Chapter 9 Oscillatory Motion

Main Points of chapter 9 The kinematics of simple harmonic motion Connection to circular motion The dynamics of simple harmonic motion s Energy Simple and physical pendulums Damped and driven harmonic motion

9-1 The Kinematics of Simple Harmonic Motion Any motion that repeats itself at regular intervals is called periodic motion Examples: circular motion, oscillatory motion We know that if we stretch a spring with a mass on the end and let it go, the mass will oscillate back and forth (if there is no friction). k m This oscillation is called Simple Harmonic Motion

The position of the object is angular frequency ω: determined by the inertia of the moving objects and the restoring force acting on it . SI: rad/s amplitude A: The maximum distance of displacement to the equilibrium point phase wt+f, phase angle (constant) f The value of A and f depend on the displacement and velocity of the particle at time t = 0 (the initial conditions)

Period T: the time for one complete oscillation (or cycle); Frequency f: number of oscillations that are completed each second.

The red curve differs from the blue curve (a) only in that its amplitude is greater (b) only in that its period is T´ = T/2 (c) only in that f = -p/4 rad rather than zero a phase difference

    T = 2/ A -      -    f

They are in phase They have a phase difference of p o A1 -A1 A2 - A2 x1 x2 T t x x o A1 -A1 A2 - A2 x1 x2 T t They are in phase They have a phase difference of p

Relations Among Position, Velocity, and Acceleration in Simple Harmonic Motion We can take derivatives to find velocity and acceleration: v(t) leads x(t) by p/2 v(t) is phase –shifted to the left from x(t) by p/2 x(t) lags behind v(t) by p/2 x(t) is phase –shifted to the right from v(t) by -p/2

a(t) is phase –shifted to the left from x(t) by p In SHM, the acceleration is proportional to the displacement but opposite in sign, and the two quantities are related by the square of the angular frequency.

9-2 A Connection to Circular Motion A reference particle P´ moving in a reference circle of radius A with steady angular velocity w . Its projection P on the x axis executes simple harmonic motion. Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs. demo

ACT A mass oscillates up & down on a spring ACT A mass oscillates up & down on a spring. Its position as a function of time is shown below. Write down the displacement as the function of time t (s) y(t)(cm) 4 2 1 y

9-3 Springs and Simple Harmonic Motion The block–spring system forms a linear simple harmonic oscillator k x m F = -kx a Hooke’s law Combining with Newton’s second law a differential equation for x(t) Simple harmonic motion is the 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 ( a restoring force).

The period of the motion is independent of the amplitude Solution The period of the motion is independent of the amplitude The initial conditions t=0；x=x0，v=v0

X=0 X=A X=-A X=A; v=0; a=-amax X=0; v=-vmax; a=0 X=-A; v=0; a=amax X=0; v=vmax; a=0 X=A; v=0; a=-amax

Another solution is ok is equivalent to where

Example A block whose mass m is 680 g is fastened to a spring whose spring constant k is 65 N/m. The block is pulled a distance x = 11 cm from its equilibrium position at x = 0 on a frictionless surface and released from rest at t = 0. (a)  What are the angular frequency, the frequency, and the period of the resulting motion? (b)  What is the amplitude of the oscillation? (c)  What is the maximum speed vm of the oscillating block, and where is the block when it occurs? (d)  What is the magnitude am of the maximum acceleration of the block? (e)  What is the phase constant f for the motion? (f)  What is the displacement function x(t) for the spring–block system?

