Chapter 13 Periodic Motion
PowerPoint ® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman
Lectures by James Pazun
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Goals for Chapter 13 • To outline periodic motion • To quantify simple harmonic motion • To explore the energy in simple harmonic motion • To consider angular simple harmonic motion • To study the simple pendulum • To examine the physical pendulum • To explore damped oscillations • To consider driven oscillations and resonance
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Homework Read 476 to 486 3, 5, 9, 15 Read 486 to 494 25, 27 Read 495 to 505 (Skip Molecular Vibration) 31, 35, 45, 49, 51, 61, 69
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Introduction • If you look to the right, you’ll see a time-lapse photograph of a simple pendulum. It’s far from simple, but it is a great example of the regular oscillatory motion we’re about to study.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Describing oscillations
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Describing oscillations
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Describing oscillations
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley In Oscillations . . . Displacement : the distance the body is from its equilibrium position. Restoring Force : the force that tends to restore the body to its equilibrium position. Amplitude : the maximum displacement from equilibrium Period : (T) the time for one cycle. Frequency : f( ) the number of cycles per unit of time. Angular Frequency : ( ω) 2 π times the frequency. = 1 = 1 (13.1) f T T f ω = π = 2π 2 f T (13.2)
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 13.1 Summary PERIODIC MOTION: PERIODIC MOTION IS MOTION THAT REPEATS ITSELF IN A DEFINITE CYCLE. IT OCCURS WHENEVER A BODY HAS A STABLE EQUILIBRIUM POSITION AND A RESTORING FORCE THAT ACTS WHEN IT IS DISPLACED FROM EQUILIBRIUM. PERIOD T IS THE TIME FOR ONE CYCLE. FREQUENCY IS THE NUMBER OF CYCLES PER UNIT TIME. ANGULAR FREQUENCY IS TIMES THE FREQUENCY.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Period, Frequency, and Angular Frequency An ultrasonic transducer (loudspeaker) used for medical diagnosis oscillates at a frequency of 6.7 MHz = 6.7 x 10 6 Hz. How much time does each oscillation take, and what is the angular frequency?
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Simple harmonic motion • An ideal spring responds to stretch and compression linearly, obeying Hooke’s Law.
• For a real spring, Hookes’ Law is a good approximation.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Definition of Simple Harmonic Motion The easiest way to view SHM is with a mass on a spring. It obeys Hooke’s Law: Fx = -kx (13.3) When the restoring force is directly proportional to the displacement from equilibrium, as given by Eq. (13.3), the oscillation is called simple harmonic motion abbreviated SHM.
2 2 The acceleration ax = d x/dt = Fx/m of a body in SHM is given by: d 2 x k a = = − x (13.4) x dt 2 m
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Simple harmonic motion viewed as a projection If you illuminate uniform circular motion (say by shining a flashlight on a ball placed on a rotating lazy-Susan spice rack), the shadow projection that will be cast will be undergoing simple harmonic motion, like in figures 13.4 below. The SHM and circular motion analogy should hold true if the amplitude, A, equals the disk radius and the angular frequency equals the disk’s angular velocity.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Comparing acceleration of shadow at P to SHM The circle the ball moves in so that its shadow matches SHM is called the circle of reference as we call Q the reference point. At a certain time, t, the vector OQ makes an angle,θ, with the horizontal. As the point Q moves around the reference circle with a constant angular speed ω, the vector rotates with the same speed, called a phasor . The x-component of the phasor at time t is just the x-coordinate of the point Q: x = A cos θ This position also refers to the position of the shadow at P, which is the projection of Q onto the x-axis. The acceleration of P is just the x-component of the acceleration of Q.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Comparing acceleration of shadow at P to SHM ω2 From centripetal acceleration, the acceleration of Q is a Q = A, directed toward O, as shown below. θ The x-component is just a x = -aQ cos ω2 θ Combining these two equations gives a x = - A cos θ ω2 From before x = A cos , so a x = - x This gives exactly what is to be expected from equation 13.4, if the angular speed of the reference point Q is related to the force constant k and mass m of the oscillating body by k k ω 2 = or ω = m m
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Equations on 481 Both omegas, one for angular velocity and one for angular frequency are the same thing. ω = k m ω 1 k f = = 2π 2π m 1 2π m T = = = 2π f ω k
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Angular frequency, frequency, and period in SHM A spring is mounted horizontally, with its left end held stationary. By attaching a spring balance to the free end and pulling toward the right, we determine that the stretching force is proportional to the displacement and that a force of 6.0 N causes a displacement of 0.030 m. We remove the spring balance and attach a 0.50 kg body to the end, pull it a distance of 0.020 m, release it, and watch it oscillate. Find the: Force constant of the spring. Angular frequency. Frequency. Period of the oscillation.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley X versus t for SHO
If at time t = 0 the phasor OQ makes an angle Φ with the x-axis then at any later time t the angle is ( ωt + Φ). The position of the oscillating object can be found by the equation: x = Acos( ωt +φ) (13.13)
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Variations on a theme—write equations on board Changing either k or m changes the period of oscillation.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley SHM phase and motion • SHM can occur with various phase angles , the angle at which t = 0.
