Physics 207, Lecture 20, Nov. 13 Agenda: Chapter 15, Finish, Chapter 16, Begin Simple pendulum Physical pendulum Torsional pendulum Energy Damping Resonance Chapter 16, Traveling Waves Assignments: Problem Set 7 due Nov. 14, Tuesday 11:59 PM Problem Set 8 due Nov. 21, Tuesday 11:59 PM Ch. 16: 3, 18, 30, 40, 58, 59 (Honors) Ch. 17: 3, 15, 34, 38, 40 For Wednesday, Finish Chapter 16, Start Chapter 17 Physics 207: Lecture 20, Pg 1 Energy of the Spring-Mass System We know enough to discuss the mechanical energy of the
oscillating mass on a spring. x(t) = A cos( t + ) Remember, v(t) = -A sin( t + ) 2 a(t) = - A cos(t + ) Kinetic energy is always K = mv2 K = m [ -A sin( t + )]2 And the potential energy of a spring is, U = k x2 U = k [ A cos (t + ) ]2 Physics 207: Lecture 20, Pg 2 Energy of the Spring-Mass System Add to get E = K + U = constant. m ( A )2 sin2( t + ) + 1/2 k (A cos( t + ))2
Recalling so, E k k 2 m m = k A2 sin2(t + ) + kA2 cos2(t + ) = k A2 [ sin2(t + ) + cos2(t + )] = k A2 with = t + E = kA2 U~cos2 K~sin2
Active Figure Physics 207: Lecture 20, Pg 3 SHM So Far The most general solution is x = A cos(t + ) where A = amplitude = (angular) frequency = phase constant For SHM without friction, k m 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. Physics 207: Lecture 20, Pg 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. z Fy = mac = T mg cos() = m v2/L Fx = max = -mg sin() If small then x L and sin() y dx/dt = L d/dt L ax = d2x/dt2 = L d2/dt2 x so ax = -g = L d2/ dt2 L d2/ dt2 - g = 0
T and = cos(t + ) or = sin(t + ) with = (g/L) m mg Physics 207: Lecture 20, Pg 5 Lecture 20, Exercise 1 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)
(A) T1 = T2 (B) T1 > T2 (C) T1 < T2 T1 T2 Physics 207: Lecture 20, Pg 6 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., sin ). z = I = -| r x F | = (L/2) mg sin() z (no torque from T) -[ mL2/12 + m (L/2)2 ] L/2 mg T -1/3 L d2/dt2 = g xCM
L The rest is for homework mg Physics 207: Lecture 20, Pg 7 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. z-axis The torque about the rotation (z) axis for R small is (sin ) d 2 MgR I xCM 2
dt = -MgR sin -MgR 2 d dt 2 2 where MgR I Mg = 0 cos(t + ) Physics 207: Lecture 20, Pg 8
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: = - where is the torsional spring constant wire I
= (/ I) Physics 207: Lecture 20, Pg 9 Torsional spring constant of DNA Session Y15: Biosensors and Hybrid Biodevices 11:15 AM2:03 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 Physics 207: Lecture 20, Pg 10 Lecture 20, Exercise 2 Period All of the following torsional pendulum bobs have the same mass and = (/I) Which pendulum rotates the slowest, i.e. has the longest period? (The wires are identical, is constant) R R (A) (B)
R R (C) (D) Physics 207: Lecture 20, Pg 11 Reviewing Simple Harmonic Oscillators Spring-mass system d2x 2 xwhere 2 dt k
F = -kx m a k m x z-axis x(t) = A cos( t + ) Pendulums d 2 dt 2 R xCM
2 Mg = 0 cos( t + ) General physical pendulum MgR I Torsion pendulum I wire I Physics 207: Lecture 20, Pg 12
Energy in SHM 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 U system undergoing SMH will always be constant ! E This is not surprising since there are only conservative -A forces present, hence mechanical energy ought be conserved.
