H A R M O N I C
M OT I O N
(PERIODIC MOTION)
PERIODIC MOTION
• Periodic Motion is the movement of an object vibrating or
oscillating back and forth, over the same path, each oscillation
taking the same amount of time
Quantities
» amplitude (A) - maximum displacement of the object
from equilibrium position
» period (T) - time it takes the object to move through one
complete cycle of motion or oscillation
» frequency (f) - number of complete cycles or vibrations
per unit of time, the reciprocal of the period (f = 1/T)
» angular frequency (⍵)
2𝜋
𝜔 = 2𝜋𝑓 =
𝑇
Simple harmonic motion
• Any oscillatory system driven by a restoring force proportional to
the negative of the displacement is called a Simple Harmonic
Oscillator
Simple harmonic motion
» SHM occurs when the net force along the direction of
motion obeys Hooke’s Law
𝐹 = −𝑘𝑥
§ Hooke’s Law: When the net force is proportional to
the displacement (F α x) from the equilibrium point and
is always directed toward the equilibrium position
(negative sign)
Mass on a spring
» The simplest form of periodic
motion is represented by an object
oscillating on the end of a uniform
coil spring
» If the mass is moved either to the equilibrium
position
right, which stretches it, or to the
left, which compresses the spring,
the spring exerts a restoring
force*
*A restoring force always pushes or pulls the
object toward the equilibrium position
Mass on a spring
» Hooke’s Law: if a spring is
stretched or compressed, it exerts
a restoring force equal to
𝐹 = −𝑘𝑥
equilibrium
x = distance the spring is displaced from equilibrium position
k = the spring constant of the spring
The negative sign indicates that the force is directed
opposite to the direction of the displacement of the
mass
Acceleration in simple harmonic motion
» Hooke’s Law 𝐹 = −𝑘𝑥
» Recall: Newton’s Second Law of Motion
𝐹 = 𝑚𝑎 −𝑘𝑥 = 𝑚𝑎
−𝑘𝑥
𝑎=
𝑚
» NOTE: The acceleration is a function of position
§ Acceleration is not constant and therefore the uniformly
accelerated motion equation cannot be applied
Motion equations in shm
Recall:
∆𝑥 𝑑𝑥
» Velocity 𝑣= =
∆𝑡 𝑑𝑡
∆𝑣 𝑑𝑣 𝑑! 𝑥 𝑑𝑣
» Acceleration 𝑎 = ∆𝑡 = 𝑑𝑡 𝑎= !=
𝑑𝑡 𝑑𝑡
Motion equations in shm
» Hooke’s Law 𝐹 = −𝑘𝑥 = 𝑚𝑎
𝑘 𝑑𝑣 𝑑! 𝑥
» Acceleration 𝑎=− 𝑥= = !
𝑚 𝑑𝑡 𝑑𝑡
𝑘 𝑘
» Angular Frequency 𝜔 = 2𝜋𝑓 = !
𝜔 =
𝑚 𝑚
𝑑! 𝑥 !
!
= −𝜔 𝑥 What mathematical function exhibits this behavior?
𝑑𝑡
Motion equations in SHM
𝑑! 𝑥 !
!
