Chapter 15Oscillatory Motion

Recall the Spring

kxFspring

xm

ka

kxma

x

x

Since F=ma, this can be rewritten as:

Negative because it is a restoring force.In other words, if x is positive, the force and acceleration is negative and vice versa.

This makes the object oscillate and therefore, it undergoes simple harmonic motion.

Now pay attention

xm

kax

2

2

dt

xda

dt

dva

dt

dxv

xm

k

dt

xd

2

2

Remember that:

can be rewritten as:So

2m

k

xm

k

dt

xd

2

2

xdt

xd 22

2

To make this differential equation easier to solve, we make

then

)cos()( tAtx

After solving, we get:

Note: This requires knowledge on how to solve differential equations. It is more important to know what the solution is.

Simple Harmonic Motion

)cos()( tAtxEquation for distance (x) as time (t) changes.General formula for simple harmonic motion.

A, ω, ϕ are constants

A is Amplitude - For springs: max value of distance (x) (positive or negative) - Maximum value the wave alternates back and forth between

ω is Angular frequency → - How rapidly oscillations occur - Units are rad/s

m

k

m

k

2Remember:

)( t

)cos()( tAtx

is the phase constant

is the phase Essentially is the shifts of the wave

These determine the starting position of the wave

General Concepts

kxF

xm

k

dt

xd

2

2

)cos()( tAtx

Anything with behaviors which have formulas that look like these are undergoing simple harmonic motion and can be measured using the same method as the spring.

Period and frequency

k

m

T

2

2

m

k

Tf

21

1

Period: time for 1 full oscillation

frequency: number of oscillations per secondMeasured in cycles per second

- Hertz (Hz)

For springs

For springs

Note: frequency (f) and angular frequency (ω) measure the same thing but with different units. They differ by a factor of 2 pi.

Velocity and Acceleration

)sin( tAdt

dxv

)cos(22

2

tAdt

xda

Velocity of oscillation

Acceleration of oscillation

Note: magnitude of maximum values are when the sin and cosine arguments equal 1

Energy of Simple Harmonic Oscillators

2

2

1kAE

2

2

1mvK

2

2

1kxU

Remember that:

UKE

After substituting the equations of velocity(v) and distance(x) for simple harmonic oscillations, we get:

Applications: Simple Pendulum

sinmg

2

2

sindt

sdmmg

maF tensiontension

mg

TThe restoring force for a pendulum is

Ls

2

2

2

2

2

2

sin

sin

sin

dt

d

L

g

dt

dmLmg

dt

Ldmmg

wherewhich is the arclength or the path the ball travels alongthus

Simple Pendulums continued

sin

2

2

L

g

dt

d

L

g

xm

k

dt

xd

2

2

L

g

dt

d

2

2

g

LT

22

Notice that almost looks like

According to the small angle approximation, which states that sinθ ≈ θ if θ is small (about less than 10°)

We can rewrite the equation to be

which is exactly in the form for simple harmonic motions

where so then

we can now use all the other formulas for simple harmonic motions for the case of a pendulum

for small angles

Applications: Torsional Pendulum

When a torsion pendulum is twisted, there exists a restoring torque which is equal to:

This looks just like kxF but in rotational form

Thus, we can apply what we know about angular motion to get information about this object’s simple harmonic oscillations

Torsion Pendulum continued

Remember: I

Idt

d

dt

dI

I

2

2

2

2

After substituting we get

whereI

and

I

T 22