2 Energy of the Spring-Mass System We know enough to discuss the mechanical energy of the oscillating mass on a spring. Remember Kinetic energy is always K ½ mv2 K ½ m -A sin( t )2 And the potential energy of a spring is U ½ k x2 U ½ k A cos (t ) 2 3 Energy of the Spring-Mass System Add to get E K U constant. ½ m ( A )2 sin2( t ) 1/2 k (A cos( t ))2 Recalling so E ½ k A2 sin2(t ) ½ kA2 cos2(t ) ½ k A2 sin2(t ) cos2(t ) ½ k A2 with q wt f Active Figure 4 SHM So Far
The most general solution is x A cos(t )
where A amplitude
For SHM without friction
The frequency does not depend on the amplitude !
We will see that 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.
5 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 mac T mg cos(q) m v2/L
S Fx max -mg sin(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
L x T m mg 6 Lecture 20 Exercise 1Simple 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 7 The Rod Pendulum
A pendulum is made by suspending a thin rod of length L and mass M at one end. Find the frequency of oscillation for small displacements (i.e. q sin q).
S tz I a - r x F (L/2) mg sin(q)
(no torque from T)
- mL2/12 m (L/2)2 a L/2 mg q
-1/3 L d2q/dt2 ½ g q
The rest is for homework
x CM L mg 8 General Physical Pendulum
Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis that we know where the CM is located and what the moment of inertia I about the axis is.
The torque about the rotation (z) axis for small is (sin )
-MgR sinq -MgR
x CM Mg 9 Torsion Pendulum
Consider an object suspended by a wire attached at its CM. The wire defines the rotation axis and the moment of inertia I about this axis is known.
The wire acts like a rotational spring.
When the object is rotated the wire is twisted. This produces a torque that opposes the rotation.
In analogy with a spring the torque produced is proportional to the displacement - k where k is the torsional spring constant
w (k / I)½
10 Torsional spring constant of DNA
Session Y15 Biosensors and Hybrid Biodevices
1115 AM203 PM Friday March 25 2005 LACC - 405
Abstract Y15.00010 Optical measurement of DNA torsional modulus under various stretching forces
Jaehyuck Choi Kai Zhao Y.-H. Lo Department of Electrical and Computer Engineering Department of Physics University of California at San Diego La Jolla California 92093-0407 We have measured the torsional spring modulus of a double stranded-DNA by applying an external torque around the axis of a vertically stretched DNA molecule. We observed that the torsional modulus of the DNA increases with stretching force. This result supports the hypothesis that an applied stretching force may raise the intrinsic torsional modulus of ds-DNA via elastic coupling between twisting and stretching. This further verifies that the torsional modulus value (C 46.5 /- 10 pN nm) of a ds-DNA investigated under Brownian torque (no external force and torque) could be the pure intrinsic value without contribution from other effects such as stretching bending or buckling of DNA chains.
DNA half gold sphere 11 Lecture 20 Exercise 2Period
All of the following torsional pendulum bobs have the same mass and w (k/I)½
Which pendulum rotates the slowest i.e. has the longest period (The wires are identical k is constant)
For both the spring and the pendulum we can derive the SHM solution and examine U and K
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 mechanical energy ought be conserved.
14 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 15 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 16 What about Friction
Friction causes the oscillations to get smaller over time
This is known as DAMPING.
As a model we assume that the force due to friction is proportional to the velocity Ffriction - b v .
17 What about Friction We can guess at a new solution. and now w02 k / m With 18 What about Friction if What does this function look like 19 Damped Simple Harmonic Motion
There are three mathematically distinct regimes
underdamped critically damped overdamped 20 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 Coefficient 60 pN-sec / m
Deformation of protein in a viscous fluid
21 Driven SHM with Resistance
Apply a sinusoidal force F0 cos (wt) and now consider what A and b do
w w0 22 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) 23 Stick - Slip Friction
How can a constant motion produce resonant vibrations
Singing / Whistling
Tacoma Narrows Bridge
24 Dramatic example of resonance
In 1940 a steady wind set up a torsional vibration in the Tacoma Narrows Bridge
25 A short clip
In 1940 a steady wind sets up a torsional vibration in the Tacoma Narrows Bridge
26 Dramatic example of resonance
Large scale torsion at the bridges natural frequency
27 Dramatic example of resonance
Eventually it collapsed
28 Lecture 20 Exercise 3Resonant Motion
Consider the following set of pendulums all attached to the same string
If I start bob D swinging which of the others will have the largest swing amplitude (A) (B) (C) 29 Waves (Chapter 16)
Movement around one equilibrium point
Look only at one point oscillations
But changes in time and space (i.e. in 2 dimensions!)
30 What is a wave
A definition of a wave
A wave is a traveling disturbance that transports energy but not matter.
Transverse The mediums displacement is perpendicular to the direction the wave is moving.
Water (more or less)
Longitudinal The mediums displacement is in the same direction as the wave is moving
32 Wave Properties
Wavelength The distance between identical points on the wave.
Amplitude The maximum displacement A of a point on the
Animation 33 Wave Properties...
Period The time T for a point on the wave to undergo one complete oscillation.
Speed The wave moves one wavelength in one period T so its speed is v / T.
Animation 34 Lecture 20 Exercise 4Wave Motion
The speed of sound in air is a bit over 300 m/s and the speed of light in air is about 300000000 m/s.
Suppose we make a sound wave and a light wave that both have a wavelength of 3 meters.
What is the ratio of the frequency of the light wave to that of the sound wave (Recall v / T f )
(A) About 1000000 (B) About 0.000001 (C) About 1000 35 Wave Forms
So far we have examined continuous waves that go on forever in each direction !
36 Lecture 20 Exercise 5Wave Motion
A harmonic wave moving in the positive x direction can be described by the equation
(The wave varies in space and time.)
v l / T l f (l/2p ) (2p f) w / k and by definition w gt 0
y(xt) A cos ( (2p / l) x - wt ) A cos (k x w t )
Which of the following equation describes a harmonic wave moving in the negative x direction
(A) y(xt) A sin ( k x - wt ) (B) y(xt) A cos ( k x wt ) (C) y(xt) A cos (-k x wt ) 37 Lecture 20 Exercise 6Wave Motion
A boat is moored in a fixed location and waves make it move up and down. If the spacing between wave crests is 20 meters and the speed of the waves is 5 m/s how long Dt does it take the boat to go from the top of a crest to the bottom of a trough (Recall v / T f )
(A) 2 sec (B) 4 sec (C) 8 sec t t Dt 38 Waves on a string
What determines the speed of a wave
Consider a pulse propagating along a string
Snap a rope to see such a pulse
How can you make it go faster
Animation 39 Waves on a string... Suppose
The tension in the string is F
The mass per unit length of the string is (kg/m)
The shape of the string at the pulses maximum is circular and has radius R
R 40 Waves on a string...
So we find
Making the tension bigger increases the speed.
Making the string heavier decreases the speed.
The speed depends only on the nature of the medium not on amplitude frequency etc of the wave.
41 Lecture 20 Recap
Agenda Chapter 15 Finish Chapter 16 Begin
Chapter 16 Traveling Waves
Problem Set 7 due Nov. 14 Tuesday 1159 PM
Problem Set 8 due Nov. 21 Tuesday 1159 PM
For Wednesday Finish Chapter 16 Start Chapter 17
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