PHY 3221: Mechanics I
Fall Term 2009
Midterm Exam 2, October 28, 2009
• This is a closed book exam lasting 50 minutes.
• Since calculators are not allowed on this
√ test, if the problem asks for a numerical answer,
answering 2 + 2 is as good as 4, and 2 is as good as 1.4142....
• There are three problems worth a total of 20 pts. All three problems appear on the second
page of this test. Begin each problem on a fresh sheet of paper. Use only one side of the
paper. Avoid microscopic handwriting.
• Put your name, the problem number and the page number in the upper right-hand corner
of each sheet.
• To receive partial credit you must explain what you are doing. Carefully labelled figures
are important! Randomly scrawled equations are not helpful.
• Draw a box around important results (or at least results which you think might be important).
• Good luck!
1
Problem 1. [8 pts] Consider a particle of mass m moving in the region x > 0 according to
v(x) = v0 2 −
x0
x
where the initial condition at x = x0 is v(x0 ) = v0 .
(a) [4 pts] Find the potential energy U (x).
(b) [2 pts] Find the equilibrium points (if any) and determine if they are maxima or minima.
(c) [2 pts] Discuss qualitatively the motion of the particle for v0 > 0 and for v0 < 0. In the
latter case, does the particle reach the origin x = 0?
Hint: a rough sketch of U (x) might be helpful for parts (b) and (c).
Problem 2. [7 pts.] Consider a particle moving in one dimension with the following equation of motion
ẍ + 4ẋ + 9x = 0,
where x is the position of the particle (in SI units it will be measured in meters m).
(a) [1 pt] What are the SI units of the coefficients 4 and 9 in the last two terms on the
left-hand side of this equation?
(b) [1.5 pts] Is the particle behaving as an underdamped, critically damped, or overdamped
oscillator? Explain your reasoning.
(c) [3 pts] Write down the solution x(t) to this equation of motion, with the initial conditions
x(t = 0) = x0 ,
ẋ(t = 0) = 0.
(d) [1.5 pts] Find the logarithmic decrement of the motion (i.e. βτ1 ).
Make sure to define carefully all symbols which you introduce in solving this problem.
Problem 3. [5 pts.] When a damped oscillator was driven at its natural frequency ω0 , its
amplitude D0 was measured to be a certain fraction R of the resonant amplitude Dmax , i.e.
D0 = RDmax , with R < 1. Find the quality factor Q of the system in terms of R.
2
Formula sheet
A·(B × C) = B·(C × A) = C·(A × B) ≡ ABC
A×(B × C) = (A · C)B − (A · B)C
(A × B) · (C × D) = A · [B × (C × D)]
= A · [(B · D)C − (B · C)D]
= (A · C)(B · D) − (A · D)(B · C)
(A × B) × (C × D) = [(A × B) · D] C − [(A × B) · C] D
= (ABD)C − (ABC)D = (ACD)B − (BCD)A
v = ṙ er + r θ̇ eθ + r sin θ φ̇ eφ
a =
+ 2ṙ φ̇ sin θ + 2r θ̇φ̇ cos θ + r θ̈ sin θ eφ
v = ṙ er + r φ̇ eφ + ż ez
a =
r̈ − r θ̇ 2 − r φ̇2 sin2 θ er + 2ṙ θ̇ + r θ̈ − r φ̇2 sin θ cos θ eθ
r̈ − r φ̇2 er + r φ̈ + 2ṙ φ̇ eφ + z̈ ez
X
k
εijk εlmk = δil δjm − δim δjl
X
εijk εljk = 2 δil
j,k
X
εijk εijk = 6
i,j,k
Time averages over one period T :
1 Z t+T
1
hsin ωti =
dt sin2 ωt =
T t
2
2
1
hcos ωti =
T
2
Z
t+T
t
3
dt cos2 ωt =
1
2
Simple harmonic oscillator:
mẍ + kx = 0
x(t) = A sin(ω0 t − δ)
x(t) = A cos(ω0 t − φ)
2π
=
ω0 = 2πν0 =
τ0
s
k
m
Damped oscillator:
b
ẍ + 2β ẋ + ω02 x = 0, 2β =
m
√ 2 2
√ 2 2 x(t) = e−βt A1 e β −ω0 t + A2 e− β −ω0 t
Underdamped motion
x(t) = Ae−βt cos(ω1 t − δ),
ω1 =
q
ω02 − β 2
Critically damped motion
x(t) = (A + Bt)e−βt
Overdamped motion
h
i
x(t) = e−βt A1 eω2 t + A2 e−ω2 t ,
Driven oscillator
ω2 =
q
β 2 − ω02
F0
ẍ + 2β ẋ + ω02 x = A cos ωt, A =
m
√ 2 2 √ 2 2
xc (t) = e−βt A1 e β −ω0 t + A2 e− β −ω0 t
xp (t) = q
A
(ω02 − ω 2 )2 + 4ω 2 β 2
δ = tan
2ωβ
2
ω0 − ω 2
−1
cos(ωt − δ)
!
q
ω02 − 2β 2
ωR
Q=
2β
ωR =
RLC circuit
VL = L
dI
dt
VR = RI
4
VC =
q
C