Analytical Physics 1B Lecture 5: Physical Pendulums and Introduction to Mechanical Waves Sang-Wook Cheong Friday, February 16 th, 2017
Two Exam 1 Questions with errors Correct answer: L = r X p = (2000 m) * (250 m/s * 4000 kg) sin(30 deg = 10 9 kg m 2 /s None of the listed answers were the correct answer. Everyone gets the point for this one
Disk: v= 4gh/3 Hoop: v= gh The disk will always come first, it does not matter what the masses and radii of the hoop or disk are. Solve for v at the bottom, it does not depend on m or R. Correct answer: This is not possible (if they both start from rest) None of the listed answers were the correct answer. Everyone gets the point for this one We will add an extra 2% to everyone s grade to compensate for time spent.
Simple Pendulums Consider the torque about the point where the string is hanging when θ is small sin(θ) θ what is I? Remember that I is defined about the axis of rotation Therefore, Notice that α is always in the opposite direction as θ, so This is SHM! 4
Simple Pendulums Given the equation of motion, let s derive θ as a function of time Our guess is: Plugging this in to the above yields: which means 5
Physical Pendulums Physical pendulums are like simple pendulums but where the mass is extended in space similar calculation as before: approximate sin(θ) θ but can t simplify expression for I! Don t forget: I is with respect to the pivot point! 6
Damped Springs Suppose we have a spring, but there is a damping force proportional to its velocity: the solution to this differential equation (when b is small) is and What does this look like? 7
Damped Springs Notice that the amplitude is not a constant (it is exponentially decreasing) Also the frequency is somewhat smaller What happens when b is big? Overdamping. Critical damping: b=2 km 8
Forced Oscillations If we apply a periodically varying force to an oscillator where the frequency of the varying force is close to the natural frequency of the oscillator ( k/m), then we can get resonance behavior b>=2 km 9
Forced Oscillations Virtual Demo https://phet.colorado.edu/sims/resonance/resonance_en.html 10
Mechanical Waves Waves resulting from the physical displacement of part of the medium from equilibrium These are to be distinguished from say electromagnetic waves or gravitational waves These wave can be transverse or longitudinal transverse waves particles undergo displacements in a direction perpendicular to the direction of wave motion longitudinal waves particles undergo displacements in a direction parallel to the direction of wave motion 11
Periodic Transverse Waves Any periodic wave can be expressed as the sum of many sinusoidal waves The waves have an amplitude, a frequency, a wavelength, and a speed with which they propagate the wave advances one wavelength in a time interval of one period wavelength 12
Period Transverse Waves 13
Ranking Frequencies The drawings represent snapshots taken of waves traveling to the right along strings. The grids shown in the background are identical. The waves all have the same speed, but their amplitudes and wavelengths vary. Rank the frequency of the waves. A. B > A > D > C B. C = D > A > B C. B > A > D = C D. C > D > A > B E. They are all the same. 14
Ranking Speeds The drawings represent snapshots taken of waves traveling to the right along strings. The grids shown in the background are identical. The waves all have the same frequency, but their amplitudes and wavelengths vary. Rank the speed of the waves on the string. A. C > D > A > B B. C = D > A > B C. B > A > D > C D. B > A > D = C E. They are all the same. 15
Mathematical Description of a Wave Consider a wave propagating on a string extended in the x direction The position in y of any point on the string depends on its location in x and what time it is i.e. the wave function depends on x and t: y(x,t) Consider the particle at point B it starts at the maximum value of y at t=0, and returns to it after t=t: where 16
Mathematical Description of a Wave We solved for a single point x=b, but how does y depend in general on x? Notice that this wave is moving left to right As we know, the wave travels with a speed λf, which means that: the motion of point C at time t is the same as the motion of point A at the earlier time t-x/v, where x is the distance between A and C 17
Mathematical Description of a Wave We can simplify the expression so that { } where k a quantity we define as the wave number For a wave traveling right to left: 18
Definitions 19
Principle of Superposition When two waves overlap, the actual displacement on the string is given by adding the together the expect locations of the waves had the other wave not been present Waves can interact constructively or destructively, resulting in a potentially larger or smaller final wave than either of the two input waves 20
Colliding Waves Rectangular transverse wave pulses are traveling toward each other along a string. The grids shown in the background are identical, and the pulses vary in height and length. The pulses will meet and interact soon after they are in the positions shown. Rank the maximum amplitude of the string at the instant that the positions of the centers of the two pulses coincide. A. A > B > C > D B. A = C > B = D C. C > D > A > B D. A > C > B > D E. They are all the same. 21
LIGO https://www.youtube.com/watch?v=s06_jrk939i 22
One more Example of SHM Consider an object freely falling through the center of the Earth It turns out that the object at radius r feels the force of gravity only from the part of the Earth within a spherical region of radius r Moreover, it feels it as if it were a point mass concentrated at the center of the earth 23
One more Example of SHM Let s assume that the Earth has a uniform density Therefore the mass contained within a sphere of radius r is So the force felt by an object at a given radius must be This is SHM! 24
One more Example of SHM We can calculate the period of the motion: Plugging in the constants, we get that the period is 84 minutes Therefore, it would take a period 42 minutes to freely fall from one side of the earth to the other, just using the force of gravity 25