Wednesday, December 2, 1998

Chapter 13: Simple Harmonic Motionvelocity vs positioncircular motionperiods of springs

There will be two more meetings of Lab:- Thursday, December 3 (Lab 10)

- Thursday, December 10 (Lab Final=Party)MLK CenterPizza & Beverages providedOrders taken in lab tomorrow5:00 - 7:00 pm

Now, sketch a plotof the height of theblock above the flooras a function of time.

h

What kind ofmathematicalfunctions (withwhich you’refamiliar) result insuch a pattern?

Height of Block vs Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Hei

gh

t (m

)

Equilibriumposition

Amplitude

period

Amplitude

This type of oscillatory behavior is known as

An object in simple harmonic motion displaysan acceleration that is proportional to thedisplacement and in the opposite direction.

Now, sketch a plotof the velocity of theblock as a function oftime as it goes throughits oscillating motion.

h

How is this plotrelated to that forthe height?

Velocity of Block vs Time

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (s)

Vel

oci

ty (

m/s

)

Height of Block vs Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Hei

gh

t (m

) Height vs Time

Velocity of Block vs Time

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (s)

Vel

oci

ty (

m/s

)

Velocity vs Time

The wave for theheight is 1/4 periodbehind the wave forthe velocity!

Using your sketch of velocity vs time, try to sketch the acceleration of the block as a function of time.

h

How is this plotrelated to thevelocity plot?

Acceleration of Block vs Time

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (s)

Acc

eler

atio

n (

m/s

/s)

Acceleration of Block vs Time

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (s)

Accele

rati

on

(m

/s/s

)

Acceleration vs Time

Velocity of Block vs Time

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (s)

Vel

oci

ty (

m/s

)

Velocity vs Time

The wave for thevelocity is 1/4 periodbehind the wave forthe velocity!

Okay, so we’ve seen that objects in simpleharmonic motion can be described bytrigonometric functions (sines and cosines).

We know that these trigonometric functionscomplete one cycle (peak-to-peak) overwhat angular displacement?

0 1

That is to say,what is 1-0?

2

So what if we observe our spring to oscillatewith a period of 3 seconds. How would wewrite our function for the height of our blockversus time?

Take a stabat it!t = 0 s t = 3 s

3 s

At

cos 23

s

FHG

IKJ

t = 0 s t = 3 s

3 s

At

cos 23

s

FHG

IKJ

What is the Period of this wave?

What is the Frequency of this wave?

3 seconds

1/3 s

ATt A f tcos cos

22

FHGIKJ b g

ATt A f tcos cos

22

FHGIKJ b g

Now…dust off the cobwebs…

Where have we seenthis quantity before? 2f

Circular motion!Angular Frequency!!!

f

Now…you’ve got to be thinking to yourself...

What doesa spring have

to do withcircularmotion?

Well…I’m gonnatell ya!

Here’s mytennis ballon a stringagain.

We’relookingdown on theplane of motion.

What if I were now to shine a spotlight on this system from the right side and look at the motion of the shadow on a wall to the left?

L

I G

H

T

!

What will a graphof the height of the shadow on the wall look like?

Just like the position vs timefor the mass ona spring!

Height of Block vs Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Hei

gh

t (m

)Height of Shadow vs Time

So, the trigonometric functions describeBOTH circular motion (as we saw in Chapter 7)and the motion of a mass on a spring! WOW!

On what quantities did the period of ourtennis ball moving in a circle at the endof a string depend?

its speed and the radius of the circle.

Tr

v

2

If we look at the motionof the shadow, what willbe the amplitude of theoscillation?

r

So we might surmise that the period for ourmass-spring system will also be related tothe amplitude of the motion!

TA

v

2

max

Where A representsthe amplitude of theoscillation.