Download Thu Mar 22

3/20/12
What kind of damping behavior should the shock
absorbers in your car have, for the most comfortable
ride?
A)  No damping is best
B)  under-damping
C)  critical damping
D)  over-damping
1
What kind of oscillator motion does this
phase space diagram describe?
A)  overdamped
B)  underdamped
C)  critically damped
D)  undamped (ideal SHM)
E)  ??? (not possible?)
2 1
3/20/12
Which phase path below best describes overdamped
motion for a harmonic oscillator released from rest?
Challenge question: How does your answer change if the oscillator is “critically
damped”?
3 Important concepts Angular frequency ω =
Frequency: f =
1
T
2π
k
=
T
m
Posi6on: x(t ) = A cos(ωt + δ )
Velocity: v(t ) = − Aω sin(ωt + δ )
2
Accelera6on: a(t ) = −ω x(t )
Energy: E =
2
mv 1 2 1 2
+ kx = kA
2
2
2
No fric0on implies conserva0on of mechanical energy 2
3/20/12
Damped Oscilla6ons In most physical situa6ons, there are nonconserva6ve forces of some sort, which will tend to decrease the amplitude of the oscilla6on. In many cases (e.g. viscous flow) the damping force is propor6onal to the speed: Fdrag = −bv 2β = b / m
F = mx = −kx − bx
x + 2βx + ω0 x = 0
2
Underdamped Oscilla6ons An underdamped oscilla/on with b<w0 : Note that damping reduces the oscilla6on frequency. 3
3/20/12
Overdamped Oscilla6ons An overdamped oscilla/on with b>w0 : Cri6cally damped Oscilla6ons A cri6cally damped oscilla/on with b=w0 : 4
3/20/12
Damped oscilla6ons What kind of motion does this phase path describe?
v
x
A)  overdamped
B)  underdamped
C)  critically damped
D)  impossible to tell
E)  This motion is impossible
10 5
3/20/12
This phase space plot (the solid line that starts at P and
ends at the origin) represents what system?
A) undamped B) under-damped C) critically damped
D) over-damped E) Not enough info to decide!!
11 This phase space plot (the solid line that starts at P and
ends at the origin) represents what system?
A) undamped B) under-damped C) critically damped
D) over-damped E) Not enough info to decide!!
12 6
3/20/12
Which could be the phase space diagram for
one full period of an undamped (ideal) 1D oscillator?
E) None of these, or MORE than 1!!
13 Which could be the phase space diagram for
one full period of an underdamped 1D oscillator?
v
v
x
x
v
v
x
E) None of these, or MORE than 1!!
x
14 7
3/20/12
d 2Q
dQ Q
L 2 +R
+ =0
dt
dt C
Given the differential equation for an RLC
circuit, which quantity is analagous to the
damping term in a mechanical oscillator?
A)  R, resistance
B)  L, inductance
C)  C, capacitance
Challenge question: What do the other two
electrical quantities “correspond to” in the
mechanical system?
15 hOp://vnatsci.ltu.edu/s_schneider/physlets/
main/osc_damped_driven.shtml 8
3/20/12
What is the most general form of the solution of
the ODE u’’+4u=et ?
A) u=C1e2t + C2e-2t + C3et
B)  u=Acos(2t-δ) + C3et
C) u=C1e2t + C2e-2t + (1/5)et
D) u=Acos(2t-δ) + (1/5)et
E)  Something else!???
17
Review
9
3/20/12
2.16
The binomial expansion is: n
(1 + ε ) ≈ 1 + nε + …
Does this mean that, for z<<a, we can write 2 −1/ 2
(a + z )
2
?
≈ a 2 − (1/2)z 2 + …
A)  Correct, but only to leading order, it will fall
apart in the next term
B)  It’s fine, it’s correct to all orders, it’s the
binomial expansion!
C) Utterly false, even to leading order.
The hollow spherical shell has mass density ρ,
inner radius a, outer radius b, total mass M
3a
What is the gravitational force on m
at point P?
2a
P
a
A) 
B) 
C) 
D) 
E) 
GMm/a2
GMm/3a2
GMm/9a2
GMm/9a2
Something else entirely!
20 10
3/20/12
The hollow spherical shell has mass density ρ,
inner radius a, outer radius b.
How does the gravitational potential
ϕ depend on r, for r>b?
b
a
A)  ~r
B)  ~r2
C) ~r -1
D)  ~r -2
E)  Something else entirely!
21 The spherical shell has mass density ρ,
inner radius a, outer radius b.
How does the gravitational potential
ϕ depend on r, for r<a?
b
a
A)  ~r
B)  ~r2
C) ~r -1
D)  ~r -2
E)  Something else entirely!
22 11
3/20/12
Consider a thin cylindrical shell with uniform mass per
unit area σ.
If we want to find the gravitational field at an arbitrary
point on the z-axis, can we simply use Gauss’ law?
z
r
A) Yes, this problem has nice cylindrical symmetry
B) No, Gauss’ law is valid but not helpful here
C) No, Gauss’ law is invalid in this case.
Consider a thin cylindrical shell with uniform mass per
unit area σ.
What is |dg| at the origin due to the small patch of
mass shown?
z
dθ
dz
A)  Gσdz dθ/r2
B)  Gσdz dθ/(r2+z2)
C)  Gσdz dθ/r2
D)  Gσdz rdθ/(r2+z2)
E)  Something else!
r
12
3/20/12
Consider the three closed paths 1, 2, and 3 in some
vector field F, where paths 2 and 3 cover the larger
path 1 as shown.
What can you say about the 3 line integrals?
 
