General Physics I. Lecture 12: Applications of Oscillatory Motion. Prof. WAN, Xin ( 万歆 )

Transcription

1 General Physics I Lecture 1: Applications of Oscillatory Motion Prof. WAN, Xin ( 万歆 ) inwan@zju.edu.cn

2 Outline The pendulum Comparing simple harmonic motion and uniform circular motion Damped oscillation and forced oscillation Vibration in molecules Elastic properties of solids

3 Simple Pendulum

4 Period of the Simple Pendulum The period and frequency of a simple pendulum depend only on the length of the string and the acceleration due to gravity. Question: Christian Huygens ( ) suggested that an unit of length could be defined as the length of a simple pendulum having a period of eactly 1 s. How long is the length?

5 Physical Pendulum If a hanging object oscillates about a fied ais that does not pass through its center of mass and the object cannot be approimated as a point mass, we cannot treat the system as a simple pendulum. In this case the system is called a physical pendulum.

6 Physical Pendulum Used to measure the moment of inertia of a flat rigid body.

7 Torsional Pendulum When the body is twisted through some angle, the twisted wire eerts on the body a restoring torque that is proportional to the angular displacement. There is no small-angle restriction in this situation as long as the elastic limit of the wire is not eceeded.

8 Circular Motion An eperimental setup for demonstrating the connection between simple harmonic motion and uniform circular motion. As the ball rotates on the turntable with constant angular speed, its shadow on the screen moves back and forth in simple harmonic motion.

9 Oscillation vs. Circular Motion

10 Oscillation vs. Circular Motion

11 Oscillation vs. Circular Motion Simple harmonic motion along a straight line can be represented by the projection of uniform circular motion along a diameter of a reference circle. Uniform circular motion can be considered a combination of two simple harmonic motions, one along the ais and one along the y ais, with the two differing in phase by 90.

12 Damped Oscillator When the retarding force is much smaller than the restoring force, the oscillatory character of the motion is preserved but the amplitude decreases in time, with the result that the motion ultimately ceases. (c) overdamped (b) critically damped (a) underdamped

13 Damped Oscillator b: damping coefficient

14 Critical Damping (c) overdamped (b) critically damped (a) underdamped Critical damping: k m = ( b m )

15 Forced Oscillation We are interested in the underdamped case. in the absence of the driving force

16 Decomposition of Motion In the presence of the driving force Transient solution Steady solution

17 Slow Drive The driving force is slow enough that the oscillator can follow the force after the transient motion decays. ω < ω 0 = k /m

18 Fast Drive The driving force is fast such that the oscillator cannot follow the force and lags behind (p out of phase). Note that the amplitude is smaller than that for slow drive. ω > ω 0 = k /m

19 At Resonance ω = ω 0 = k /m The amplitude quickly grows to a maimum. After the transient motion decays and the oscillator settles into steady state motion, the displacement p/ out of phase with force (displacement lags the force).

20 Forced Oscillation Steady state:

21 Resonance Frequency For small damping, the amplitude becomes very large when the frequency of the driving force is near the natural frequency of oscillation. The dramatic increase in amplitude near the natural frequency w 0 is called resonance, and for this reason w 0 is sometimes called the resonance frequency of the system. At resonance the applied force is in phase with the velocity and that the power transferred to the oscillator is a maimum.

22 Lennard Jones Potential The potential energy associated with the force between a pair of neutral atoms or molecules can be modeled by the Lennard Jones potential energy function:

23 The Equilibrium [ U () = 4ϵ ( σ 1 ) ( σ 6 ) ] du d = 0 0 =1/6 σ 1.1σ We can approimate the comple atomic/molecular binding forces as tiny springs.

24 Force Near the Equilibrium U () = 4ϵ[ ( σ 1 ) ( σ 6 ) ] [ ( = ϵ )1 ( 0 )6 0 ] F () = du () d = 1 ϵ 0 [ ( 0 )13 ( )7 0 ] = d U d = 0 ( 0 )+O (( 0 ) ) 7 ϵ 0 ( 0 )

25 Vibration Frequency Effective spring constant: ω = 7ϵ μ 0 k = 7 ϵ 0 Reduced mass! Eample: Vibration of two water molecules σ = m ϵ = J μ 9m proton Why? f = ω π = 1 7 ϵ π μ Hz

26 Block-Spring System Revisit F 1 U ( ) k du ( ) F d k m d dt F d dt = ω ω 0 cos( wt ) k m

27 Two Harmonic Oscillators Two Harmonic Oscillators ) ( ' k k dt d m 1 ) ( ' 1 k k dt d m ) ( ' ) ( 1 1 m k dt d ) ( ' ) ( 1 1 m k k dt d

28 Two Harmonic Oscillators d m dt d m dt 1 Assume k' 1 k( 1 ) 1 mw 10 ( k' k) 10 k0 k k( ) i ' 1 i0 cos( wt ) mw 0 k 10 ( k' k) 0

29 Two Harmonic Oscillators 1 Assume m i i0 cos( wt ) w 10 ( k' k) 10 k0 mw 0 k10 ( k' k) 0 Determinant be zero k' k mw k k k' k mw

30 Vibrational Mode 1 Solution 1: w k' k k ' 0 m k m / k' Vibration with the reduced mass.

31 Translational Mode 1 Solution 1: w k m ' k 0 ' Translation!

32 Two Harmonic Oscillators Two Harmonic Oscillators ' ' m k k k k k k w 1 In mathematics language, we solved an eigenvalue problem. The two eigenvectors are orthogonal to each other. Independent!

33 Mode Counting N-atom linear molecule Translation: 3 Rotation: Vibration: 3N 5 N-atom (nonlinear) molecule Translation: 3 Rotation: 3 Vibration: 3N 6

34 Vibrational Modes of CO N = 3, linear Translation: 3 Rotation: Vibration: 3N 3 = 4

35 Vibrational Modes of H O N = 3, planer Translation: 3 Rotation: 3 Vibration: 3N 3 3 = 3

36 Solids Microscopically, a solid can be regarded as an array of atoms connected by springs (atomic forces). Macroscopically, therefore, it is possible to change the shape or the size of a solid by applying eternal forces. As these changes take place, however, internal forces in the object resist the deformation.

37 Elastic Properties for sufficiently small stresses. Stress: A quantity that is proportional to the force causing a deformation; more specifically, stress is the eternal force acting on an object per unit cross-sectional area. Strain: A measure of the degree of deformation. Elastic modulus: The constant of proportionality depends on the material being deformed and on the nature of the deformation.

38 Elasticity in Length Young s Modulus:

39 Elasticity of Shape Shear Modulus:

40 Volume Elasticity Bulk Modulus:

41 Typical Values for Elastic Modulus