Download PHYS2330 Intermediate Mechanics Fall 2010 Final Exam

Name:
PHYS2330
Intermediate Mechanics
Final Exam
Fall 2010
Tuesday, 21 Dec 2010
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. An object with mass m moves vertically near the Earth’s surface. If y measures
the vertical coordinate, with y increasing moving upward, the Hamiltonian is
A.
B.
C.
D.
E.
1
my 2
2
1
my 2
2
1
mẏ 2
2
1
mẏ 2
2
1
mẏ 2
2
+ mgy
− mgy
+ mgy
− mgy
+ mg ẏ
2. Which of the following fields can represent a conservative force?
A. F(x, y) = xyx̂ + xŷ
B. F(x, y) = yx̂ − xyŷ
C. F(x, y) = xy(x̂ + ŷ)
D. F(x, y) = x2 x̂ + y 2 ŷ
E. F(x, y) = yx̂
3. A force in one dimension is given by F (x) = x2 . The potential energy function is
A. 2x
B. −2x
C. −x2
D. x3 /3
E. −x3 /3
4. A 2 kg mass is attached to a spring with stiffness constant k = 50 N/m. The
mass is also subject to a linear drag force, with drag coefficient b = 16 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
5. 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 yz plane is
A. zero.
B. σdAx̂
C. σdAŷ
D. σdAẑ
√
E. σdA(ŷ + ẑ)/ 2
6. The following diagram plots time dependent responses x(t) for a nonlinear system,
as a function of some kind of driver r:
For which values of r do you expect x(t) to eventually be completely different for only
slight changes in the initial conditions?
A. r ≤ 3.45
B. r ≈ 3.62
C. 3.59 ≤ r ≤ 3.61
D. 3.81 ≤ r ≤ 3.82
E. r = 3.45, 3.55, 3.57, . . .
7. A particle of mass m moves in a plane, subject to a force F = −3kr2 where r is
the distance from the origin. If φ measures the particle’s angular position with respect
to the x-axis, the Lagrangian is
A. mṙ2 /2 + mφ̇2 /2 − kr3
B. mṙ2 /2 + mφ̇2 /2 + kr3
C. mṙ2 /2 + mr2 φ̇2 /2 − kr3
D. mṙ2 /2 + mr2 φ̇2 /2 + kr3
E. mṙ2 /2 + mr2 φ̇2 /2 − 3kr2
8. An asteroid is in a circular orbit about the Sun. The radius of the orbit is about
nine times the distance of the Earth to the Sun. How long does it take the asteroid to
complete one orbit?
A. Three years.
B. Nine years.
C. Eighteen years.
D. Twenty-seven years.
E. Eighty-one years.
9. A pendulum bob hangs from a string in a laboratory located precisely at the
Earth’s equator. The bob is not moving. It hangs
A. directly towards the center of the Earth.
B. nearly towards the center of the Earth, but slightly north.
C. nearly towards the center of the Earth, but slightly south.
D. nearly towards the center of the Earth, but slightly east.
E. nearly towards the center of the Earth, but slightly west.
10. An object of mass m moves according to the Hamiltonian H(x, p) = Ap2 + Bx4
where A and B are constants. The Hamiltonian equations of motion are
A. ẋ = 2Ap and ṗ = Bx4
B. ẋ = p/m and ṗ = −Bx4
C. ẋ = 2Ap and ṗ = −Bx4
D. ẋ = p/m and ṗ = −4Bx3
E. ẋ = 2Ap and ṗ = −4Bx3
11. An inviscid, incompressible fluid flows horizontally through a pipe of constant
cross sectional area with a uniform velocity of 10 m/s. If the fluid flows from left to
right, the difference in pressure between two points 100 m apart is closest to which of
the following? (Assume the density of the fluid is 1000 kg/m3 .)
A. zero.
B. 10 Pa
C. 103 Pa
D. 105 Pa
E. 107 Pa
12. A rigid body rotates with angular velocity ω = ωx̂ in a certain (x, y, z) coordinate
system. The moment-of-inertia tensor, as a matrix in the same coordinate system, is


