Chapter 15Oscillatory Motion

PHYS 2211

Recall the Spring

kxFspring

xm

ka

kxma

x

x

Since F=ma, this can be rewritten as:

The direction of the force is negative because it is a restoring force.In other words, if x is positive, the force is negative and vice versa.

This makes the object oscillate and as we will see, it undergoes simple harmonic motion.

Using the kinematic definitions…

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 write

then

)cos()( tAtx

A valid solution of this equation is:

Note: Solving differential equations is beyond the scope of this class. It is more important for us to know what the solution is.

Simple Harmonic Motion

)cos()( tAtxAn equation for distance (x) as time (t) changes;sin and cos are the basic components of any formula describing simple harmonic motion.

Here A, ω, ϕ are constantsA 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

The phase constant determines the starting position of the wave at the timet = 0.

General Concepts

kxF

xm

k

dt

xd

2

2

)cos()( tAtx

Anything with behaviors which have formulas that look like these (not including the constants) are undergoing simple harmonic motion and can be described using the same method as we used for 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: magnitudes of maximum values are when the sine/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

tantan

sindt

sdmmg

maF gentialgential

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