#### Transcript PHYS 342: Modern Physics

Periodic Motion • We are surrounded by oscillations – motions that repeat themselves (periodic motion) – Grandfather clock pendulum, boats bobbing at anchor, oscillating guitar strings, pistons in car engines • Understanding periodic motion is essential for the study of waves, sound, alternating electric currents, light, etc. • An object in periodic motion experiences restoring forces or torques that bring it back toward an equilibrium position • Those same forces cause the object to “overshoot” the equilibrium position • Think of a block oscillating on a spring or a pendulum swinging back and forth past its equilibrium position Review of Springs • Classic example of periodic motion: – Spring exerts restoring force on block: Fs kx (Hooke’s Law) – k = spring constant (a measure of spring stiffness) – “Slinky” has k = 1 N/m; auto suspensions have k = 105 N/m – Movie of vertical spring: 1 2 • Elastic potential energy stored in spring: U el kx 2 – – – – Uel = 0 when x = 0 (spring relaxed) Uel is > 0 always We do not have freedom to pick where x = 0 Uel conserves mechanical energy CQ 1: The diagram below shows two different masses hung from identical Hooke’s law springs. The Hooke’s law constant k for the springs is equal to: A) B) C) D) 2 N/cm 5 N/cm 10 N/cm 20 N/cm Example Problem #13.67 A 3.00-kg object is fastened to a light spring, with the intervening cord passing over a pulley (see figure). The pulley is frictionless, and its inertia may be neglected. The object is released from rest when the spring is unstretched. If the object drops 10.0 cm before stopping, find (a) the spring constant of the spring and (b) the speed of the object when it is 5.00 cm below its starting point. Solution (details given in class): (a) 588 N/m (b) 0.700 m/s Periodic Motion • Sequence of “snapshots” of a simple oscillating system: • Frequency ( f ) = number of oscillations that are completed each second – Units of frequency = Hertz 1 Hz = 1 oscillation per second = 1 s–1 • Period = time for one complete oscillation (or cycle) = T = 1/f Simple Harmonic Motion • For the motion shown in the previous slide, a graph of the displacement x as a function of time looks like the following: Position vs Time (Look familiar?? See previous slide) • Written as a function: x(t) = Acos(wt + f) – A = Amplitude of the motion – (wt + f) = Phase of the motion – f = Phase constant (or phase angle) value depends on the displacement and velocity of particle at time t = 0 – w = Angular frequency = 2p/T = 2pf (measures rate of change of an angular quantity in rad/s) • Simple harmonic motion (SHM) = periodic motion is a sinusoidal function of time (represented by sine or cosine function) Simple Harmonic Motion • Affect of changes in the amplitude, period, and phase Fundamentals of Halliday, angle on curves of displacement vs. time: Physics, Resnick, and Walker, 6th ed. Red A > Blue A Red T < Blue T Red f < Blue f • The velocity of a particle moving with SHM is given by (from conservation of mechanical energy ½ kA2 = ½ mv2 + ½ kx2): k 2 2 v A m x • From Hooke’s Law coupled with Newton’s 2nd Law (–kx = ma), the acceleration of a particle moving with SHM is: a kx / m Simple Harmonic Motion • Since v and a both depend on x, they also are sinusoidal functions of time (see figure at right) • The relationship between displacement, velocity, and acceleration in SHM is demonstrated by the following: – When the magnitude of the displacement is greatest, the magnitude of the velocity is least and vice–versa – When displacement has its greatest positive value, acceleration has its greatest negative value, and vice– versa – When displacement = 0, acceleration = 0 Simple Harmonic Motion • From Hooke’s Law, we have another definition of SHM: – Motion executed by a particle of mass m subject to a force that is proportional to the displacement of the particle but opposite in sign • We can further analyze SHM by comparing it to uniform circular motion – For example, when a ball is attached to a turntable rotating with constant angular speed, the shadow of the ball moves back and forth with SHM • The angular frequency (w and period (T) are: – w used due to strong similarity k w 2 p f between SHM and circular motion m m T 2p k Energy in Simple Harmonic Motion • In Chapter 5 we saw that the total mechanical energy of a linear oscillator (mass on a spring) was conserved if the motion proceeded without friction • We can now see directly how both the kinetic and potential energies vary with time, yet the total mechanical energy remains constant in time E vs. t Fundamentals of Physics, Halliday, Resnick, and Walker, 6th ed. CQ 2: Interactive Example Problem: Mass on a Spring Which animation shows the correct graph of position vs. time for the ball? A) B) C) D) Animation 1 Animation 2 Animation 3 Animation 4 (Physlet Physics Problem #16.2, copyright Prentice–Hall publishing) CQ 3: Interactive Example Problem: Measuring Young Tarzan’s Mass What is Tarzan Jr.’s mass? A) B) C) D) 14.5 kg 41.4 kg 55.7 