physics 207: lecture 19, pg 1 physics 207, lecture 19, nov. 5goals: chapter 14 chapter 14 ...
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Physics 207: Lecture 19, Pg 1
Physics 207, Lecture 19, Nov. 5Goals:Goals:
• Chapter 14Chapter 14 Understand and use energy conservation in oscillatory systems. Understand the basic ideas of damping and resonance.
• Chapter 15Chapter 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
• AssignmentAssignment HW8, Due Wednesday, Nov. 12th Monday: Read all of Chapter 15.
Physics 207: Lecture 19, Pg 2
SHM So Far The most general solution is x(t) = A cos(t + )
where A = amplitude
= (angular) frequency = 2 f = 2/T
= 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
mk
Velocity: v(t) = -A sin(t + )
Acceleration: a(t) = -2A cos(t + )
Simple Pendulum:L
g
Physics 207: Lecture 19, Pg 3
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?
A. It increases the amplitude.B. It decreases the amplitude.C. It has no effect on the amplitude.
Hint: At equilibrium, both the cart and the bag are moving at their maximum speed. By dropping the bag at this point, energy (specifically the kinetic energy of the bag) is lost from the spring-cart system. Thus, both the elastic potential energy at maximum displacement and the kinetic energy at equilibrium must decrease
Physics 207: Lecture 19, Pg 4
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 cart’s maximum distance from equilibrium have on the amplitude of your oscillation?
A. It increases the amplitude.
B. It decreases the amplitude.
C. It has no effect on the amplitude. Hint: Dropping the bag at maximum
distance from equilibrium, both the cart
and the bag are at rest. By dropping the
bag at this point, no energy is lost from
the spring-cart system. Therefore, both the
elastic potential energy at maximum displacement
and the kinetic energy at equilibrium must remain constant.
Physics 207: Lecture 19, Pg 5
The shaker cart What effect does jettisoning the sandbag at the cart’s maximum
distance from equilibrium have on the maximum speed of the cart?
A. It increases the maximum speed.B. It decreases the maximum speed.C. It has no effect on the maximum speed.
Hint: Dropping the bag at maximum distance from equilibrium, both the cart and the bag are at rest. By dropping the bag at this point, no energy is lost from the spring-cart system. Therefore, both the elastic potential energy at maximum displacement and the kinetic energy at equilibrium must remain constant.
Physics 207: Lecture 19, Pg 7
Exercise Simple Harmonic Motion
A mass oscillates up & down on a spring. It’s 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
t
y(t)
(a)
(b)
(c)
y(t) = A cos( t + )
v(t) = -A sin( t + )
a(t) = -A cos( t + )
Physics 207: Lecture 19, Pg 8
Example
A mass m = 2 kg on a spring oscillates with amplitude
A = 10 cm. At t = 0 its speed is at a maximum, and is v=+2 m/s What is the angular frequency of oscillation ? What is the spring constant k ?
General relationships E = K + U = constant, = (k/m)½
So at maximum speed U=0 and ½ mv2 = E = ½ kA2
thus k = mv2/A2 = 2 x (2) 2/(0.1)2 = 800 N/m, = 20 rad / sec
k
x
m
Physics 207: Lecture 19, Pg 10
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 = (g / L)½
(A) T1 = T2
(B) T1 > T2
(C) T1 < T2 T1 T2
Physics 207: Lecture 19, Pg 11
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.
-A A0x
U
U
KE
Physics 207: Lecture 19, Pg 12
SHM and quadratic potentials
SHM will occur whenever the potential is quadratic. For small oscillations this will be true: For example, the potential between
H atoms in an H2 molecule lookssomething like this:
-A A0x
U
U
KEU
x
Physics 207: Lecture 19, Pg 13
See: http://hansmalab.physics.ucsb.edu
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
U
x
Physics 207: Lecture 19, Pg 14
What about Friction?A velocity dependent drag force
2
2
dtxd
mdtdxbkx
We can guess at a new solution.
)(cos expA )(2
txmbt
With,2
22
22
mb
mb
mk
o
02
2
xmk
dtdx
mb
dtxd
and now 02 ≡ k / m
Look
Physics 207: Lecture 19, Pg 15
What about Friction?
What does this function look like?
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
t
A)(cos
2expA )( )( tt
mbtx mbo 2/if
Physics 207: Lecture 19, Pg 16
Variations in the damping
Small damping time constant (b/2m)
Low friction coefficient, b << 2m
Moderate damping time constant (b/2m)
Moderate friction coefficient (b < 2m)
Physics 207: Lecture 19, Pg 17
Damped Simple Harmonic Motion
A downward shift in the angular frequency There are three mathematically distinct regimes
22 )2/( mbo
mbo 2/ mbo 2/
underdamped critically damped overdamped
mbo 2/
Physics 207: Lecture 19, Pg 18
Physical properties of a globular protein (mass 100 kDa)
Mass 166 x 10-24 kg Density 1.38 x 103 kg / m3 Volume 120 nm3
Radius 3 nm Drag Coefficient60 pN-sec / m
Deformation of protein in a viscous fluid
Physics 207: Lecture 19, Pg 19
Driven SHM with Resistance Apply a sinusoidal force, F0 cos (t), and now consider what A and b do,
2220
2
0
)()(
/
mb
mFA
b small
b middling
b large
tm
Fx
m
k
dt
dx
m
b
dt
xd cos 2
2
Physics 207: Lecture 19, Pg 20
Microcantilever resonance-based DNA detection with nanoparticle probes
Change the mass of the cantilever and change the resonant frequency and the mechanical response.
Su et al., APPL. PHYS. LETT. 82: 3562 (2003)
Physics 207: Lecture 19, Pg 21
Stick - Slip Friction
How can a constant motion produce resonant vibrations?
Examples: Strings, e.g. violin Singing / Whistling Tacoma Narrows Bridge…
Physics 207: Lecture 19, Pg 22
Dramatic example of resonance In 1940, a steady wind set up a torsional vibration in the
Tacoma Narrows Bridge
Physics 207: Lecture 19, Pg 23
Dramatic example of resonance
Eventually it collapsed
Physics 207: Lecture 19, Pg 24
Exercise Resonant Motion
Consider the following set of pendulums all attached to the same string
D
A
B
C
If I start bob D swinging which of the others will have the largest swing amplitude ?
(A) (B) (C)