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Oscillations and Simple Oscillations and Simple Harmonic Motion:Harmonic Motion:
AP Physics C: Mechanics
Oscillatory MotionOscillatory Motion
Oscillatory Motion is repetitive back and forth motion about an equilibrium position
Oscillatory Motion is periodic.
Swinging motion and vibrations are forms of Oscillatory Motion.
Objects that undergo Oscillatory Motion are called Oscillators.
Simple Harmonic MotionSimple Harmonic Motion
The time to complete one full cycle of
oscillation is a Period.
T 1
f
f 1
TThe amount of
oscillations per second is called frequency and is measured in Hertz.
What is the oscillation period for the broadcast of a 100MHz FM radio station?
What is the oscillation period for the broadcast of a 100MHz FM radio station?
Heinrich Hertz produced the first artificial radio waves back
in 1887!
Heinrich Hertz produced the first artificial radio waves back
in 1887!
T 1
f
1
1108Hz110 8s 10ns
Simple Harmonic MotionSimple Harmonic Motion
The most basic of all types of oscillation is depicted on
the bottom sinusoidal graph. Motion that follows
this pattern is called simple harmonic motion or SHM.
Simple Harmonic MotionSimple Harmonic Motion
An objects maximum displacement from its equilibrium position is
called the Amplitude (A) of the motion.
What shape will a What shape will a velocity-time graph velocity-time graph have for SHM?have for SHM?
Everywhere the slope (first derivative) of the position graph is zero, the velocity
graph crosses through zero.
2cos
tx t A
T
We need a position function to describe the motion
above.
Mathematical Models of SHMMathematical Models of SHM
2cos
tx t A
T
cos 2x t A ft
cosx t A t
1T
f
2
T
x(t) to symbolize position as a function of
time
A=xmax=xmin
When t=T, cos(2π)=cos(0)
x(t)=A
Mathematical Models of SHMMathematical Models of SHM
sinv t A t
cosx t A t
d x tv t
dt
In this context we will call omega Angular
Frequency
What is the physical meaning of the product (Aω)?
maxv AThe maximum speed of an oscillation!
Example:Example:
An airtrack glider is attached to a spring, pulled 20cm to the right, and
released at t=0s. It makes 15 oscillations in 10 seconds.
An airtrack glider is attached to a spring, pulled 20cm to the right, and
released at t=0s. It makes 15 oscillations in 10 seconds.
What is the period of oscillation?What is the period of oscillation?
1510sec
11.5oscilationsf Hz
T
1 10.67
1.5T s
f Hz
Example:Example:
An airtrack glider is attached to a spring, pulled 20cm to the right, and
released at t=0s. It makes 15 oscillations in 10 seconds.
An airtrack glider is attached to a spring, pulled 20cm to the right, and
released at t=0s. It makes 15 oscillations in 10 seconds.
What is the object’s maximum speed?What is the object’s maximum speed?
max
2Av A
T
max
0.2 21.88 /
0.67
mv m s
s
Example:Example:
An airtrack glider is attached to a spring, pulled 20cm to the right, and
released at t=0s. It makes 15 oscillations in 10 seconds.
An airtrack glider is attached to a spring, pulled 20cm to the right, and
released at t=0s. It makes 15 oscillations in 10 seconds.
What are the position and velocity at t=0.8s?
What are the position and velocity at t=0.8s?
cos 0.2 cos 0.8 0.0625x t A t m s m
sin 0.2 sin 0.8 1.79 /v t A t m s m s
Example:Example:
A mass oscillating in SHM starts at x=A and has period T. At what time,
as a fraction of T, does the object first pass through 0.5A?
A mass oscillating in SHM starts at x=A and has period T. At what time,
as a fraction of T, does the object first pass through 0.5A?
2cos
( ) 0.5
tx t A
T
x t A
20.5 cos
tA A
T
1cos 0.52
Tt
2 3
Tt
6
Tt
Model of SHMModel of SHM
When collecting and modeling data of SHM your mathematical model had a value as shown below:
When collecting and modeling data of SHM your mathematical model had a value as shown below:
x(t) Acos t
x(t) Acos t C What if your clock didn’t start at x=A or x=-A?What if your clock didn’t start at x=A or x=-A?
This value represents our initial conditions. We call it the phase angle:
This value represents our initial conditions. We call it the phase angle:
x(t) Acos t
SHM and Circular MotionSHM and Circular Motion
Uniform circular motion projected onto one dimension is simple harmonic motion.
Uniform circular motion projected onto one dimension is simple harmonic motion.
SHM and Circular MotionSHM and Circular Motion
x(t) Acos
ddt
t
x(t) Acos t
Start with the x-component of position of the particle in UCMStart with the x-component of position of the particle in UCM
End with the same result as the spring in SHM!
