康涅狄格州大学：《普通物理》课程PPT教学课件（英文版）Lecture 22 The Simple Pendulum

Physics 121: Lecture 22 Today's Agenda Announcements No class next weel Homework 9: due Friday Nov 18@ 6: 00 PM Chap.9:#1,7,9,13,17,23,24,27,29,37,39,and40. Today 's topics SHM Pendulum Damped oscillations Resonance Waves Physics 121: Lecture 22, Pg

Physics 121: Lecture 22, Pg 1 Physics 121: Lecture 22 Today’s Agenda Announcements No class next week Homework 9: due Friday Nov. 18 @ 6:00 PM. Chap. 9: # 1, 7, 9, 13, 17, 23, 24, 27, 29, 37, 39, and 40. Today’s topics SHM Pendulum Damped oscillations Resonance Waves

The simple Pendulum A pendulum is made by suspending a mass m at the end of a string of length L Find the frequency of oscillation for small displacements 0 L mg Physics 121: Lecture 22, Pg 2

Physics 121: Lecture 22, Pg 2 The Simple Pendulum A pendulum is made by suspending a mass m at the end of a string of length L. Find the frequency of oscillation for small displacements.  L m mg z

Aside: sin e and cos e for small e A Taylor expansion of sin 0 and cos 0 about 0=0 gives sine=0 and cos0=1 3!5! So forθ<<1,sin≈0 and cOS0≈1 Physics 121: Lecture 22, Pg 3

Physics 121: Lecture 22, Pg 3 Aside: sin  and cos  for small  A Taylor expansion of sin  and cos  about  = 0 gives: ... 3! 5! sin 3 5 −  +   =  − ... 2! 4! cos 1 2 4 −  +  and  = − So for  <<1, sin   and cos 1

The Simple Pendulum. Recall that the torque due to gravity about the rotation(z) axis Isτ=la=-mgd d=Lsin0≈ Le for smal|0 soτ=-mgL0 melo= mLa Butτ=lo,/=mL2 a=-00 where VL m Same equation as for simple harmonic motion d 0=0ocos(ot+φ) mg Physics 121: Lecture 22, Pg 4

Physics 121: Lecture 22, Pg 4 The Simple Pendulum... Recall that the torque due to gravity about the rotation (z) axis is  = Ia = -mgd. d = Lsin   L for small  so  = -mg L But  = Ia  I = mL2  L d m mg z  a 2 − mgL = mL a   2 = −  = g L where Same equation as for simple harmonic motion !  = 0 cos(t + )

The rod pendulum a pendulum is made by suspending a thin rod of length L and mass M at one end Find the frequency of oscillation for small displacements X CM Physics 121: Lecture 22, Pg 5

Physics 121: Lecture 22, Pg 5 The Rod Pendulum A pendulum is made by suspending a thin rod of length L and mass M at one end. Find the frequency of oscillation for small displacements.  L mg z x CM