SHM Notesheet (AIS 2015)

8/17/2019 SHM Notesheet (AIS 2015)

http://slidepdf.com/reader/full/shm-notesheet-ais-2015 1/9

An __________________ is a back and forth motion over the same path. Requires a

____________ ____________.

It is said to be ______________ if each cycle of the motion takes place in equal periods of time.

__________ ______________ ____________ : a projection of circular motion on one axis.

Types of !"# "ass on a sprin$% simple pendulum% and a particle in a bo&l

The Relation between SHM and Circular Motion An object moving with constant speed in a circular path observed from a distant point willappear to be ________________with ________________________________________.

The ___________ of a pendulum bob moves with SHM when the pendulum itself is either _____________ or moving in a circle with constant speed.

For any SHM there is a corresponding _____________ motion.

# the radius of the circle is equal to the ____________ of the SHM.# the time period of the circular motion is equal to the __________________ of the

SHM.

The relationship of circular motion and SHM is

earranging!

a

r

T

'' (π =

r T

a

=

'

'(π

recall T

π

ω

'

=Therefore!

The constant of proportionalit" between acceleration and displacement for an object movingwith SHM is equal and opposite to the square of the angular velocit" of correspondingcircular motion.So! to find the value of the constant for a given oscillation! we simpl" measure the

___________________ and then use the relation

Simple Harmonic MotionSimple Harmonic Motion



Characteristics of SHM

# A particle is disturbed awa" from a fi#ed equilibrium position and e#perience anacceleration that is proportional and _____________ to its ______________.

• Requirements:

) __________ equilibrium position

) $article moved awa" from equilibrium position! ____________________ is bothproportional to the _______________ and in the opposite direction.

• Characteristics

) $eriod and amplitude are ___________

) $eriod is _______________ of the amplitude

) %isplacement! velocit"! and acceleration are _______ or ________ functionsover time.

Spring Review

Hookes !aw

# Fs&restoring force of a spring

# ' & spring constant

# # & displacement of the spring

# The friction free motion shown above is 'nown as Simple Harmonic Motion (SHM)

!" $raph

# _______________ motion

# max. stretchin$ distance *x+ from the equilibrium position is equal

to the __________________ of the $raph.

!" and ,ircular "otion

• !" has displacement% velocity and acceleration.

-isplacement:

_____________ *T+ ) time required to complete one cycle.

_____________ *f+ ) number of cycles per second# nits ) !ert/ *!/+



0hysics for the I1 -iploma 2th 3dition *Tsokos

0hase difference *6+ ) amount by &hich one curve is shifted relative to another curve

7elocity

# 7elocity of the shado& is the vx of vT.

At x 84 m v à vmax Therefore% 9 8 4

Acceleration# Acceleration of the shado& is the ax of ac.

At x 8 r% a à amax Therefore% 9 8 ;4

"raphs #escribing S$H$M$

%isplacement against time % & ________________

*elocit" against time v & _________________

Acceleration against time a & ___________________

All these graphs are drawn assuming that! at

t & +! the oscillating bod" is at the equilibrium position.

SHM (,- Formulation) Alternative form

# # & Acos(t/ϕ) v & 0Asin(t/ϕ) a & A1cos(t/ϕ)where A and ϕ are both constants

A is the amplitude and ϕ is the phase angle,- shows both

# # & #+sin(t) or # & #+cos(t)# v & v+cos(t) or v & 0v+sin(t)



<ote: ine form of displacement &hen at t84 the displacement is /ero *particle at equilibrium+

,osine form of displacement &hen at t84 the displacement is the amplitude.

SHM of Springs

-erivation of =requency of !"

# !ooke>s ?a&: =8@kx

#

t r m

k t r ω ω ω sinsin

'

−=−

= 8 ma 8 @kx à

# =requency%

# 0eriod%

Elastic Potential Energy (PEelastic)

# 3ner$y that a sprin$ contains by bein$ stretched or compressed.

SHM Energy

,f an oscillation is SHM! then the total energ" possessed b"

the oscillating bod" does _____ _____________ with time.ecall!

Total Mechanical 2nerg" & ______ / ______ An oscillation in which the total energ" decreases with time (usuall" because of air resistanceor something similar) is described as a ___________ oscillation.

'inetic energ( against time _______________

Since! 2p & 2 0 2'

$otential energ" graph has the same form as the

'inetic energ" graph but is ____________.)otential energ( against time

The elastic constant of the spring is '.