 
 
3
2
A) ∫ F ⋅ d s > ∫ F ⋅ d s + ∫ F ⋅ d s
1
2
3
 
 
 
B) ∫ F ⋅ d s < ∫ F ⋅ d s + ∫ F ⋅ d s
1
2
3
 
 
 
C) ∫ F ⋅ d s = ∫ F ⋅ d s + ∫ F ⋅ d s
1
2
3

D) There is no way to decide without knowing F
1 25 Do you agree or disagree with the following statements?
1) “For a conservative force, the magnitude of the force is
related to potential energy. The larger the potential
energy, the larger the magnitude of the force.”
2) “For a conservative force, the magnitude of the force
is related to potential energy. For any equipotential
contour line, the magnitude of the force must be the
same at every point along that contour.”
A) Agree with 1 and 2
B) Agree only with 1
C) Agree only with 2
D) Disagree with both
13
3/20/12
Can you come up with equipotential lines for
the 3 force fields below?
Draw it if possible
F=(-y, -x2)
Is this force field conservative? A) Y, B) N, C) ?
14
3/20/12
Consider the E-field at Point A, just outside a solenoid, which
has a B-field that is increasing with time. Does this E-field
produce a conservative force?
(Hint: Consider the motion of positive charge in a wire loop
that passes through point A.)
A) Yes, it produces a conservative force
B)  No, it is not conservative
29 An object moves with a “square-root” drag force:
| fdrag | = k | v |
When dropped, what terminal speed will it reach?
(To think about: is this the SAME as terminal velocity?)
A)
mg/k
B) mg / k
C) (mg/k)2
D) Some other formula
E) There's not enough information to decide!!
30 15
3/20/12
vfuel
v
A rocket travels with velocity v with respect to an
(inertial) NASA observer.
It ejects fuel at velocity vexh in its own reference frame.
Which formula correctly relates these two velocities with
the velocity vfuel of a chunk of ejected fuel with respect to
an (inertial) NASA observer?
A) 
B) 
C) 
D) 
E)
vfuel = vexh + v
vfuel = vexh - v
vfuel = -vexh + v
vfuel = -vexh - v
Other/not sure??
31 A point mass m is near a closed cylindrical gaussian
surface. The closed surface consists of the flat end
caps (labeled A and B) and the curved barrel surface

(C). What is the sign of ∫∫ g ⋅ dA through surface C?
C
A) +
B) -
C) zero
D) ????
(the direction of the surface vector is the direction of the outward normal.)
32 16
3/20/12
A point mass m is near a closed cylindrical gaussian
surface. The closed surface consists of the flat end
caps (labeled A and B) and the curved barrel surface

(C). What is the sign of ∫∫ g ⋅ dA through surface C?
C
A) +
B) -
C) zero
D) ????
(the direction of the surface vector is the direction of the outward normal.)
33 A particle oscillates in a 1-D potential.
How many (which?) of the following properties
guarantee SHM?
i) Period T is independent of the amplitude A
ii) Potential U(x)~x2
iii) Force F= –kx (Hooke’s law)
iv) Position is sinusoidal in time: x=Acos(wt-δ)
A) only 1 of these properties guarantees SHM
B) exactly 2 properties guarantee SHM
C) exactly 3 properties guarantee SHM
D) all (any one of these guarantees SHM)
34 17
3/20/12
Consider a super ball which bounces up and down on
super concrete. After the ball is dropped from an initial
height h, it bounces with no dissipation and executes
an infinite number of bounces back to height h.
Is the motion of the ball in z simple harmonic motion?
A)  yes
B)  no
C)  ???
35 18