical, but slowly moves forward of vertical
E. initially forward of vertical, but slowly moves behind the vertical
15. Which of the following fields can represent a conservative force?
A. F(x, y) = yx̂
B. F(x, y) = xx̂ + yŷ
C. F(x, y) = x(x̂ + ŷ)
D. F(x, y) = xyx̂ + x2 ŷ
E. F(x, y) = y 2 x̂ − xyŷ
16. Let F (x) = −1/x be a force in one dimension. The potential energy function is
A. 1/x
B. 1/x2
C. −1/x2
D. ln x
E. constant.
17. A 4 kg mass is attached to a spring with stiffness constant k = 100 N/m. The
mass is also subject to a linear drag force, with drag coefficient b = 24 N/(m/sec). At
what (angular) frequency does this system oscillate?
A. 2/sec
B. 3/sec
C. 4/sec
D. 5/sec
E. 6/sec
18. In a certain coordinate system, the matrix representation of a stress tensor in a
continuous medium has Σxy = Σyx = σ while all other elements are zero. The force on
an area element dA lying in the xy plane is
A. zero.
B. σdAx̂
C. σdAŷ
D. σdAẑ
√
E. σdA(ŷ + ẑ)/ 2
19. The Lagrangian for a system of particles is a function of the spatial coordinates
and their velocities. All coordinates are translated by the same fixed amount and the
Lagrangian does not change. What physical quantity is a constant of the motion?
A. Energy
B. Momentum
C. Kinetic energy
D. Angular momentum
E. Effective potential energy
20. What is the name of this course?
A. Fluid Mechanics
B. Quantum Mechanics
C. Statistical Mechanics
D. Theoretical Mechanics
E. Automobile Mechanics
Part II: Short Problems (10 points each)
Problem 1. An object with mass m is launched vertically upward against gravity with
speed v0 . It is subject to a linear drag force bv where v is the velocity and b is constant.
Find the time it takes to reach its highest point, in terms of m, b, v0 , and g. Show that you
get the correct answer for bv0 mg.
Problem 2. Two stars orbit about their common center of mass. One star has the mass
of the Sun, and the other is three times as massive as the Sun. The elliptical orbit has
eccentricity 1/2, and the stars are 3 AU apart at their largest separation. (One AU is the
mean distance from the Earth to the Sun.) Find the orbital period in years.
Problem 3. As shown in class, the equation of motion for an elastic deformation u(r, t) is
∂ 2u
1
ρ 2 = K + G ∇(∇ · u) + G∇2 u
∂t
3
where K and G are the bulk and shear moduli, respectively. Show that the spherically
symmetric radial deformation wave u(r, t) = (r̂/r) cos(kr − kct), in the usual spherical
coordinates and where k is a constant, solves this equation in the “far distance” limit kr 1.
Find the wave speed c in terms of K, G, and ρ. You are welcome to use vector calculus
forms in spherical coordinates without deriving them.
Problem 4. A pendulum bob with mass m hangs from a massless string of length ` attached to a cart of mass M . The cart
moves on a horizontal frictionless track. Using the coordinates
x and φ as shown, construct the Lagrangian and then reduce it
to a linear form for small angles φ. Calculate the normal mode
frequencies. (One of the eigenmodes is rather peculiar, but it
makes sense if you think about it for a moment.) Explain why
your answer is correct in the limit M m.
x
Problem 5. An elementary particle of mass M , at rest, decays into two lighter particles of
masses m1 and m2 . Find the (total) energy of particle #2 in terms of M , m1 , m2 , and c.
Check that you get the answer you expect for m1 = m2 .
Problem 6. At a specific instant, a disk of mass M and radius R lies in the xy plane
with center at the origin. It rotates with a fixed angular velocity vector ω lying in the yz
plane and makes an angle α with respect to the z-axis. At this instant, find the vectors
(a) angular momentum L, and (b) torque Γ using unit vectors in this (x, y, z) coordinate
system. Express your answers in terms of M , R, α, and ω ≡ |ω|.
