Goals:

1
Lecture 20

• Goals

• Chapter 14
• Compare and contrast different systems with SHM.
• Understand energy conservation (transfer) in SHM.
• Understand the basic ideas of damping and
  resonance.
• Chapter 15
• Understand pressure in liquids and gases
• Use Archimedes principle to understand buoyancy
• Understand the equation of continuity
• Use an ideal-fluid model to study fluid flow.
• Investigate the elastic deformation of solids
  and liquids

• Assignment
• HW9, Due Wednesday, Apr. 7th
• Tuesday (after break!!!) Read all of Chapter 15

2
SHM So Far

• The most general solution is x A cos(?t ?)
• where A amplitude
• ? (angular) frequency
• ? phase constant
• For SHM without friction,
• The frequency does not depend on the amplitude !
• This is true of all simple harmonic motion!
• The oscillation occurs around the equilibrium
  point where the force is zero!
• Energy is a constant, it transfers between
  potential and kinetic

3
The Simple Pendulum

• A pendulum is made by suspending a mass m at the
  end of a string of length L. Find the frequency
  of oscillation for small displacements.
• S Fy may T mg cos(q) mac m vT2/L
• S Fx max -mg sin(q)
• where x L tan q
• If q small then x ? L q and sin(q) ? q
• 0 tan 0.00 sin 0.00 0.00
• 5 tan 0.09 sin 0.09 0.09
• 10 tan 0.17 sin 0.17 0.17
• 15 tan 0.26 0.27
• sin 0.26 0.26

4
The Simple Pendulum

• A pendulum is made by suspending a mass m at the
  end of a string of length L. Find the frequency
  of oscillation for small displacements.
• S Fy may T mg cos(q) mac m vT2/L
• S Fx max -mg sin(q)
• where x L tan q
• If q small then x ? L q and sin(q) ? q
• dx/dt L dq/dt
• ax d2x/dt2 L d2q/dt2
• so ax -g q L d2q / dt2 ? L d2q / dt2 g q
  0
• and q q0 cos(wt f) or q q0 sin(wt
  f)
• with w (g/L)½

5
SHM So Far

• The most general solution is x(t) A cos(?t ?)
• where A amplitude
• ? (angular) frequency 2p f 2p/T
• ? phase constant

Velocity v(t) -?A sin(?t ?) Acceleration
a(t) -?2A cos(?t ?)
Spring constant Inertia
Hookes Law Spring
Simple Pendulum
6
SHM So Far

• For SHM without friction
• The frequency does not depend on the amplitude !
• The oscillation occurs around the equilibrium
  point where the force is zero!
• Mechanical Energy is constant, it transfers
  between potential and kinetic energies.

7
The shaker cart

• You stand inside a small cart attached to a
  heavy-duty spring, the spring is compressed and
  released, and you shake back and forth,
  attempting to maintain your balance. Note that
  there is also a sandbag in the cart with you.
• At the instant you pass through the equilibrium
  position of the spring, you drop the sandbag out
  of the cart onto the ground.
• What effect does jettisoning the sandbag at the
  equilibrium position have on the amplitude of
  your oscillation?
• It increases the amplitude.
• It decreases the amplitude.
• It has no effect on the amplitude.

8
The shaker cart

• Instead of dropping the sandbag as you pass
  through equilibrium, you decide to drop the
  sandbag when the cart is at its maximum distance
  from equilibrium.
• What effect does jettisoning the sandbag at the
  carts maximum distance from equilibrium have on
  the amplitude of your oscillation?
• It increases the amplitude.
• It decreases the amplitude.
• It has no effect on the amplitude.

9
The shaker cart

• What effect does jettisoning the sandbag at the
  carts maximum distance from equilibrium have on
  the maximum speed of the cart?
• It increases the maximum speed.
• It decreases the maximum speed.
• It has no effect on the maximum speed.

10
What about Vertical Springs?

• For a vertical spring, if y is measured from the
  equilibrium position
• Recall force of the spring is the negative
  derivative of this function
• This will be just like the horizontal case-ky
  ma

j
k
y 0
F -ky
m
Which has solution y(t) A cos( ?t ?)
where
11
Exercise Simple Harmonic Motion

• A mass oscillates up down on a spring. Its
  position as a function of time is shown below.
  At which of the points shown does the mass have
  positive velocity and negative acceleration ?
• Remember velocity is slope and acceleration is
  the curvature

y(t) A cos( ?t ?) v(t) -A ? sin( ?t ?)
a(t) -A ?2 cos( ?t ?)
12
Home Exercise

• A mass m 2 kg on a spring oscillates (no
  friction) with amplitude A 10 cm. At t 0
  its speed is at a maximum, and is v2 m/s
• What is the angular frequency of oscillation ? ?
• What is the spring constant k ?
• General relationships E K U constant, w
  (k/m)½
• So at maximum speed U0 and ½ mv2 E ½ kA2
• thus k mv2/A2 2 x (2) 2/(0.1)2 800 N/m, w
  20 rad / sec

13
Home Exercise Initial Conditions

• A mass hanging from a vertical spring is lifted a
  distance d above equilibrium and released at t
  0.
• Which of the following describe its velocity and
  acceleration as a function of time (upwards is
  positive y direction)?