End with the same result as the spring in SHM!
Notice it started at angle zeroNotice it started at angle zero
Initial conditions:Initial conditions:
t 0
We will not always start our clocks at one amplitude.
We will not always start our clocks at one amplitude.
x(t) Acos t 0
vx (t) Asin t 0
vx (t) vmax sin t 0
The Phase Constant:The Phase Constant:
t 0
Phi is called the phase of the oscillation
Phi is called the phase of the oscillation
Phi naught is called the phase constant or phase shift. This
value specifies the initial conditions.
Phi naught is called the phase constant or phase shift. This
value specifies the initial conditions.
Different values of the phase constant correspond to different starting points on the circle and thus to
different initial conditions
Different values of the phase constant correspond to different starting points on the circle and thus to
different initial conditions
Phase Shifts:Phase Shifts:
An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of
equilibrium and moving to the left. What are its position and direction of motion at t=2s?
An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of
equilibrium and moving to the left. What are its position and direction of motion at t=2s?
x(t) Acos t 0
x0 5cm Acos 0 Initial conditions:Initial conditions:
0 cos 1 x0
A
cos 1 5cm
10cm
120
2
3 rads
From the period we get:From the period we get:
2T
2
0.8s7.85rad /s
An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of
equilibrium and moving to the left. What are its position and direction of motion at t=2s?
An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of
equilibrium and moving to the left. What are its position and direction of motion at t=2s?
x(t) Acos t 0
7.85rad /s
0 2
3 rads
A 0.1m
t 2s
x(t) 0.1cos 7.85 2 2
3
x(t) 0.05m
We have modeled SHM mathematically.We have modeled SHM mathematically. Now comes the physics. Now comes the physics.
Total mechanical energy is conserved for our SHM example of a spring with
constant k, mass m, and on a frictionless surface.
Total mechanical energy is conserved for our SHM example of a spring with
constant k, mass m, and on a frictionless surface.
E K U 1
2mv2
1
2kx2
The particle has all potential energy at x=A and x=–A, and the particle has purely kinetic energy at x=0.
The particle has all potential energy at x=A and x=–A, and the particle has purely kinetic energy at x=0.
At turning points:At turning points:
E U 1
2kA2
At x=0:At x=0:
E k 1
2mvmax
2
From conservation:From conservation:
1
2kA2
1
2mvmax
2
Maximum speed as related to amplitude:
Maximum speed as related to amplitude:
vmax k
mA
From energy considerations:From energy considerations:
From kinematics:From kinematics:
Combine these:Combine these:
vmax k
mA
vmax A
k
m
f 1
2k
m
T 2m
k
a 500g block on a spring is pulled a distance of 20cm and released. The subsequent oscillations are measured to
have a period of 0.8s. at what position or positions is the block’s speed 1.0m/s?
a 500g block on a spring is pulled a distance of 20cm and released. The subsequent oscillations are measured to
have a period of 0.8s. at what position or positions is the block’s speed 1.0m/s?
The motion is SHM and energy is conserved.The motion is SHM and energy is conserved.
1
2mv2
1
2kx2
1
2kA2
kx2 kA2 mv2
x A2 m
kv2
x A2 v2
2
2T
2
0.8s7.85rad /s
x 0.15m
Dynamics of SHMDynamics of SHM
Acceleration is at a maximum when the particle is at maximum and minimum displacement from x=0.
Acceleration is at a maximum when the particle is at maximum and minimum displacement from x=0.
ax dvx (t)
dt
d Asin t dt
2Acos t
Dynamics of SHMDynamics of SHM
Acceleration is proportional to the
negative of the displacement.
Acceleration is proportional to the
negative of the displacement.
ax 2Acos t
ax 2x
x Acos t
Dynamics of SHMDynamics of SHM
As we found with energy considerations:
As we found with energy considerations:
ax 2x
F max kx
max kx
ax k
mx
According to Newton’s 2nd Law:
According to Newton’s 2nd Law:
ax d2x
dt 2
Acceleration is not constant:
Acceleration is not constant:
d2x
dt 2 k
mx
This is the equation of motion for a mass on a spring. It is of a general
form called a second order differential equation.
This is the equation of motion for a mass on a spring. It is of a general
form called a second order differential equation.
22ndnd-Order Differential Equations:-Order Differential Equations:
Unlike algebraic equations, their solutions are not numbers, but functions.
Unlike algebraic equations, their solutions are not numbers, but functions.
In SHM we are only interested in one form so we can use our solution for many objects undergoing SHM.
In SHM we are only interested in one form so we can use our solution for many objects undergoing SHM.