K U 0 A Physics 207: Lecture 20, Pg 13 x 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 looks something like this: U E
U x K U -A 0 A Physics 207: Lecture 20, Pg 14 x SHM and quadratic potentials Curvature reflects the spring constant or modulus (i.e., stress vs. strain or force vs. displacement)
U x Measuring modular proteins with an AFM See: http://hansmalab.physics.ucsb.edu Physics 207: Lecture 20, Pg 15 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 . 1.2 1 0.8 0.6 0.4
A 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 t Physics 207: Lecture 20, Pg 16 What about Friction? dx d 2x kx b m 2 dt dt d 2 x b dx k
x 0 2 dt m dt m We can guess at a new solution. x A exp( 2btm ) cos(t ) and now 02 k / m With, k m 2 b 2
o 2m b 2 m 2 Physics 207: Lecture 20, Pg 17 What about Friction? x(t ) A exp( 2btm ) cos(t )
if o b / 2 m What does this function look like? 1.2 1 0.8 0.6 0.4 A 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
t Physics 207: Lecture 20, Pg 18 Damped Simple Harmonic Motion o2 (b / 2m) 2 There are three mathematically distinct regimes o b / 2 m underdamped o b / 2m critically damped o b / 2 m overdamped Physics 207: Lecture 20, Pg 19 Physical properties of a globular protein (mass 100 kDa)
Mass Density Volume Radius Drag Coefficient 166 x 10-24 kg 1.38 x 103 kg / m3 120 nm3 3 nm 60 pN-sec / m Deformation of protein in a viscous fluid Physics 207: Lecture 20, Pg 20 Driven SHM with Resistance Apply a sinusoidal force, F0 cos (t), and now consider what A and b do, d 2 x b dx k F
x cos t 2 dt m dt m m A F0 / m b 2 ( ) ( ) m 2 b small 2 2 0 b middling b large
Physics 207: Lecture 20, Pg 21 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 20, Pg 22 Stick - Slip Friction How can a constant motion produce resonant vibrations? Examples: Violin Singing / Whistling Tacoma Narrows Bridge
Physics 207: Lecture 20, Pg 23 Dramatic example of resonance In 1940, a steady wind set up a torsional vibration in the Tacoma Narrows Bridge Physics 207: Lecture 20, Pg 24 A short clip In 1940, a steady wind sets up a torsional vibration in the Tacoma Narrows Bridge Physics 207: Lecture 20, Pg 25 Dramatic example of resonance Large scale torsion at the bridges natural frequency
Physics 207: Lecture 20, Pg 26 Dramatic example of resonance Eventually it collapsed Physics 207: Lecture 20, Pg 27 Lecture 20, Exercise 3 Resonant Motion Consider the following set of pendulums all attached to the same string A D B C
If I start bob D swinging which of the others will have the largest swing amplitude ? (A) (B) (C) Physics 207: Lecture 20, Pg 28 Waves (Chapter 16) Oscillations: Movement around one equilibrium point Waves: Look only at one point: oscillations But: changes in time and space (i.e., in 2 dimensions!) Physics 207: Lecture 20, Pg 29
What is a wave ? A definition of a wave: A wave is a traveling disturbance that transports energy but not matter. Examples: Sound waves (air moves back & forth) Stadium waves (people move up & down) Water waves (water moves up & down) Light waves (an oscillating electromagnetic field) Animation Physics 207: Lecture 20, Pg 30 Types of Waves Transverse: The mediums displacement is perpendicular to the direction the wave is moving. Water (more or less) String waves Longitudinal: The mediums displacement is in
the same direction as the wave is moving Sound Slinky Physics 207: Lecture 20, Pg 31 Wave Properties Wavelength: The distance between identical points on the wave. Amplitude: The maximum displacement A of a point on the wave. Wavelength A
Animation Physics 207: Lecture 20, Pg 32 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. v T Animation Physics 207: Lecture 20, Pg 33
Lecture 20, Exercise 4 Wave Motion The speed of sound in air is a bit over 300 m/s, and the speed of light in air is about 300,000,000 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 1,000,000 (B) About 0.000,001 (C) About 1000 Physics 207: Lecture 20, Pg 34 Wave Forms So far we have examined v
continuous waves waves that go on forever in each direction ! We can also have pulses caused by a brief disturbance of the medium: And pulse trains which are somewhere in between. v v Physics 207: Lecture 20, Pg 35 Lecture 20, Exercise 5 Wave Motion
A harmonic wave moving in the positive x direction can be described by the equation (The wave varies in space and time.) v = / T = f = () ( f) = / k and, by definition, > 0 y(x,t) = A cos ( (2 /) x - t ) = A cos (k x t ) Which of the following equation describes a harmonic wave moving in the negative x direction ? (A) y(x,t) = A sin (k x t ) (B) y(x,t) = A cos ( k x t ) (C) y(x,t) = A cos (k x t ) Physics 207: Lecture 20, Pg 36 Lecture 20, Exercise 6 Wave 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 t 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 + t Physics 207: Lecture 20, Pg 37 Waves on a string
What determines the speed of a wave ? Consider a pulse propagating along a string: v Snap a rope to see such a pulse How can you make it go faster ? Animation Physics 207: Lecture 20, Pg 38 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 F R Physics 207: Lecture 20, Pg 39 Waves on a string... So we find:
v Animation F v tension F mass per unit length 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. Physics 207: Lecture 20, Pg 40 Lecture 20, Recap Agenda: Chapter 15, Finish, Chapter 16, Begin Simple pendulum Physical pendulum Torsional pendulum Energy Damping Resonance Chapter 16, Traveling Waves Assignments: Problem Set 7 due Nov. 14, Tuesday 11:59 PM Problem Set 8 due Nov. 21, Tuesday 11:59 PM For Wednesday, Finish Chapter 16, Start Chapter 17 Physics 207: Lecture 20, Pg 41
Talents Unlimited Model (Schlichter, Palmer, & Palmer, 1993) A, think of many varied things you could do, Think more carefully about each alternative. Choose one alternative that you think is best, Give many varied reasons for your choice.
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