PHY 3221: Mechanics I
Fall Term 2010
Midterm Exam 2, October 27, 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 four problems worth a total of 20 pts. All four 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. [4 pts] In Einstein’s theory of general relativity the Schwarzschild radius Rs is
the radius at which a spherical body (such as a star or a planet) would become a black hole.
For example, if you take a star of mass M and squeeze all of its matter within a sphere of radius
Rs (or smaller), the gravitational pull on the surface becomes so strong that even light cannot
escape from it, and the star becomes a black hole.
Knowing nothing about general relativity, use dimensional analysis to derive a formula for
the Schwarzschild radius Rs in terms of the mass of the star M, Newton’s gravitational constant
G and the speed of light c.
Problem 2. [6 pts.] A block of mass m is suspended vertically from a fixed support by a
spring of force constant k.
(a) [1 pt.] Find the extension ∆L ≡ L − L0 of the spring from its relaxed length L0 . You
can ignore the mass of the spring.
(b) [5 pts.] The block is now subjected to a vertical driving force F0 cos ωt. Given that the
spring will yield when its extension exceeds a certain value ∆Lmax , find the range of angular
frequencies which can safely be applied. You can ignore air resistance.
Problem 3. [5 pts.] A pendulum whose period in vacuum is T0 is now submerged in a
resistive medium. Its amplitude on each swing is observed to be half that on the previous
swing. What is the new period?
Problem 4. [5 pts.] An oscillator satisfies the equation
ẍ + 10ẋ + 16x = 0
(a) [1 pt.] Is this an underdamped, a critically damped, or an overdamped oscillator?
Explain your reasoning.
(b) [4 pts.] At time t = 0 the particle is projected from the point x = 1 towards the origin
with speed u, i.e. x(0) = 1 and ẋ(0) = −u. Find x(t).
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 θφ̇ 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 θ̇2 − 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
i
h
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
A
cos(ωt − δ)
xp (t) = q
(ω02 − ω 2 )2 + 4ω 2β 2
2ωβ
2
ω0 − ω 2
δ = tan
−1
!
q
ω02 − 2β 2
ωR
Q=
2β
ωR =
RLC circuit
VL = L
dI
dt
VR = RI
4
VC =
q
C