(A) v(t) - vmax sin( wt ) a(t) -amax
cos( wt )
k
y
(B) v(t) vmax sin( wt ) a(t) amax
cos( wt )
d
t 0
(C) v(t) vmax cos( wt ) a(t) -amax
cos(wt )
0
(both vmax and amax are positive numbers)
14
Home Exercise Initial Conditions

• A mass hanging from a vertical spring is lifted a
  distance d above equilibrium and released at t
  0. Which of the following describe its velocity
  and acceleration as a function of time (upwards
  is positive y direction)

(A) v(t) - vmax sin( wt ) a(t) -amax
cos( wt )
k
y
(B) v(t) vmax sin( wt ) a(t) amax
cos( wt )
d
t 0
(C) v(t) vmax cos( wt ) a(t) -amax
cos(wt )
0
(both vmax and amax are positive numbers)
15
The Torsional Pendulum

• A pendulum is made by suspending a mass m at the
  end of a string of length L. Find the frequency
  of oscillation for small displacements.
• S tz Iaz -mg sin(q) L
• S tz mL2az -mg q L
• L (d2q /dt2) -g q
• d2q /dt2 (-g/L) q
• with q(t) q0 sin wt or q0 cos wt
• and w (g/L)½
• or if a true horizontal torsional pendulum
• Iaz -k q with w (k/I)½

16
Exercise Simple Harmonic Motion

• You are sitting on a swing. A friend gives you a
  small push and you start swinging back forth
  with period T1.
• Suppose you were standing on the swing rather
  than sitting. When given a small push you start
  swinging back forth with period T2.
• Which of the following is true recalling that w
  (g / L)½

(A) T1 T2 (B) T1 gt T2 (C) T1 lt T2
17
Energy in SHM

• For both the spring and the pendulum, we can
  derive the SHM solution using energy
  conservation.
• The total energy (K U) of a system undergoing
  SMH will always be constant!
• This is not surprising since there are only
  conservative forces present, hence energy is
  conserved.

18
SHM and quadratic potentials

• SHM will occur whenever the potential is
  quadratic.
• For small oscillations this will be true
• For example, the potential betweenH atoms in an
  H2 molecule lookssomething like this

U
x
19
SHM and quadratic potentials

• Curvature reflects the spring constant
• or modulus (i.e., stress vs. strain or
• force vs. displacement)
• Measuring modular proteins with an AFM

See http//hansmalab.physics.ucsb.edu
20
What about Friction?A velocity dependent drag
force (A model)
We can guess at a new solution.
and now w02 k / m
Note
With,
21
What about Friction?
A damped exponential
if
22
Variations in the damping
Small damping time constant (m/b) Low friction
coefficient, b ltlt 2m Moderate damping time
constant (m/b) Moderate friction coefficient (b lt
2m)
23
Damped Simple Harmonic Motion

• A downward shift in the angular frequency
• There are three mathematically distinct regimes

underdamped
critically damped
overdamped
24
Driven SHM with Resistance

• Apply a sinusoidal force, F0 cos (wt), and now
  consider what A and b do,

Not Zero!!!
b/m small
steady state amplitude
b/m middling
b large
w
?
w ? w0
25
Resonance-based DNA detection with nanoparticle
probes
Change the mass of the cantilever change the
resonant frequency Su et al., APL 82 3562
(2003)
26
Exercise Resonant Motion

• Consider the following set of pendulums all
  attached to the same string

B
A
D
C
If I start bob D swinging which of the others
will have the largest swing amplitude
? (A) (B) (C)
27
Chapter 15, Fluids

• This is an actual photo of an iceberg, taken by a
  rig manager for Global Marine Drilling in St.
  Johns, Newfoundland. The water was calm and the
  sun was almost directly overhead so that the diver

28
Fluids (Ch. 15)

• At ordinary temperature, matter exists in one of
  three states
• Solid - has a shape and forms a surface
• Liquid - has no shape but forms a surface
• Gas - has no shape and forms no surface
• What do we mean by fluids?
• Fluids are substances that flow. substances
  that take the shape of the container
• Atoms and molecules are free to move.
• No long range correlation between positions.

29
Fluids

• An intrinsic parameter of a fluid
• Density

units kg/m3 10-3 g/cm3
r(water) 1.000 x 103 kg/m3 1.000
g/cm3 r(ice) 0.917 x 103 kg/m3
0.917 g/cm3 r(air) 1.29 kg/m3
1.29 x 10-3 g/cm3 r(Hg) 13.6
x103 kg/m3 13.6 g/cm3
30
Fluids

• Another parameter Pressure

• Any force exerted by a fluid is perpendicular to
  a surface of contact, and is proportional to the
  area of that surface.
• Force (a vector) in a fluid can be expressed in
  terms of pressure (a scalar) as

31
What is the SI unit of pressure?

1. Pascal
2. Atmosphere
3. Bernoulli
4. Young
5. p.s.i.

Units 1 N/m2 1 Pa
(Pascal) 1 bar 105 Pa 1 mbar 102 Pa 1
torr 133.3 Pa
1 atm 1.013 x105 Pa 1013 mbar
760 Torr 14.7 lb/ in2 (PSI)
32
Pressure vs. Depth

• For a uniform fluid in an open container
  pressure same at a given depth independent of the
  container

• Fluid level is the same everywhere in a connected
  container, assuming no surface forces
• Why is this so? Why does the pressure below the
  surface depend only on depth if it is in
  equilibrium?

• Imagine a tube that would connect two regions at
  the same depth.

• If the pressures were different, fluid would
  flow in the tube!

33
Lecture 20

• Assignment
• HW9, Due Wednesday, Apr. 7th
• Tuesday Read all of Chapter 15