Solution At equilibrium point

Correct phase constant is1550 Example At t = 0, the displacement x(0) of the block in a linear oscillator is -8.50 cm. The block's velocity v(0) then is -0.920 m/s, and its acceleration a(0) is +47.0 m/s2. (a)  What is the angular frequency w of this system? (b)  What are the phase constant f and amplitude A? Solution 155 -25 Correct phase constant is1550

Additional Constant Forces Solution Simple harmonic motion with the same frequency, but equilibrium point is shifted from x=0 to x=x1

Vertical Springs Choose the origin at equilibrium position Fs mg Solution Simple harmonic motion with equilibrium point at y=0

ACTA mass hanging from a vertical spring is lifted a distance d above equilibrium and released at t = 0. Which of the following describes its velocity and acceleration as a function of time? (a) v(t) = -vmax sin(wt) a(t) = -amax cos(wt) (b) v(t) = vmax sin(wt) a(t) = amax cos(wt) k y (c) v(t) = vmax cos(wt) a(t) = -amax cos(wt) d t = 0 m (both vmax and amax are positive numbers)

9-4 Energy and Simple Harmonic Motion This is not surprising since there are only conservative forces present, hence the total energy is conserved.

(a)Potential energy U(t), kinetic energy K(t), and mechanical energy E as functions of time t for a linear harmonic oscillator. They are all positive. U(t) and K(t) peak twice during every period (b)Potential energy U(x), kinetic energy K(x), and mechanical energy E as functions of position x for a linear harmonic oscillator with amplitude xm. For x = 0 the energy is all kinetic, and for x = ±xm it is all potential. The mechanical energy is conserved

Note The potential energy and the kinetic energy peak twice during every period The mechanical energy is conserved for a linear harmonic oscillator The dependence of energy on the square of the amplitude is typical of Simple Harmonic Motion

ACT In Case 1 a mass on a spring oscillates back and forth ACT In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation is the same as in Case 1. In which case is the maximum potential energy of the mass and spring the biggest? A. Case 1 B. Case 2 C. Same Look at time of maximum displacement x = A Energy = ½ k A2 + 0 Same for both!

It’s Not Just About Springs Besides springs, there are many other systems that exhibit simple harmonic motion. Here are some examples:

Almost all systems that are in stable equilibrium exhibit simple harmonic motion when they depart slightly from their equilibrium position For example, the potential between H atoms in an H2 molecule looks something like this: U x

since x0 is minimum of potential If we do a Taylor expansion of this function about the minimum, we find that for small displacements, the potential is quadratic: since x0 is minimum of potential U x x0 U x  then Restoring force

Identifying SHM c, c’ positive constant

He gets back 84 minutes later, at 1:24 p.m. Transport Tunnel A straight tunnel with a frictionless interior is dug through the Earth. A student jumps into the hole at noon. What time does he get back? x g = 9.81 m/s2 RE = 6.38 x 106 m He gets back 84 minutes later, at 1:24 p.m.

Strange but true: The period of oscillation does not depend on the length of the tunnel. Any straight tunnel gives the same answer, as long as it is frictionless and the density of the Earth is constant. Another strange but true fact: An object orbiting the earth near the surface will have a period of the same length as that of the transport tunnel. g = 2R 9.81 = 2 6.38(10)6 m  = .00124 s-1 so T = = 5067 s 84 min

Example The potential energy of a diatomic molecule whose two atoms have the same mass, m, and are separated by a distance r is given by the formula where A and e are positive constants. (a) Find the equilibrium separation of the two atoms. (b) Show that if the atoms are slightly displaced, then they will undergo simple harmonic motion about the equilibrium position. Calculate the angular frequency of the harmonic motion. Solution (a) The equilibrium separation occurs where the potential energy is a minimum, so we set

(b) We do a Taylor expansion of this U(r) function about the equilibrium separation This has the form of the elastic potential energy, so the motion will be simple harmonic The spring constant The reduced mass The angular frequency is

9-5 The Simple Pendulum a simple pendulum consists of a pointlike mass m (called the bob of the pendulum) suspended from one end of an unstretchable, massless string of length l that is fixed at the other end T mg If q is small

Solution with The motion of a simple pendulum swinging through only small angles is approximately SHM. The period of small-amplitude pendulum is independent of the amplitude --- the pendulum clock The horizontal displacement

The energy of a simple pendulum: For small q The total energy is conserved

ACT 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: (a) T1 = T2 (b) T1 > T2 (c) T1 < T2 You make a pendulum shorter, it oscillates faster (smaller period)