= −𝜔 𝑥 Sinusoidal Function
𝑑𝑡
» Position 𝑥 = 𝐴 cos 𝜔𝑡
𝑑𝑥
» Velocity 𝑣= = −𝐴𝜔 sin 𝜔𝑡
𝑑𝑡
𝑑! 𝑥
» Acceleration𝑎 = ! = −𝐴𝜔! cos 𝜔𝑡
𝑑𝑡
Describing simple harmonic motion
» Angular frequency 𝑘
𝜔 = 2𝜋𝑓 =
𝑚
» Frequency of a Spring-Mass System
𝜔 1 𝑘
𝑓= =
2𝜋 2𝜋 𝑚
» Period of a Spring-Mass System
1 𝑚
𝑇 = = 2𝜋
𝑓 𝑘
Seatwork (5pts) pair or threes. 10 minutes
1. 𝑁
𝑘 = 10
𝑚
𝐴 = 10 𝑐𝑚 = 0.1𝑚
𝑚 = 2𝑘𝑔
𝜇! = 0
Find 𝑣!"# and 𝑎!"#
2. Given 𝑥 𝑡 = 10𝑐𝑚 cos(𝜔𝑡)
Find amplitude A, frequency f, Period T, and x (t=2s)
Verification of sinusoidal nature
» This experiment shows the sinusoidal
nature of simple harmonic motion
» The spring mass system oscillates in
simple harmonic motion
» The attached pen traces out the
sinusoidal motion
Energy transformations in spring-mass system
mass m
• A block of mass m slides on a frictionless horizontal surface
with constant velocity vi and collides with a coiled spring
» In the absence of friction, the total mechanical energy of
the spring-mass system remains constant (Conservation
of Energy)
Energy transformations in spring-mass system
mass m
• RECALL:
» Kinetic Energy 𝐾𝐸 = $%𝑚𝑣 %
» Gravitational Potential Energy 𝑈& = 𝑚𝑔𝑦
» Elastic Potential Energy 𝑈' = $%𝑘𝑥 %
Energy transformations in spring-mass system
𝐸 = 𝐾𝐸 + 𝑈" + 𝑈#
𝐸 = $!%&! " + 0 + 0
Energy transformations in spring-mass system
𝐸 = 𝐾𝐸 + 𝑈" + 𝑈#
𝐸 = $!%& " + 0 + $!'( "
Energy transformations in spring-mass
system
𝐸 = 𝐾𝐸 + 𝑈" + 𝑈#
𝐸 = 0 + 0 + $!'(# "
Energy transformations in spring-mass system
𝐸 = 𝐾𝐸 + 𝑃𝐸" + 𝑃𝐸#
𝐸 = $!%&! " + 0 + 0
Conservation of Energy in spring-mass system
𝐸 = $!%& " + 0 + $!'( " 𝐸 = 0 + 0 + $!'(# "
$
Since energy is conserved
!
%& " + $!'( " = $!')"
where A is the maximum extension
𝑘 % (A = xm)
Solving for v 𝑣=± 𝐴 − 𝑥%
𝑚
Velocity as a function of position
𝑘 !
𝑣=± 𝐴 − 𝑥!
𝑚
• Speed is a maximum at x = 0
• Speed is zero at x = ±A
• The ± indicates the object can be traveling in either direction
Simple pendulum
• The simple pendulum is another Cable,
example of a system that exhibits Rope, or
String
simple harmonic motion
Mass or Bob
Simple pendulum
» The forces acting on the bob are the
tension T exerted by the string and the
gravitational force mg
» The tangential component mg sin θ of
the gravitational force always acts
toward θ = 0 (opposite the
displacement of the bob from the
lowest position)
» The tangential component is a
restoring force
Remember another restoring force? Hooke’s Law: F = –kx
Simple pendulum
» Is the motion of a pendulum simple
harmonic? Does it obey Hooke’s Law?
𝐹 = −𝑘𝑥 𝐹* = −𝑚𝑔 sin 𝜃 NO!
» What if θ is small? sin 𝜃 ≈ 𝜃 (θ in radians)
Since s = L θ
𝑚𝑔
𝐹* = −𝑚𝑔 sin 𝜃 ≈ −𝑚𝑔𝜃 ≈ − 𝑠
𝐿
Simple pendulum
» Is the motion of a pendulum simple
harmonic for small values of θ?
𝑚𝑔
𝐹 = −𝑘𝑥 𝐹* = − 𝑠 YES!
𝐿
» Comparing the pendulum with the
spring mass system
𝑚𝑔
𝑘 (Constant)
𝐿
𝑚𝑔
𝑘
Simple pendulum 𝐿
Mass-Spring Simple Pendulum
𝑚 𝑚 𝐿
Period 𝑇 = 2𝜋 𝑇 = 2𝜋 !" = 2𝜋
𝑘 # 𝑔
1 1 𝑘 1 1 𝑔
Frequency 𝑓= = 𝑓= =
𝑇 2𝜋 𝑚 𝑇 2𝜋 𝐿
Angular 𝑘 𝑔
𝜔 = 2𝜋𝑓 = 𝜔 = 2𝜋𝑓 =
Frequency 𝑚 𝐿
Simple pendulum
𝐿
𝑇 = 2𝜋
𝑔
» The period of a simple
pendulum is independent of the
amplitude and the mass
» The period depends on the
length of the pendulum and the
acceleration of gravity at the
location of the pendulum
Homework (4pts)