1 3 5
2
I = mR  3 2 4 

5 4 6
The angular momentum vector L is
A. mR2 ωx̂
B. mR2 ω(2ŷ + 6ẑ)
C. mR2 ω(x̂ + 2ŷ + 6ẑ)
D. mR2 ω(x̂ + 3ŷ + 5ẑ)
E. mR2 ω(5x̂ + 4ŷ + 6ẑ)
13. To a very good approximation, the speed of light is one foot per nanosecond. In
a certain reference frame, event A occurs 4 feet away and 5 nanoseconds before event
B. In another reference frame, events A and B occur simultaneously. How far apart,
spatially, are they separated in the second reference frame?
A. 3 feet.
B. 4 feet.
C. 5 feet.
D. They happen at the same place.
E. In no reference frame can A and B occur simultaneously.
14. A simple pendulum is made from a bob of mass m hanging from a massless rod
of length �. It swings in a plane and θ measures the angle from the vertical. If θ never
gets very large, the potential energy function U (θ) is
A. mg�θ
B. (g/�)θ2
C. g�2 θ2 /2
D. mg�θ2 /2
E. mg� sin θ
15. Three equal masses m are located at the corners of an equilateral triangle of side
length a. The moment of inertia for rotations about an axis parallel to the plane of
the triangle, and passing through two of the corners, is
A. ma2 /2
B. 3ma2 /4
C. ma2
D. 2ma2
E. 3ma2
16. Which of the following is not a characteristic of a system in chaotic motion?
A. Positive Liapunov exponent.
B. Nonlinear equation of motion.
C. Extreme sensitivity to initial conditions.
D. Fractal behavior in a bifurcation diagram.
E. A lack of periodic motion in response to a periodic driving term.
17. A small planet interacts with a massive star through their mutual gravitational
attraction. Which of the following is not a possible shape of its orbit?
A. a circle.
B. an ellipse.
C. a parabola.
D. a hyperbola.
E. a straight line.
18. Which of the following functions satisfies the wave equation ∂ 2 f /∂x2 = ∂ 2 f /∂t2 ?
A. f (x, t) = x2 − t2
B. f (x, t) = x2 + t3
C. f (x, t) = sin(xt)
D. f (x, t) = cos(x2 − t2 )
E. f (x, t) = ln[(x − t)3 /(x + t)2 ]
19. A uniform circular disk of mass M and radius R rolls without slipping on a
horizontal surface with speed v. Its kinetic energy is
A. M v 2 /2
B. 3M v 2 /4
C. M v 2
D. 5M v 2 /4
E. 3M v 2 /2
20. Which of the following names is not associated with a key contribution to the
development of classical mechanics?
A. Euler
B. Hamilton
C. Lagrange
D. Napolitano
E. Newton
Part II: Short Problems (10 points each)
−0.65
−0.7
−0.75
Effective potential
Problem 1. The plot shows the effective
potential for a planet orbiting a massive
star, as a function of the distance of the
planet from the star, in certain units. A
planet in this potential takes one year to
execute a circular orbit. For an elliptical orbit of a planet with the same mass,
but with total energy E = −1, find the
eccentricity and the orbital period.
−0.8
−0.85
−0.9
−0.95
−1
−1.05
−1.1
2
3
4
5
6
Distance from star
7
8
Problem 2. A particle of mass m moves in one dimension x, and is acted on by a force
F (t) = F0 e−bt . The particle starts from rest at t = 0 with initial position x = 0. Find its
position x(t) as a function of time.
Problem 3. Apply the analysis concepts we used for the damped driven pendulum to
analyze a (under-)damped driven linear oscillator, namely the differential equation
φ̈ + 2β φ̇ + ω02 φ = γω02 cos ωt
In particular, (a) sketch a “bifurcation” diagram, (b) find the Liapunov exponent, (c) sketch
the state space orbit, and (d) draw a Poincaré section. It is not necessary to put in values
for the various parameters, but you are welcome to do so if you like. Clearly label the axes
of all plots, however. I am looking for just the general behavior, but any detail you would
like to provide can help me understand if things don’t look quite right.
Problem 4. A simple plane pendulum consists of a massless rod of length � and a bob
of mass m. Let φ measure the displacement from vertical. The upper end of the rod is
attached to a point which accelerates horizontally with constant acceleration a. Construct
the Lagrangian function L(φ, φ̇, t) and determine the differential equation of motion for φ.
Show that the equation of motion linearizes to that of a simple harmonic oscillator for small
φ. Find the frequency and equilibrium angle φ0 for the small oscillations.
Problem 5. A long, thin cylindrical rod has mass M , length L and cross sectional radius R.
Give values for (all three) principle moments of inertia. (You are welcome to derive them,
but you may also quote them from other sources.) At time t = 0 the rod has angular velocity
ω0 at an angle θ with respect to the rod, find the angular velocity ω3 (t) along the long axis
of the rod as a function of time.
Problem 6. The “circulation” of a fluid with velocity field v(x, t) is defined as
ΓC =
�
C
v · d�
for some closed loop C. Consider the steady flow of an incompressible inviscid fluid which
has zero circulation everywhere, with gravity being the only active volume force. Show that
�
�
1
∇ ρv 2 + ρgz + p = 0
2
(Note that this is the same quantity we studied in Bernoulli’s theorem, which stated that
this is a constant in time. You are asked here to show that it is constant
�
� in space.) You may
1 2
use without proof the vector calculus theorem v × (∇ × v) = ∇ 2 v − (v · ∇)v.
Extra Paper #1: Please show the problem on which you are working!
Extra Paper #2: Please show the problem on which you are working!