“Effective g” is larger when accelerating upward ACT A pendulum is hanging vertically from the ceiling of an elevator. Initially the elevator is at rest and the period of the pendulum is T. Now the pendulum accelerates upward. The period of the pendulum will now be 1. greater than T 2. equal to T 3. less than T “Effective g” is larger when accelerating upward (you feel heavier)

9-6 More About Pendulums The Physical Pendulum Any object, if suspended and then displaced so the gravitational force does no run through the center of mass, can oscillate due to the torque. If q is small T Mg

with Solution The period of a physical pendulum is independent of it’s total mass—only how the mass is distributed matters For a simple pendulum

ACT A pendulum is made by hanging a thin hoola-hoop of diameter D on a small nail. What is the angular frequency of oscillation of the hoop for small displacements? (ICM = mR2 for a hoop) (a) (b) (c) pivot (nail) D

Example In Figure below, a meter stick swings about a pivot point at one end, at distance h from its center of mass. (a)  What is its period of oscillation T? (b)  What is the distance L0 between the pivot point O of the stick and the center of oscillation of the stick? Solution

Example In Figure below , a penguin (obviously skilled in aquatic sports) dives from a uniform board that is hinged at the left and attached to a spring at the right. The board has length L = 2.0 m and mass m = 12 kg; the spring constant k is 1300 N/m. When the penguin dives, it leaves the board and spring oscillating with a small amplitude. Assume that the board is stiff enough not to bend, and find the period T of the oscillations.

y Solution Choose the equilibrium position as the origin mg F O T is independent of the board’s length

Example A block of mass m is attached to a spring of constant k through a disk of mass M which is free to rotate about its fixed axis. Find the period of small oscillations M Solution Choose the equilibrium position as the origin T’ T T’ T o x k m mg

9-7 Damped Harmonic Motion A pendulum does not go on swinging forever. Energy is gradually lost (because of air resistance) and the oscillations die away. This effect is called damping. Look at drag force that is proportional to velocity; b is the damping coefficient: Then the equation of motion is:

Damping factor If a is small Solution Natural frequency

The larger the value of t ,the slower the exponential Life time The larger the value of t ,the slower the exponential

As b increases, w’ decreases When Some systems have so much damping that no real oscillations occur. The minimum damping needed for this is called critical damping

critical damping Over (heavy) damping (light) damping x o critical damping heavy damping Over (heavy) damping light damping (light) damping The time of the critical damping takes for the displacement to settle to zero is a minimum

Example For the damped oscillator: m = 250 g, k = 85 N/m, and b = 70 g/s. (a)  What is the period of the motion? (b)  How long does it take for the amplitude of the damped oscillations to drop to half its initial value? (c)  How long does it take for the mechanical energy to drop to one-half its initial value?

Solution

9-8 Driven Harmonic Motion In damped harmonic motion, a mechanism such as friction dissipates or reduces the energy of an oscillating system, with the result that the amplitude of the motion decreases in time. Now, applying a driving force Equation of motion becomes: solution

After long times

The condition for the maximum of A If b=0 In the absence of damping, if the frequency of the force matches the natural frequency of the system , then the amplitude of the oscillation reaches a maximum. This effect is called resonance

For small b, the total width at half maximum peak becomes broader as b increases:

The role played by the frequency of a driving force is a critical one The role played by the frequency of a driving force is a critical one. The matching of this frequency with a natural frequency of vibration allows even a relatively weak force to produce a large amplitude vibration Examples Breaking glass The collapse of the Tacoma Narrows Bridge Turbulent winds set up standing waves in the Tacoma Narrows suspension bridge leading to its collapse on November 7, 1940, just four months after it had been opened for traffic demo

Summary of chapter 9 Simple Harmonic Motion Springs

Summary of chapter 9 Cont. Energy Simple and physical pendulums

Summary of chapter 9 Cont. Damped and driven harmonic motion If b=0 resonance