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461

CHAPTER

14

SUMMARY

Simple harmonic motion:If the restoring force in

periodic motion is directly proportional to the displace- mentx, the motion is called simple harmonic motion (SHM). In many cases this condition is satis“ed if the displacement from equilibrium is small. The angular frequency, frequency, and period in SHM do not depend on the amplitude, but only on the mass mand force con- stantk. The displacement, velocity, and acceleration in SHM are sinusoidal functions of time; the amplitude A and phase angle of the oscillation are determined by the initial position and velocity of the body. (See Exam- ples 14.2, 14.3, 14.6, and 14.7.)fF x Energy in simple harmonic motion:Energy is conserved in SHM. The total energy can be expressed in terms of the force constant kand amplitude A. (See Examples

14.4 and 14.5.)

Angular simple harmonic motion:In angular SHM, the frequency and angular frequency are related to the moment of inertia

Iand the torsion constant .k

(14.3) (14.4) (14.10) (14.11) (14.12) (14.13) x=Acos1vt+f2T=1

ƒ=2pAmk

ƒ=v

2 p=12pAkm v=Ak m a x =F x m=-kmxF x =-kx (14.21) E= 1 2 m v x2 1 2 kx 2 1 2 kA 2 =constant (14.24) v=Ak I and ƒ=1 2 pAkI Simple pendulum:Asimple pendulum consists of a point massmat the end of a massless string of length L. Its motion is approximately simple harmonic for suf“- ciently small amplitude; the angular frequency, fre- quency, and period then depend only on gandL, not on the mass or amplitude. (See Example 14.8.) (14.32) (14.33) (14.34) T=2p v=1ƒ=2pALg

ƒ=v

2 p=12pAgL v=Ag L Periodic motion:Periodic motion is motion that repeats itself in a de“nite cycle. It occurs whenever a body has a stable equilibrium position and a restoring force that acts when it is displaced from equilibrium. Period Tis the time for one cycle. Frequency is the number of cycles per unit time. Angular frequency is times the frequency. (See Example 14.1.)2pvƒ (14.1) (14.2) v=2pƒ=2p

Tƒ=1

T T=1 F x a x xxn mgy n mgy F x a x xn mgy x 2

Ax = 0

x,0x.0x = A x 2 TOA T tx 2 A

Energy

xE5K1U OA2AU K ut z

SpringBalance wheel

Spring torque

t zopposes angular displacement u L T mgsinumgmg cos uu

Physical pendulum:Aphysical pendulum is any body

suspended from an axis of rotation. The angular fre- quency and period for small-amplitude oscillations are independent of amplitude, but depend on the mass m, distancedfrom the axis of rotation to the center of grav- ity, and moment of inertia

Iabout the axis. (See Exam-

ples 14.9 and 14.10.) (14.38) (14.39)

T=2pAI

mgd v= B mgd I dz mgsinu mgmg cosucg O dsinuu Driven oscillations and resonance:When a sinusoidally varying driving force is added to a damped harmonic oscillator, the resulting motion is called a forced oscilla- tion. The amplitude is a function of the driving fre- quency and reaches a peak at a driving frequency close to the natural frequency of the system. This behav- ior is called resonance.v d (14.46)A=F max 2 1 k-mv d2 2 2 +b 2 v d2

Damped oscillations:When a force propor-

tional to velocity is added to a simple harmonic oscilla- tor, the motion is called a damped oscillation. If (called underdamping), the system oscil- lates with a decaying amplitude and an angular fre- quency that is lower than it would be without damping. If (called critical damping) or (called overdamping), when the system is displaced it returns to equilibrium without oscillating.b721km b=21km v¿b622km F x =-bv x(14.42) (14.43) v¿= B k m-b 2 4 m 2 x=Ae -1b>2m2t cos1v¿t+f2

462CHAPTER 14Periodic Motion

O 2AA T 0 2T 0 3T 0 4T 0 5T 0 t Ae 2(b 2 m t x b

50.1km

b

50.4km

Fmax k2F max k3F max k4F max k 5 Fmaxquotesdbs_dbs27.pdfusesText_33

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