kg 130.2 kg (ActivPhysics online Problem #9.4, copyright Addison Wesley) The Simple Pendulum • A simple pendulum consists of a particle of mass m (bob) suspended from one end of an unstretchable, massless string of length L fixed at the other end • The component of gravity tangent to the path of the bob provides a restoring torque about the pivot point: Pendulum vs. Block-Spring t = –L(mg sinq) = Ia • If q is small ( 15°) then sinq q: a = –(mgL / I )q • This equation is the angular equivalent of the condition for SHM (a = –w2 x), so: (Note that T = period here!) w = (mgL / I )½ and T = 2p(I / mgL)½ • Since I = mL2 in this case: T 2p L g (independent of mass and amplitude!) CQ 4: Which of the following would most accurately demonstrate the kinetic energy of a pendulum? A) B) C) D) Figure A Figure B Figure C Figure D CQ 5: Interactive Example Problem: Risky Pendulum Walk At what constant speed must the person walk in order to move safely under the pendulum? A) B) C) D) 0.9 m/s 1.8 m/s 2.5 m/s 3.5 m/s (ActivPhysics online Problem #9.11, copyright Addison Wesley) Shock Absorbers • Shock absorbers provide a damping of the oscillations – A piston moves through a viscous fluid like oil – The piston has holes in it, which creates a (reduced) viscous force on the piston, regardless of the direction it moves (up or down) – Viscous force reduces amplitude of oscillations smoothly after car hits bump in road – When oil leaks out of the shock absorber, the damping is insufficient to prevent oscillations • Shock absorber is example of an underdamped oscillator (see also critically damped and overdamped) Wave Motion • The wave is another basic model used to describe the physical world (the particle is another example) • Any wave is characterized as some sort of “disturbance” that travels away from its source • In many cases, waves are result of oscillations – For example, sound waves produced by vibrating string • For now, we will concentrate on mechanical waves traveling through a material medium – For example: water, sound, seismic waves – The wave is the propagation of the disturbance: they do not carry the medium with it • Electromagnetic waves do not require a medium • All waves carry momentum and energy Types of Waves • A traveling wave is a disturbance (pulse) that travels along the medium with a definite speed • A transverse wave produces particles in the medium that move perpendicular to the motion of the wave pulse • A longitudinal wave produces particles that move parallel to the motion of the wave pulse • Both transverse and longitudinal waves can be represented by waveforms: 1–D snapshots at particular instant in time (transverse) (longitudinal) Types of Waves • In solids, both transverse and longitudinal waves can exist – Transverse waves result from shear disturbance – Longitudinal waves result from compressional disturbance • Only longitudinal waves propagate in fluids (they can be compressed but do not sustain shear stresses) – Transverse waves can travel along surface of liquid, though (due to gravity or surface tension) • Sound waves are longitudinal – Each small volume of air vibrates back and forth along direction of travel of the wave • Earthquakes generate both longitudinal (4–8 km/s P waves) and transverse (2–5 km/s S waves) seismic waves – Also surface waves which have both components Properties of Waves • Consider traveling waves on a continuous rope: y A = wave amplitude A x l = wavelength wave speed = v fl (of pattern) • For the particular case of a transverse wave on a stretched string (under tension): F v – F = tension (restoring force) m – m = mass per unit length (property of medium) • Multiple traveling waves can meet and pass through each other without being destroyed or altered – We can hear multiple voices in a crowded room • When multiple waves overlap, the wave in the overlap region is determined by the superposition principle Properties of Waves • Superposition principle: The overlap of 2 or more waves (having small amplitude) results in a wave that is a point-by-point summation of each individual wave (constructive interference) (destructive interference) Properties of Waves • Traveling waves can both reflect and transmit across a boundary between 2 media – Reflected wave pulse is inverted (not inverted) if wave reaches a boundary that is fixed (free to move) Reflection of Waves Wave Pulse One End Fixed Wave Pulse Sliding Support CQ 6: Waves A and B, pictured below, may or may not be in phase. If wave A and wave B are superimposed, the range of possible amplitudes for the resulting wave will be: A) B) C) D) from 0 cm to 3 cm. from 0 cm to 9 cm. from 3 cm to 6 cm. from 3 cm to 9 cm. Example Problem #13.59 A 2.65-kg power line running between two towers has a length of 38.0 m and is under a tension of 12.5 N. (a)What is the speed of a transverse pulse set up on the line? (b)If the tension in the line was unknown, describe a procedure a worker on the ground might use to estimate the tension. Partial solution (details given in class): (a) 13.4 m/s