Extra Paper #1: Please show the problem on which you are working!
Extra Paper #2: Please show the problem on which you are working!
Solutions
Multiple Choice
1. D 2. C 3. E
11. A 12. C 13. D
4. C
14. C
5. C
15. B
6. D 7. C 8. A
16. D 17. C 18. A
9. B 10. D
19. B 20. D
Problem 1.R mdv/dt = −mg − bv = −mg(1 + av) with a ≡ b/mg, so −gt = v00 dv/(1 + av),
or t = (1/g) 0v0 dv/(1 + av) = (1/ag) ln(1 + av0 ) = (m/b) ln(1 + bv0 /mg). Since ln(1 + x) = x
for x 1, t → (m/b)(bv0 /mg) = v0 /g for (bv0 /mg) 1, correct for the case of zero drag.
R
Problem 2. τ 2 = 4π 2 a3 /G(M1 + M2 ) = (4π 2 /GM )a3 /(m1 + m2 ) for solar mass M and m1 ,
m2 in solar masses. For Earth, m1 = 1 m2 so (4π 2 /GM ) = 1 if a in AU and τ in years.
√
Here m1 + m2 = 4 and a = c/(1 − 2 ) = rmax /(1 + ) = 3/(3/2) = 2, so τ = (8/4)1/2 = 2.
Problem 3. No dependence on φ or θ, and ur = f (kr − kct)/r and uθ = uφ = 0. Also,
radial derivatives of r̂ are zero. The equation of motion becomes
f 00
∂ 1 ∂
1 ∂2
1
∂ f
f0
f 00
1
2
k 2 c2 ρr̂
(rf
)
+G
(r̂f
)
=
K
+
G
r̂
+
k
+k
Gr̂
K + G r̂
r
3
∂r r2 ∂r
r ∂r2
3
∂r r2
r
r
"
#
"
#
The derivative of the brackets gives kf 0 /r2 − 2f /r3 − kf 0 /r2 + k 2 f 00 /r = (k 2 r2 f 00 − 2f )/r3 .
Since f 00 = −f , only the first term survives for kr
q 1. The equation of motion then
2
becomes c ρ = (K + G/3) + G = K + 4G/3 so c = (K + 4G/3)/ρ. This is the same as for
a compressional plane wave, which is consistent with the far distance approximation.
Problem 4. L = 12 M ẋ2 + 12 m(ẋ2b + ẏb2 )−mgyb = 12 M ẋ2 + 21 m(ẋ2 +2`ẋφ̇ cos φ+`2 φ̇2 )−mg`(1−
"
#
x
1
1
1 T
1 T
2
2 2
2
cos φ) → 2 [(M + m)ẋ + 2m`ẋφ̇ + m` φ̇ ] − 2 mg`φ = 2 q̇ Mq̇ − 2 q Kq where q =
,
φ
"
#
"
#
M + m m`
0 0
m
M=
. Then |K − ω 2 M| = 0 gives ω 2 = g` 1 + M
and
2 , and K =
m`
m`
0 mg`
ω 2 = 0. The first gives ω 2 = g/` for large M , correct. The second is just φ = 0, for any x.
Problem 5. P = p1 + p2 so p1 = P − p2 and m21 c2 = M 2 c2 + m22 c2 − 2P · p2 In the rest
frame, P · p2 = M c(E2 /c), so E2 = (M 2 + m22 − m21 )c2 /(2M ), correct (M c2 /2) for m1 = m2 .
Problem 6. Principal axes are xyz with Ixx = Iyy = M R2 /4 and Izz = M R2 /2 and others
zero. So L = Ixx ωx x̂ + Iyy ωy ŷ + Izz ωz ẑ = (M R2 /4)ω sin αŷ + (M R2 /2)ω cos αẑ. The torque
is Γ = dL/dt = ω × L = (ω sin αŷ + ω cos αẑ) × L = (M R2 ω 2 /4) sin α cos α(2x̂ − x̂) =
(M R2 ω 2 /4)x̂ sin α cos α.