Solutions to these diff. eqns. are unique (there is only one). One common method of solving is guessing the
solution that the equation should have…
Solutions to these diff. eqns. are unique (there is only one). One common method of solving is guessing the
solution that the equation should have…
d2x
dt 2 k
mx
From evidence, we expect
the solution:
From evidence, we expect
the solution:
x Acos t 0
22ndnd-Order Differential Equations:-Order Differential Equations:
Let’s put this possible solution into our equation and see if we guessed right!
Let’s put this possible solution into our equation and see if we guessed right!
d2x
dt 2 k
mx
IT WORKS. Sinusoidal oscillation of SHM is a result of Newton’s laws!
IT WORKS. Sinusoidal oscillation of SHM is a result of Newton’s laws!
x Acos t 0
d2x
dt 2 2Acos t
dx
dt Asin t
2Acos t k
mAcos t
2 k
m
What about vertical oscillations of What about vertical oscillations of a spring-mass system??a spring-mass system??
Fnet kL mg 0Hanging at rest:Hanging at rest:
kL mg
L m
kg
this is the equilibrium position of the system.this is the equilibrium position of the system.
Now we let the system oscillate. At maximum:Now we let the system oscillate. At maximum:
But:But:
Fnet k L y mg
Fnet kL mg ky
kL mg 0So:So:
Fnet ky
Everything that we have learned about horizontal oscillations is equally valid for
vertical oscillations!
Everything that we have learned about horizontal oscillations is equally valid for
vertical oscillations!
The PendulumThe Pendulum
Fnet t mgsin ma t
d2s
dt 2 gsin
Equation of motion for a pendulum
Equation of motion for a pendulum
s L
Small Angle Approximation:Small Angle Approximation:
d2s
dt 2 gsin
When θ is about 0.1rad or less, h and
s are about the same.
When θ is about 0.1rad or less, h and
s are about the same.
sin
cos 1
tan sin 1
d2s
dt 2 g
s
L
Fnet tm
d2s
dt 2 mgs
L
The PendulumThe Pendulum
Equation of motion for a pendulum
Equation of motion for a pendulum
d2s
dt 2 gs
L
g
L
(t) max cos t 0
x(t) Acos t 0
A Pendulum ClockA Pendulum Clock
What length pendulum will have a period of exactly 1s?What length pendulum will have a period of exactly 1s?
g
L
T 2L
g
gT
2
2
L
L 9.8m/s2 1s
2
2
0.248m
Conditions for SHMConditions for SHM
Notice that all objects that we look at are described
the same mathematically.
Notice that all objects that we look at are described
the same mathematically.
Any system with a linear restoring force will undergo simple
harmonic motion around the equilibrium position.
Any system with a linear restoring force will undergo simple
harmonic motion around the equilibrium position.
A Physical PendulumA Physical Pendulum
d2dt 2
mgl
I
I mgd mglsin
when there is mass in the
entire pendulum, not just the bob.
when there is mass in the
entire pendulum, not just the bob.
Small Angle Approx.Small Angle Approx.
mgl
I
Damped OscillationsDamped Oscillations
All real oscillators are damped oscillators. these are any that slow
down and eventually stop.
All real oscillators are damped oscillators. these are any that slow
down and eventually stop.
a model of drag force for slow objects:
a model of drag force for slow objects:
Fdrag bv
b is the damping constant (sort of like a coefficient of friction).
b is the damping constant (sort of like a coefficient of friction).
Damped OscillationsDamped Oscillations
F Fs Fdrag kx bv ma
kx bdx
dt m
d2x
dt 20
Another 2nd-order diff eq.Another 2nd-order diff eq.
Solution to 2nd-order diff eq:
Solution to 2nd-order diff eq:
x(t) Ae bt / 2m cos t 0
k
m
b2
4m2
02
b2
4m2
Damped OscillationsDamped Oscillations
x(t) Ae bt / 2m cos t 0
A slowly changing line that provides a border to
a rapid oscillation is called the envelope of
the oscillations.
A slowly changing line that provides a border to
a rapid oscillation is called the envelope of
the oscillations.
Driven Driven OscillationsOscillations
Not all oscillating objects are disturbed from rest then allowed to move undisturbed.
Not all oscillating objects are disturbed from rest then allowed to move undisturbed.
Some objects may be subjected to a periodic external force.
Some objects may be subjected to a periodic external force.
DrivenDrivenOscillationsOscillations
All objects have a natural frequency at which they tend to vibrate when disturbed.
All objects have a natural frequency at which they tend to vibrate when disturbed.
Objects may be exposed to a periodic force with a particular driving frequency.
Objects may be exposed to a periodic force with a particular driving frequency.
If the driven frequency matches
the natural frequency of an
object, RESONANCE occurs
If the driven frequency matches
the natural frequency of an
object, RESONANCE occurs
THE
END