A Chapter 17 Oscillations
Chapter 17 Oscillations
17-1 Oscillating Systems Each day we encounter many kinds of oscillatory a motion, such as swinging pendulum of a clock, a person bouncing on a trampoline, a vibrating guitar string, and a mass on a spring e They have common properties 1. The particle oscillates back and forth about a equilibrium position. The time necessary for one complete cycle (a complete repetition of the motion)is called the period I
17-1 Oscillating Systems Each day we encounter many kinds of oscillatory motion, such as swinging pendulum of a clock, a person bouncing on a trampoline, a vibrating guitar string, and a mass on a spring. They have common properties: 1. The particle oscillates back and forth about a equilibrium position. The time necessary for one complete cycle (a complete repetition of the motion) is called the period T
2. No matter what the direction of the displacement the force always acts in a direction to restore the system to its equilibrium position. Such a force is called a" restoring force(恢复力) 3. The number of cycles per unit time is called the ●“ frequency (17-1) Unit: period(s) frequency (Hz, sI unit), 1 Hz =1 cycle/s 4. The magnitude of the maximum displacement from equilibrium is called the amplitude of the motion
2. No matter what the direction of the displacement, the force always acts in a direction to restore the system to its equilibrium position. Such a force is called a “restoring force(恢复力)”. 3. The number of cycles per unit time is called the “frequency” f. (17-1) Unit: period (s) frequency(Hz, SI unit), 1 Hz = 1 cycle/s T f 1 = 4. The magnitude of the maximum displacement from equilibrium is called the amplitude of the motion
17-2/3 The simple harmonic oscillator and its motion 1. Simple harmonic motion An oscillating system which can be described in e terms of sine and cosine functions is called a "simple harmonic oscillator" and its motion is called e simple harmonic motion". 2. Equation of motion of the simple harmonic oscillator Fig 17-5 shows a simple harmonic oscillator consisting of a spring of force constant K acting on
17-2/3 The simple harmonic oscillator and its motion 1. Simple harmonic motion An oscillating system which can be described in terms of sine and cosine functions is called a “simple harmonic oscillator” and its motion is called “simple harmonic motion”. 2. Equation of motion of the simple harmonic oscillator Fig 17-5 shows a simple harmonic oscillator, consisting of a spring of force constant K acting on
a body of mass m that slides on a frictionless horizontal e surface. the body moves in x direction Fg17-5 origin is chosen at here Relaxed state ∑F=-k 2 -kx= m k (17-4)
a body of mass m that slides on a frictionless horizontal surface. The body moves in x direction. Fig 17-5 x x m m F o o • • Relaxed state origin is chosen at here F kx x = − 2 2 dt d x ax = 2 2 dt d x − kx = m 0 2 2 + x = m k dt d x (17-4)
-a Eq(17-4)is called the equation of motion of the e simple harmonic oscillator".It is the basis of many complex oscillator problems 3. Find the solution of Eg(17-4) Rewrite Eq(17-4)as k (17-5) We write a tentative solution to Eq(17-5)as x=xm cos(at +o) (17-6)
Eq(17-4) is called the “equation of motion of the simple harmonic oscillator”. It is the basis of many complex oscillator problems. Rewrite Eq(17-4) as (17-5) We write a tentative solution to Eq(17-5) as (17-6) x m k dt d x ( ) 2 2 = − 3. Find the solution of Eq. (17-4) x = x cos(t +) m
We differentiate Eq(17-6) twice with respect to the Time 2 @xm cos(at+o) Putting this into Eq(17-5)we obtain k @xm cos(at +o)=--xm cos(@t +o) Therefore, if we choose the constant O such that k (17-7) e Eq(17-6)is in fact a solution of the equation of motion of a simple harmonic oscillator
We differentiate Eq(17-6) twice with respect to the Time. Putting this into Eq(17-5) we obtain Therefore, if we choose the constant such that (17-7) Eq(17-6) is in fact a solution of the equation of motion of a simple harmonic oscillator. cos( ) 2 2 2 = − x t + dt d x m cos( ) cos( ) 2 − + = − x t + m k x t m m m k = 2
a): If we increase the time by 2n in Eq(17-6), then 2丌 x=xm[cos a(t+)+o]=xm cos(at +o Therefore/ is the period of the motion T 2丌 2丌 nk (17-8) 11k T 2 Vm (179) The quantity a is called the angular frequency 2f
a) : If we increase the time by in Eq(17-6), then Therefore is the period of the motion T. 2 2 k m T 2 2 = = (17-8) (17-9) ) ] cos( ) 2 [cos ( x = xm t + + = xm t + m k T f 2 1 1 = = The quantity is called the angular frequency. = 2f
e b xm is the maximum value of displacement. We call it the amplitude of the motion c)at+o and o The quantity at +o is called phase of the motion is called" phase constant(常相位)” Xm and o are determined by the initial position and velocity of the particle. a is determined by the system
b) : is the maximum value of displacement. We call it the amplitude of the motion. c) and : The quantity is called phase of the motion. is called “phase constant （常相位)”. and are determined by the initial position and velocity of the particle. is determined by the system. m x m x t + t + m x
How to understand 8x=xm cos(at+o) xx-/图 0元 =0
How to understand ? x = x cos(t +) m 2 = = 0 = T x t m x o x −t 图 m − x