3hen!2p is a ma#imum! 2' is a minimum (2' & +)2' is a ma#imum! 2p is a minimum (2p & +)Energy in SHM (IB Formulation)

• Elastic Potential Energy (Ep) Kinetic Energy (Ek )

• Total Mechanical Energy (E)

• Total energy is conserve

# hen mass is released from rest and the extension is the amplitude of the motion A

• Solving for velocity

• For this motion, ω2

=k

m

• So,

• Maximum velocity occurs at x !

• "t the extremes of the motion, x # ", an v !

$otal Mechanical Energy

elastic PE

al gravitaion PE

rotational KE

nal translatio KE TME +++=

Energy Pro%lem

An object of mass m 8 4.'44 k$ is vibratin$ on a hori/ontal frictionless table as sho&n. The sprin$ has

a sprin$ constant k 8 2(2 <Bm. It is stretched initially to xo 8(.24 cm and released from rest.

-etermine the final translational speed vf of the object &hen the final displacement of the sprin$ is *a+xf 8 '.'2 cm and *b+ xf 84 cm.

E=1

k x2+1

m v2=constant

E=1

k x2+1

m v2=1

k A2

v=±

√ k

m √ A2

− x2

vmax=ωA



$he Simple Pen&ulum

• hen displaced from its equilibrium position by an an$le 9 and released it s&in$s

back and forth.

• 0lottin$ the motion reveals a pattern similar to the sinusoidal motion of _______ • The $ravitational force *=$x+ provides the torque.

• This restorin$ force *=$x+:

• ince the displacement and restorin$ force act in opposite directions.

• The torque of the pendulum is

• =or small an$les *____or smaller+

•

9 8 sin9• here kC is a constant independent of 9

• This form is similar to =8@kx *!ooke>s ?a&+

• m

k =ω

I

mgL

I

k =

′=ω

'mL I = o%

à

=or __________ an$les D<?E *F4or smaller+

L

g f == π ω '

&in$in$ 0roblem

-etermine the len$th of a simple pendulum that &ill s&in$ back and forth in !" &ith

a period of F.44 s

2nerg" of a $endulum

TM2 & ________________

$2ma# à 42 & ____

θ sinmg F −=



____________ à $2 & +

$article in a -owl

# The gravitational force (Fg#) provides the restoring force.

# This restoring force (Fg#)5 F =ma=−mgsinθ

# So the acceleration is5 a=−gsinθ

# The displacement is5 θ= x

r

# For small angles (6+7or smaller) , sin( xr )≈ x

r

# So! a=−g x

r=−g

r x which is in the form a=−ω

2

x

# Therefore! ω2

=g

r

# The period is

8ote5 onl" for small angles (6+7or smaller)

#amped Simple Harmonic Motion

# ,n real life! energ" is lost causing each oscillation to be less than the previous.

# ______________ 9 reduction of an oscillation motion b" an e#ternal force.

# Modeled below where damping force is

bv F d −=and the net force is

makxbv F net =−−=

%amping

# %amped oscillations 9 resistance forces

# : t"pes of damping

) ;nder0damping

$h"sics for the ,- %iploma <th 2dition (Tso'os



0hysics for the I1 -iploma 2th 3dition *Ts

) ?ritical damping

) @ver0damping

# ;nder0damping

) Small resistance force

) Amplitudes graduall" decrease

# ight %amping

# Heav" %amping

# @scillations die out quic'er

# arger period than lighter damping

# ___________ 9 determines how quic'l" the oscillations will die out in an underdampeds"stem.

#

# ?riticall" damped

) ,deal damping

) eturns the s"stem to the equilibrium position as fast as possible

) 8o oscillations

# @ver0damping

) S"stem returns to equilibrium without oscillations

) Much slower than criticall" damped case.

esonance# _____________frequenc" () 9 angular frequenc" at which a s"stem will vibrate after

an e#ternal disturbance

Q=2 π energy stored∈acycle

energy dissipated∈acycle



# ___________ 9 condition when the amplitude increases due to a periodic e#ternalforce where

Angular frequenc" of driving force (d)

# For a small degree of damping! the pea' of the curve occurs at the natural frequenc"!f +

# The lower the degree of damping! the higher and narrower the curve

# As the amount of damping increases! the pea' shift to lower frequencies

# At ver" low frequencies! the amplitude is essentiall" constant.

SHM Notesheet (AIS 2015)

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