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