Waves and Oscillations
Physics
Physics 105/2105
Fall 2024
Web ref provided on slides
Harmonic Motion
Ref: google image
Simple Harmonic Motion
Periodic Motion: A motion which repeats itself in equal intervals of time is periodic
motion. For example, the motion of the hands of a clock, the motion of the wheels of a
car and the motion of a merry-go-round.
Oscillatory Motion: An oscillatory motion is a periodic motion in which an object
moves to and fro about its equilibrium position. The object performs the same set of
movements again and again after a fixed time. One such set of movements is an
Oscillation. The motion of a simple pendulum, the motion of leaves vibrating in a
breeze and the motion of a cradle are all examples of oscillatory motion.
SHM: To-and-fro motion under the action of a restoring force. Simple harmonic motion
is the simplest example of oscillatory motion.
Ref: toppr.com
Simple Harmonic Motion: Definition
Definitions of some related
quantities for y = A sin(ωt+φ)
Amplitude: The amplitude of the
motion, denoted by A, is the
maximum magnitude of
displacement from the equilibrium
position. It is always positive
Period: The period T, is the time
required for one oscillation.
Frequency: The frequency, f, is
the number of cycles in a unit tine.
Ref: google
Simple Harmonic Motion: Graphs
Ref: google image
Simple Harmonic Motion
Ref: toppr.com
Simple Harmonic Motion: Equation
Hooke’s Law: The extension of an elastic object is
directly proportional to the force applied to it. Or,
The restoring force applied to an elastic object (such as a
spring) is proportional to the displacement (or, extension)
and in the opposite direction of that displacement. k is called the spring constant,
which is characteristic of
a spring which is defined as
the ratio of the force affecting
the spring to the displacement
caused by it.
Ref: google
Simple Harmonic Motion: Equation
We can combine the constants k and m by
making the substitution:
=ω02, which results
d 2x
2 + ω02 x = 0.
dt
Some solutions of this equation are:
x = A sin(ω0t+φ)
x = A cos(ω0t+φ)
This solutions can be proved to be
the solutions of the above
differential equation (see lecture).
Ref: google image
Here,
For t = 0 at x = 0 mean or equilibrium position
0 = a sin(0+ Ø)
a sinØ = 0
Ø =0
x = a sinωt
If x = ± a at t = 0
± a = ± a sin Ø
sin Ø = ± 1
∅= ±
𝑥 = asin(ω𝑡 + ± )
𝑥 = acos ω𝑡
Simple Harmonic Motion: Graphs
Ref: google image
Simple Harmonic Motion: Graphs
Ref: google image
Simple Harmonic Motion: Equation
Another Method:
F ∝ -x
or, F= -kx,
where x is the displacement from equilibrium and k is called the spring
constant, which is characteristic of a spring which is defined as the ratio of
the force affecting the spring to the displacement caused by it.
Since the acceleration:
dv d2x
a = = 2,
dt dt
Newton's second law becomes:
d 2x
-kx = m 2 ,
dt
which is called a second-order differential equation because it contains a
second derivative.
For vertical motion:
d 2y
m dt2 + ω2 y = 0.
Some solutions of this equation are:
y = A sin(ωt+φ)
y = A cos(ωt+φ)
Ref: csbsju.edu
Simple Harmonic Motion: Energy
Ref: google
Simple Harmonic Motion: Energy
Differential Equation of SHO/SHM from
Conservation of Energy for a mass spring system
Ref: google
Simple Harmonic Motion: Energy
Ref: google image
Simple Harmonic Motion: Energy
Ref: google image
Simple Harmonic Motion: Energy
Energy vs. Time Graph
Energy vs. Displacement Graph
Ref: google image
