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Simple Harmonic Motion: A Comprehensive Guide for Students, Lecture notes of Physics

The spring is neither stretched nor compressed. (b) The object is displaced and the spring is stretched. • When the restoring force is directly proportional to ...

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2021/2022

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SIMPLE'HARMONIC'MOTION!
Created!by:!Binh!Cao!
!!! ! !!
Simple'harmonic' motion!(SHM)!is!a!type!of!periodic!motion.!Two!simple!systems!of!SHM!
that! are! mainly! discussed! in! college! are! an' ideal' spring!and! a' simple' pendulum.! But!
before!discussing!these!2!systems,!it!is!essential!to!go!over!periodic!motion!first.'
Periodic'motion!or!oscillation!refers!to!kinds!of! motion!that!repeat!themselves!over!and!
over.!For!example,!a!clock!pendulum.!
Understanding! periodic! motion! is! critical! for! understanding! more! complicated! concepts!
like!mechanical!waves!(e.g.!sound)!and!electromagnetic!waves!(e.g.!light)!
Oscillation! is! characterized! by! an! equilibrium!and! a! restoring' force.! At! equilibrium,!
restoring! force! on! the! object! is! zero.! When! the! object! is! displaced! from! equilibrium,! a!
restoring!force!acts!on!the!object!to!restore!its!equilibrium.!For!example,!a!spring.!
!
(a) The!object!is!at!equilibrium.!The!spring!is!neither!stretched!nor!compressed.!
(b) The!object!is!displaced!and!the!spring!is!stretched.!
When! the! restoring! force! is! directly! proportional! to! displacement! from! equilibrium,! the!
oscillation!is!called!simple'harmonic'motion!(SHM).!!
Important!characteristics!of!any!periodic!motion:!
o Amplitude'(A)!is!maximum!magnitude!of!displacement!from!equilibrium!
o Period'(T)!is!the!time!to!complete!one!cycle!(unit:!s)!
o Frequency!(f)!is!the!number!of!cycles!in!a!unit!of!time!(unit:!sN1)!
o Period!and!frequency!are!related!by!the!following!relationship:!
𝑇=
1
𝑓!𝑜𝑟!𝑓=
1
𝑇!
o Angular'frequency!(ω):!!
𝜔=2𝜋𝑓 =!!
!!(rad/s)!
A/'Ideal'spring'
𝐹
!"#$%!&'( =𝑘𝑥!!
pf3

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Download Simple Harmonic Motion: A Comprehensive Guide for Students and more Lecture notes Physics in PDF only on Docsity!

Created by: Binh Cao

  • Simple harmonic motion (SHM) is a type of periodic motion. Two simple systems of SHM that are mainly discussed in college are an ideal spring and a simple pendulum. But before discussing these 2 systems, it is essential to go over periodic motion first.
  • Periodic motion or oscillation refers to kinds of motion that repeat themselves over and over. For example, a clock pendulum.
  • Understanding periodic motion is critical for understanding more complicated concepts like mechanical waves (e.g. sound) and electromagnetic waves (e.g. light)
  • Oscillation is characterized by an equilibrium and a restoring force. At equilibrium, restoring force on the object is zero. When the object is displaced from equilibrium, a restoring force acts on the object to restore its equilibrium. For example, a spring. (a) The object is at equilibrium. The spring is neither stretched nor compressed. (b) The object is displaced and the spring is stretched.
  • When the restoring force is directly proportional to displacement from equilibrium, the oscillation is called simple harmonic motion (SHM).
  • Important characteristics of any periodic motion: o Amplitude (A) is maximum magnitude of displacement from equilibrium o Period (T) is the time to complete one cycle (unit: s) o Frequency (f) is the number of cycles in a unit of time (unit: s-­‐^1 ) o Period and frequency are related by the following relationship: 𝑇 =

o Angular frequency (ω): 𝜔 = 2 𝜋𝑓 = !! ! (rad/s) A/ Ideal spring 𝐹!"#$%!&'( = −𝑘𝑥

Created by: Binh Cao k: spring constant x: displacement from equilibrium 𝜔 = ! !

! !!

! !! ! !

! !

!! !

! ! 𝑥 = 𝐴 cos(𝜔𝑡 + 𝛷) Φ: initial angular displacement 𝐸 = ! !

𝑚𝑣!^!^ +

! !

𝑘𝑥!^ =

! !

𝑘𝐴!^ = 𝑐𝑜𝑛𝑠𝑡

E: mechanical energy of the system 𝑣!: velocity of mass m at x (m/s) B/ Simple pendulum 𝐹!"#$%!&'( = −𝑚𝑔𝑠𝑖𝑛𝜃 ≅ −𝑚𝑔𝜃 (when θ is small) Θ: angular displacement form equilibrium 𝜔 = ! !

! !!

! !! ! !

! !

!! !

! ! L: string length (m) C/ Examples 1/ When a body of unknown mass is attached to an ideal spring with force constant 120 N/m, it is found to vibrate with a frequency of 6.00 Hz. Find (a) The period of the motion; (b) The angular frequency; (c) The mass of the body. Solution: k = 120 N/m; f = 6.00 Hz (a) 𝑇 = ! !

! !.!!

(b) 𝜔 = 2 𝜋𝑓 = 2 𝜋× 6. 00 = 37. 7 (rad/s) (c) 𝜔 = ! ! or 𝑚 = ! !!^

!"# !".!!^