Chapter 13 periodic motion Collapse of the Tacoma Narrows suspension bridge in America in 1940 (p 415)

oscillation SHM Energy Damped oscillation Forced oscillation kinematics dynamics Kinematics equation Dynamic equation Circle of reference Energy Superposition of shm

Key terms: periodic motion / oscillation restoring force amplitude cycle period frequency angular frequency simple harmonic motion harmonic oscillator circle of reference phasor phase angle simple pendulum

physical pendulum Damping Damped oscillation Critical damping overdamping underdamping driving force forced oscillation natural angular frequency resonance chaotic motion chaos

§1 Dynamic equation 1)　dynamic equation Ideal model: A) spring mass system

B) The Simple Pendulum Small angle approximation sin

C) physical pendulum

Example: Tyrannosaurus rex and physical pendulum the walking speed of tyrannosaurus rex can be estimated from its leg length L and its stride length s

Conclusion:Equation of SHM Solution:

Example:A particle dropped down a hole that extends from one side of the earth, through its center, to the other side. Prove that the motion is SHM and find the period. Solution:

Example:An astronaut on a body mass measuring device (BMMD),designed for use on orbiting space vehicles,its purpose is to allow astronauts to measure their mass in the ‘weight-less’ condition in earth orbit. The BMMD is a spring mounted chair,if M is mass of astronaut and m effective mass of the BMMD,which also oscillate, show that

Example:the system is as follow,prove the block will oscillate in SHM Solution: We have

Take a derivative of y with respect to x Alternative solution (1) (2) Take a derivative of y with respect to x

§ 2 kinematic equation 2.1 Equation Solution:

2.2) 　the basic quantity——amplitude、period,phase A) Basic quantity: 1) Amplitude (A): the maximum magnitude of displacement from equilibrium. 2) Angular frequency(): Spring-mass: Simple pendulum: Caution: is not angular frequency rather than velocity .it depends on the system

Lag in phase Ahead in phase Out of phase In phase 3) Phase angle (  = t+ ): the status of the object. Lag in phase Ahead in phase Out of phase In phase

B) The formula to solve: A, ,  1)  is predetermined by the system. 2) A and  are determined by initial condition: if t=0, x=x0, v=v0 , Caution: Is fixed by initial condition

An object of mass 4kg is attached to a spring of k=100N. m-1 An object of mass 4kg is attached to a spring of k=100N.m-1. The object is given an initial velocity of v0=-5m.s-1 and an initial displacement of x0=1. Find the kinematics equation Solution:

Circle of reference method

Angle between OQ and axis-x Compare SHM with UCM x(+), v(-), a(-) In first quadrant  Angle between OQ and axis-x Phase Angular Velocity Angular Frequency  Projection Displacement x Radius UCM Amplitude SHM A

Example:Find the initial phase of the two oscillation x(m) x(cm) 0.8 6 3 o 1 t(s) o 1 t(s) 2 1 6 1 3 /3 o x o 3 x 4 2

From circle of reference SHM: x-t graph,find 0 , a , b , and the angular frequency Solution: x (m) From circle of reference 2 a b 1 t (s) -2

§ 3 Energy in SHM Potential energy: Total energy of the system: Kinetic energy: Potential energy: Total energy of the system:

Example:Spring mass system Example:Spring mass system.particle move from left to right, amplitude A1. when the block passes through its equilibrium position, a lump of putty dropped vertically on to the block and stick to it. Find the kinetic equation suppose t=0 when putty dropped on to the block Solution: k O X M

Example:A wheel is free to rotate about its fixed axle,a spring is attached to one of its spokes a distance r from axle.assuming that the wheel is a hoop of mass m and radius R,spring constant k. a) obtain the angular frequency of small oscillations of this system b) find angular frequency and how about r=R and r=0

§ 4. Superposition of SHM 4.1 mathematics method

x= x1+x2= Acos( t+ ) x x1 =A1cos( t+1 ) x2 =A2cos( t+2 ) A B) circle of reference x1 =A1cos( t+1 ) x2 =A2cos( t+2 ) ω M x= x1+x2= Acos( t+ ) M2 A2 A A2 (2-1) A1 M1  2 1 o x x1 x2 x

Example:x1=3cos(2t+)cm, x2=3cos(2t+/2)cm, find the superposition displacement of x1 and x2. Solution: Draw a circle of reference,

If damping force is relative small § 5 Damped Oscillations 5.1 Phenomena 5.2 equation If damping force is relative small

overdamping No oscillation underdamping Critical damping Amplitude decrease

:§6 Forced Oscillations drive an oscillator with a sinusoidally varying force: The steady-state solution is where 0=(k/m)½ is the natural frequency of the system. The amplitude has a large increase near 0 - resonance

Computation physics

Projectile motion with air resistance (case study:p147)

2. Tracing problem A plane moves in constant velocity due eastward,a missile trace it,suppose at anytime the missile direct to plane,speed is u,u>v,draw the path of missile (X,Y) (X0,h) v h (x,y) u x

(X,Y) y v h (x,y) u O x y(0)=0, x(0)=0 Y=h,X(0)=0

3. Planets trajectory Example:the orbits of satellites in the gravitational field v Solution: r ms me

We get: reference :《大学物理》吴锡珑 p 149

Solution1: Newton’s laws 3. Planets trajectory Solution1: Newton’s laws

Solution2: conservation of mechanical energy