• For a given phase we can examine position, velocity, and acceleration.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley SHM equations x = Acos( ωt +φ) When t = 0: = −ω ω +φ vx Asin( t ) = φ xo Acos = −ω 2 ω +φ ax Acos( t ) v = −ωAsinφ ox Do these equations fit with equation 13.4: a = −ω 2 Acosφ 2 ox = d x = − k ax 2 x 2 dt m vo A = x2 + x o ω 2
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Watch variables change for a glider example Keep problems solving strategy on 485 in mind when solving the following problem. Let’s return to the system of mass and horizontal spring we considered previously, with k = 200 N/m and m = 0.50 kg. This time we give the body an initial displacement of +0.015 m and an initial velocity of +0.40 m/s. a) Find the period, amplitude, and phase angle of the motions. b) Write equations for the displacement, velocity, and accelerations as functions of time.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 13.2 Summary Simple harmonic motion: If the restoring force in periodic motion is directly proportional to the displacement x, the motion is called simple harmonic motion (SHM). In many cases this condition is satisfied if the displacement from equilibrium is small. The angular frequency, frequency, and period in SHM do not depend on the amplitude, but only on the mass m and force constant k. The displacement, velocity, and acceleration in SHM are sinusoidal functions of time; the amplitude A and phase angle of the oscillation are determined by the initial position and velocity of the body. (See Examples 13.2, 13.3, 13.6, and 13.7.)
Homework: Read pages 486 to 492 On page 507: 3, 5, 9, 15
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy in SHM
• Energy is conserved during SHM and the forms (potential and kinetic) interconvert as the position of the object in motion changes. 12+ 1 2 = 1 2 (13.21) E= 2mvx 2 kx 2 kA • Page 487 goes through a proof of this relationship. • We can rewrite equation 13.21 for a velocity at any displacement: k v = ±A2 − x 2 x m (13.22) • Using this equation we can find the maximum velocity (when displacement is zero): k k (13.23) v =AAA2 = = ω max m m Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy in SHM
• You can use equation 13.22 to calculate the velocity at each point in the figure below.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy in SHM II • Figure 13.13 shows the interconversion of kinetic and potential energy with an energy versus position graphic.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy in SHM II • Refer to Problem-Solving Strategy on page 488 while solving the following problem. • In the oscillation described in the previous example, k = 200 N/m, m = 0.50 kg, and the oscillating mass is released from rest at x = 0.020 m. a) Find the maximum and minimum velocities attained by the oscillating body. b) Compute the maximum acceleration. c) Determine the velocity and acceleration when the body has moved halfway to the center from its original position. d) Find the total energy, potential energy, and kinetic energy at this position. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy and momentum are related in SHM A block with mass M attached to a horizontal spring with force constant k is moving with simple harmonic motion having amplitude A1. At the instant when the block passes through its equilibrium position, a lump of putty with mass m is dropped vertically onto the block from a very small height and sticks to it. Find the new amplitude and period. Find the amplitude and period of the putty is dropped onto the block at its maximum displacement A1.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 13.3 Summary Energy in simple harmonic motion: Energy is conserved in SHM. The total energy can be expressed in terms of the force constant k and amplitude A. (See Examples 13.4 and 13.5.) 12+ 1 2 = 1 2 E= 2mvx 2 kx 2 kA
Homework: Read pages 495 to 505 (Skip molecular vibration) On page 508: 25 and 27
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Applications of SHM When we have a mass hanging from a spring, or a mass compressing a spring, the new equilibrium position is when the spring’s restoring force balances out the force of gravity on the mass: k∆l = mg
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Vertical SHM We can set x = 0 at this new equilibrium position and the equation for the new net force is:
Fnet = k(∆l – x) + (–mg ) = –kx
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Vertical SHM in an old car The shock absorbers in an old car with mass 1000 kg are completely worn out. When a 980 N person climbs slowly into the car to its center of gravity, the car sinks 2.8 cm. When the car, with the person aboard, hits a bump, the car starts oscillating up and down in SHM. Model the car and person as a single body on a single spring, and find the period and frequency of the oscillation.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Angular SHM Watches keep time based on regular oscillations of a balance wheel initially set in motion by a spring. Figure 13.18 illustrates the delicate inner mechanism of a clock or watch. The wheel has an moment of inertia I about its axis. A coil spring exerts a restoring torque τ that is proportional to the angular displacement θ from the equilibrium position. τ κθ κ We write z = - where is called the torsion constant. Using the rotational analog to Newton’s θ α τ = = 2 2 second law: ∑ zI z Id dt 2 -κθ = I α or d θ = − κ θ dt 2 I This is very similar to our definition of SHM.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Angular SHM Angular simple harmonic motion: In angular SHM, the κ frequency and angular ω = frequency are related to the I moment of inertia I and the torsion constant . 1 κ f = 2π I ω φθ = Θcos()t +
Homework on 508: 31 and 37
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley The Simple Pendulum A simple pendulum consists of a point mass m at the end of a massless string of length L. Its motion is approximately simple harmonic for sufficiently small amplitude; the angular frequency, frequency, and period then depend only on g and L, not on the mass or amplitude. (See Figure 13.20.)
