PSC 151Laboratory Activity 4

Graphical Analysis IIB Nonlinear Graphs 2

andThe Motion of a Simple Pendulum

Using Graphical Analysis to Investigate the Motion of a

Simple Pendulum

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are needed to see this picture.

The composition and motion of a pendulum can be described in terms of four measurable quantities.

Independent Variables

MassLengthAmplitude

Dependent Variable

Period

On which of the independent variables is the period dependent?

Only two variables can be investigated at a time.

Period versus MassLength & Amplitude constant

Period versus AmplitudeLength & Mass constant

Period versus LengthMass & Amplitude constant

Investigating the dependence of the Period on the Mass of the pendulum.

Varying the mass while keeping the amplitude and length constant

Experimental Set-Up

As the pendulum swings the CBR emits sound waves which reflect off of the pendulum and return to the CBR.

The CBR calculates the distance to the pendulum and sends the data to the TI-83 which then plots the position of the pendulum versus time.

CBRTI-83

Sound Waves

CBR / TI-83 Set-UPConnect the CBR to the TI-83

Press: APPS

Press “4” CBL/CBR

Press “Enter”

Press “2” Data Logger

Data Logger Set-Up

Probe

# SAMPLES

INTRVL (SEC)

UNITS

PLOT

DIRECTNS

GO...

Sonic

Enter: 75

Enter: .02

Select: m

Select: REAL TIME

Select: ON

Press "Enter"

Press: Enter

CBR/CBL Set-Upcontinued

Press “2” CBR

After CBR-CBL link has been tested:Press: “Enter”

After “Status OK”:Press: “Enter”

When you are ready to begin taking data”Press: “Enter”

After data collection is complete the TI-83 will plot a graph of the pendulum’s position versus time.

During Data Acquisition

Taking Data From the Graph:Determining the Period

Time

Position

Cursor starts here

*

Trough X = 0.00 Y = 0.00X = 0.11 Y = 0.34

*

X = 0.18 Y = 0.41

*X = 0.22 Y = 0.63

*X = 0.48 Y = 1.01

*

t1 =0.22st2 =0.48s

T =0.26s

t2 =0.48s

X = 0.76 Y = 1.88*

t3 =0.76s

T =0.28s

Trial 1

Trial 2

Measuring the PeriodUse this key to advance cursor

Dependence of the Period on the Mass of the Pendulum

Each group will use a different mass and determine the pendulum’s period. Then the period related to each mass will be recorded in a composite data table.

From this composite data table each group will determine whether or not the period of a pendulum depends on its mass.

Length (constant) cmAmplitude (constant) cm

(Your) Mass gTrial # Period, s

123

Average

Data Table 1 Dependence on Mass

7510

0.370.390.370.38

Dependence of the Period on the Amplitude and Length of the Pendulum

We will now use Interactive Physics to simulate the motion of a simple pendulum and determine the dependence of the Period on Amplitude and Length.

Mass = 6.0kg Length = 10.0mAmplitude, m Period, s

1.02.03.04.05.06.0

Does the Period of a Pendulum depend on its Amplitude?

6.336.336.336.336.336.33

We will first exam the dependence on the Amplitude by choosing a mass (6kg) and a length (10m) and holding them constant while varying the amplitude.

Does the Period depend on the Length?Choose a mass (6kg) and amplitude (10m) and hold them constant while varying the length.

Mass = 6.0kg Amplitude = 10.0mLength, cm Period, s

520406080100

Does the Period of a Pendulum depend on its Length?

What is the mathematical relationship between Period and Length?

4.508.95

12.6515.5017.9020.05

We will begin by plotting a graph of Period, T versus Length, L.

If this graph is a straight line we then determine its slope and y-intercept and use the general slope-intercept equation to determine the relationship between T and L. T =mL+b

Length, m

Period versus Length

If the graph of period versus length is not a straight line we must determine what function of L to graph next.

Study the various graph shapes to determine which one most resembles the graph of Period versus Length.

Once a new function of L has been chosen, create a new column in the data table for that function.

01

2

3

4

5

6

7

Y

1 2 3 4 5 6 7X

y=x

Y=mX+ b

Y∝X

70

10

20

30

40

Y

1 2 3 4 5 6X

=y x2

Y∝X2

Y=mX2 +b

0

0.5

1

1.5

2

2.5

Y

2 4 6 8X

= y x1/2

Y ∝ X

Y =m X +b

Revised Data Table

Length-L, m Period-T, s5 4.50

20 8.95

40 12.65

60 15.50

80 17.90

100 20.05

Convert lengths based on new function.

Next, plot a new graph of T versus L

New Function of LSqrt L, √L

If this graph is a straight line determine is slope and intercept, and use the general slope-intercept equation to determine the relationship.

If this graph is not a straight line, continue the process with a different function of L until a straight line graph is achieved.

Sqrt L, L

Period versus Sqrt L

T =m⋅ L +b