Chapter 14 Periodic Motion PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson
Exam 3 results Class Average - 57 (Approximate grade boundaries) A 85-100 B 68-84 C 46-67 D 3-45 F 0-31 1 10 Exam 3 8 Frequency 6 4 Frequency 0 0 8 16 4 3 40 48 56 64 7 80 88 96 raw score
Goals for Chapter 14 To describe oscillations in terms of amplitude, period, frequency and angular frequency To do calculations with simple harmonic motion To analyze simple harmonic motion using energy To apply the ideas of simple harmonic motion to different physical situations To analyze the motion of a simple pendulum To examine the characteristics of a physical pendulum To explore how oscillations die out To learn how a driving i force can cause resonance
SHM differential equation: F=ma d x F = kx, F = max = m dt d x kx = m dt x( t) = Asin( ω t) + B cos( ωt) = C sin( ωt + φ)
Angular SHM A coil spring (see Figure 14.19 19 below) exerts a restoring torque τ z = κθ, where κ is called the torsion constant of the spring. The result is angular simple harmonic motion.
d θ t C t B t A t k dt d I φ ω ω ω θ θ θ τ κθ τ + = + = = = = ) cos( ) cos( ) sin( ) (, I κ ω φ = ) ( ) ( ) ( ) ( I
Vibrations of molecules Figure 14.0 shows two atoms having centers a distance r apart, with the equilibrium point at r = R 0. If they are displaced a small distance x from equilibrium, the restoring force is F r = (7U 0 /R 0 )x, so k = 7U 0 /R 0 and the motion is SHM.
Binomial Expansion Binomial Expansion for u <1 n n( n 1) n( n 1)( n ) ( 1+ u) = 1+ nu + u + u! 3! 3 +
The simple pendulum A simple pendulum consists of a point mass (the bob) suspended by a massless, unstretchable string. If the pendulum swings with a small amplitude θ with the vertical, its motion is simple harmonic. (See Figure 14.1 at the right.)
Simple Pendulum d x Fθ = mg sinθ, F = max = m = mg dt mg d x x = m L dt x( t) = Asin( ω t) + B cos( ωt) = C cos( ωt + φ) mg k g ω = = L = m m L x L
The physical pendulum A physical pendulum is any real pendulum that uses an extended dbody instead of a point-mass bob. For small amplitudes, its motion is simple harmonic. (See Figure 14.3 at the right.)
Physical Pendulum d θ τ z = ( mg ) d sin θ = ( mgd ) θ, τ = I α = I = ( mgd ) θ dt d θ ( mgd) θ = I dt θ ( t) = Asin( ωt) + B cos( ωt) = C cos( ωt + φ) mgd ω = = I
Harmonic oscillation with damping F x = kx bv dx 0 = kx + b + dt x x, F = d x m dt ( b m) t ( t) = Ae cos( ω' t ma x + φ) = m d x dt = kx b dx dt ω ' = k m b 4mm
Damped oscillations Real-world systems have some dissipative forces that decrease the amplitude. The decrease in amplitude is called damping and the motion is called damped oscillation. Figure 14.6 at the right illustrates an oscillator with a small amount of damping. The mechanical energy of a damped oscillator decreases continuously.
Forced oscillations and resonance A forced oscillation occurs if a driving force acts on an oscillator. Resonance occurs if the frequency of the driving force is near the natural frequency of the system. (See Figure 14.8 below.)
Tacoma Narrows Bridge collapse An Example of driven harmonic motion with resonance. https://www.youtube.com/watch?v=xox9bvsu7ok
Chapter 15 Mechanical Waves PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson
Goals for Chapter 15 To study the properties and varieties of mechanical waves To relate the speed, frequency, and wavelength of periodic waves To interpret periodic waves mathematically To calculate the speed of a wave on a string To calculate the energy of mechanical waves To understand dthe interference of mechanical waves To analyze standing waves on a string To investigate the sound produced by stringed instruments
Introduction Earthquake waves carry enormous power as they travel through the earth. Other types of mechanical waves, such as sound waves or the vibration of the strings of a piano, carry far less energy. Overlapping waves interfere, which helps us understand musical instruments.
Types of mechanical waves A mechanical wave is a disturbance traveling through a medium. Figure 15.1 below illustrates transverse waves and longitudinal waves.
The Wave Equation
Solutions to the wave equation November 13 01 Physics 08 3
Periodic transverse waves For the transverse waves shown here in Figures 15.3 and 15.4, the particles move up and down, but the wave moves to the right.
Periodic longitudinal waves For the longitudinal waves shown here in Figures 15.6 and 15.7, the particles oscillate back and forth along the same direction that the wave moves. Follow Example 15.1.
The solutions to the Wave Equation what do they look like??? November 13 01 Physics 08 6
November 13 01 Physics 08 7
November 13 01 Physics 08 8
The Wave Equation for y(x,t) x y = μ T t y the solutions for y(x, t) are f(t-x/v) and g(t + x/v) with v = T μ
Choosing our favorite solution t kx A t x y ω = ) cos( ), ( x with kx t v x t ω = f f k ω π d where k f v f k ω λ π ω λ π = = = = and,