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley The Simple Pendulum Equations
ω =k = mg L = g m m L ω 1 g f = = 2 ππ 2 L 2π 1 L T = = = 2π ω f g
The equations on the left are only true for small amplitude oscillations, as shown in the graph on the right.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley A simple pendulum Find the frequency of a simple pendulum 1.000 m long at a location where g = 9.800 m/s 2.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Physical Pendulums Like a simple pendulum, the physical pendulum’s angular frequency depends on the distance from the axis of rotation to the mass. With the physical pendulum, the mass is spread out. It is not a point mass. The frequency (and hence period) depends on the distance between the axis of rotation and the center of gravity, d, and the moment of inertia of the body, I. mgd I ω = T = 2π I mgd
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley The physical pendulum • A physical pendulum is any real pendulum that uses an extended body in motion. Figure 13.22 illustrates a physical pendulum and you may wish to refer to it as you consider the next example. • Suppose the object to the right is a uniform rod with length L pivoted at one end. Find the period of its motion.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Dinosaurs, long tails, and the physical pendulum All walking animals, including humans, have a natural walking pace, a number of steps per minute that is more comfortable than a faster or slower pace. Suppose this natural pace is equal to the period of the leg, viewed as a uniform rod pivoted at the hip joint. a) How does the natural walking pace depend on the length L of the leg, measured from hip to foot? b) Fossil evidence shows that Tyrannosaurus rex, had a leg length L = 3.1 m and a stride length of 4 m. Estimate the walking speed of Tyrannosaurus rex.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Damped oscillations Consider an oscillating system with a damping force due to friction: F x = -bv x where v x = dx/dt and b is constant that describes the strength of the damping force. The net force would include this damping force and the oscillatory restoring force: -kx – bv x
From Newton’s second law: -kx – bv x = ma x 2 or −kx − bdx = m dx dt dt 2 Solving with differential calculus yields: − x= Ae(b2 m) t cos (ω φ′ t + )
2 ω′ =k − b m4 m2 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Kinds of Damping − The amplitude with x= Ae(b2 m) t cos (ω φ′ t + ) damped oscillations decreases with time. k b2 ω ω′ = − The angular frequency ’ 2 is zero when b= 2 km m4 m This situation is called critically damped . An oscillation is overdamped when b> 2 km and underdamped when b< 2 km
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Damped oscillations Classify either line as overdamped, underdamped, or critically damped.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Damped oscillations II
When a force F x = -bv x proportional to velocity is added to a simple harmonic oscillator, the motion is called a damped oscillation. If b< 2 km (called underdamping), the system oscillates with a decaying amplitude and an angular frequency ω’ that is lower than it would be without damping. If b= 2 km (called critical damping) or b > 2 km (called overdamping), when the system is displaced it returns to equilibrium without oscillating.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Forced (driven) oscillations and resonance A force applied “in synch” F with a motion already in A = max progress will resonate and 2 2 2 2 add energy to the ()k− m ωω + b d d oscillation. A singer can shatter a glass with a pure tone in tune with the natural “ring” of a thin wine glass.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Forced (driven) oscillations and resonance II
• The Tacoma Narrows Bridge suffered spectacular structural failure after absorbing too much resonant energy (refer to Figure 13.28).
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Homework On Page 509: 45, 49, 51, 61, 69
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Homework Read 476 to 486 3, 5, 9, 15 Read 486 to 494 25, 27 Read 495 to 505 (Skip Molecular Vibration) 31, 35, 45, 49, 51, 61, 69
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley