Circles

Jacob Yambasu Physics write up =Title: Moving In Circles! Criteria Assessed: DCP and CE Aim: To Investigate the relationship between orbital period and ; a. the radius of the circular orbit with a constant centripetal force b. the centripetal force on the body for a constant orbital radius Raw Data Experiment A Investigating the relationship between orbital period and the radius of the circular orbit with a constant centripetal force Mass Of Weight / g ± 0.1 g Mass Of Rubber Bung / g ± 0.01 g Radius Of Circular Orbit / cm ± 0.01cm

50.0 18.60 Time for 20 Revolutions/s ± 0.01s Length Increasing 11.90 16.00 19.41 18.29 21.15

25.10 44.90 59.70 64.60 84.30

Length Decreasing 11.93 15.67 18.90 18.35 22.03

Observation As the orbital radius increased the system in circular motion was going slower and slower. Therefore, it became increasingly easier to count the number of revolutions as well as stop the stop clock when all the 20 revolutions were made. When rotating the system, it was difficult to keep the rotating system perfectly horizontal. Also as the experiment progressed, the experimenters hand became tired making it difficult to perform the rotation.

Experiment B Investigating the relationship between orbital period and the centripetal force on the body for a constant orbital radius

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Jacob Yambasu Physics write up Radius Of Circular Orbit / cm ± 0.01 Mass Of Rubber Bung / g ± 0.01 g Mass Of Weight / g ± 0.1 g 50.0 70.0 100.0 130.0 140.0

64.60 18.60 Time for 20 Revolutions Length Increasing Length Decreasing 18.29 18.33 15.97 15.88 13.47 13.44 11.78 11.87 11.47 11.69

Observation As the weights were increased the system in circular motion was going faster and faster. Therefore, it became increasingly difficult to count the number of revolutions as well as stop the stop clock when all the 20 revolutions were made.

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Jacob Yambasu Physics write up Processed Data in Tables Experiment A : Mass Of Load / g ± 0.1 g

50.0

Mass Of Rubber Bung / g ± 0.01 g

18.60

Radius Of Circular Orbit / cm ± 0.05cm

Radius Of Circular Orbit /m ± 0.0005m

Centripetal Force /N ±0.001 N

Time for 20 Revolutions/s ± 0.01s

Length Increasing

Average Time For 20 Length Revolutions Decreasing /s ± 0.01s

0.4905

Period , T /s Relative uncertainty in Period , T

Period2 , T2 / s2

∆Perod2 , T2 / ± s2

25.10

0.2510

11.9

11.93

11.92

0.59575

0.000839

0.3549

0.0006

44.90

0.4490

16

15.67

15.84

0.79175

0.000632

0.6269

0.0008

59.70

0.5970

17.54

17.81

17.68

0.88375

0.000566

0.7810

0.0009

64.60

0.6460

18.29

18.35

18.32

0.91600

0.000546

0.8391

0.0009

84.30

0.8430

21.15

22.03

21.59

1.07950

0.000463

1.1653

0.001

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Jacob Yambasu Physics write up Experiment B : Radius Of Circular Orbit / cm ± 0.05cm

64.60

Mass Of Rubber Bung / g ± 0.01 g

18.60

Mass Of Weights / g ± 0.1 g

Mass Of Load / kg ±0.0001 kg

Centripetal Force /N ±0.001 N

50.0

0.05

70.0 100.0

Time for 20 Revolutions/s ±0.01s

Average Time For 20 Revolutions /s ± 0.01s

Period , T /s ± 0.0005s

Relative uncertainty in Period , T

Length Increasin g

Length Reducin g

0.491

18.29

18.33

18.31

0.9155

0.000546

0.07 0.1

0.687 0.981

15.97 13.47

15.88 13.44

15.93 13.46

0.7963 0.6728

0.000628 0.000743

130.0

0.13

1.275

11.78

11.87

11.83

0.5913

0.000846

140.0

0.14

1.373

11.47

11.69

11.58

0.5790

0.000864

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Period-2 1÷T s-1

Uncertainty in Period-2

1.1931 1.577 2.209

0.001

2.861 2.983

0.005

0.002 0.003 0.005

Jacob Yambasu Physics write up Calculation of Results and Uncertainties For Processed Table of Results • •

• •

•

Radius of circular orbit in meters = Radius of circle orbit in centimeters ÷ 100 ∆ Radius of circular orbit in meters = ∆ Radius of circle orbit in centimeters ÷ 100 = 0.05cm ÷ 100 = ± 0.0005 m Mass of weights in kilograms = Mass of load in grams ÷ 1000 ∆Mass of weights in kilograms = ∆Mass of load in grams ÷ 1000 = 0.1g ÷ 1000 = ± 0.0001kg

Since the mass of the attached weight provides the centripetal force, the centripetal force is given by Centripetal Force , FC.P = Weight of attached weight on string = Mass of weights in kilograms × g = Mass of weights in kilograms × 9.81

•

∆ Centripetal Force , FC.P = ∆Mass of weights in kilograms × 9.81 = 0.0001kg × 9.81 = 0.00098 N = ± 0.001 N •

Average Time For 20 Revolutions = Time for 20 Revolutionslength increasing + Time for 20 Revolutionslength increasing = 2 • ∆ Average Time For 20 Revolutions = ∆Time for 20 Revolutionslength increasing + ∆Time for 20 Revolutionslength increasing = 2 • •

Time for 20 Revolutionslength increasing 20 ∆Time for 20 Revolutionslength increasing ∆ Period , T = 20 0.01s = 20 = ± 0.0005s Period , T =

∆Period Period

•

Relative Uncertainty in Period =

• •

Period2, T2 = Period × Period ∆ Period2, T2 = Period2 × (Relative Uncertainty in Period + Relative Uncertainty in Period)

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Jacob Yambasu Physics write up ∆Period ∆Period + ) Period Period 1 Inverse Of Period, T2 = Period 2 1 ∆ Inverse Of Period, T2 = × (Relative Uncertainty in Period + Relative Uncertainty in Period) Period 2 1 ∆Period ∆Period = + ) 2 × ( Period Period Period = Period2 × (

• •

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Jacob Yambasu Physics write up

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Derivation of equation relating; Centripetal force, mass in orbit, Orbital Radius, Orbital Period From Fundamentals When rotating an object of mass, m in a horizontal circular motion with a orbital radius, R at a velocity, V the centripetal force is given by ;

F

C.P

mv = R

2

But v = Rω

1

and

ω=

2π T

Where ω is the angular velocity and T is the orbital period

∴ V = Substituting

2πR T

V =

2πR in T

1

2

F

C .P

 2πR  = m  ÷R  T 

∴

F

C .P

4mπ 2 R = T2

Hence, we can see that the centripetal force is inversely proportional to the square of the orbital period 1 ( F C.P ∞ 2 ) T Also

4mπ 2 R T = F C .P 2

Hence, the square of the orbital period is directly proportional the orbital radius

(T 2∞R ) Therefore to investigate these relationships the following graphs will be plotted; a) Orbital radius against the square of the orbital period 1

www.sidney.k12.oh.us/schools/tenneb/Labs/CentripetalForce.pdf

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Jacob Yambasu Physics write up b) Centripetal force against the square of the orbital period Processed Results Relationship between Period and Orbital Radius

F C .P R =T × 4mπ 2 2

Of the form y = mx + c

Where y = Square of Period, T 2 m , gradient =

F C.P 4 mπ 2

x = Orbital Radius, R c=0 Graph of Square of Orbital Period against of Radius of Circular Orbit

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Jacob Yambasu Physics write up

y 0.8 0.7

0.6 0.5 0.4

0.3 0.2 rclbt,/m fC adiusO R

0.1 x

-0.05

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

Square of Orbital Period, T2 / s2

To judge the success in the experiment, the centripetal force obtained using the gradient of the graph will be compared with the centripetal force calculated from the mass of the weight. The gradient of the best fit line calculated using the graphing software. The gradient was 0.7435 m s-2 Therefore as F C.P 4mπ 2 = Gradient × 4mπ 2

The gradient =

F C .P But gradient = 0.7435 m T-2 mass m = 0.01860 kg 2 F C . P from gradient of graph == (0.7435) × 4(0.01860)π F C . P from gradient of graph = 0.5460 N 9

Jacob Yambasu Physics write up and F C . P from mass of weight = 9.81 × 0.05kg = 0.4905 N Hence percentage difference of derived centripetal force and actual measured centripetal force = FC.P from mass of weight - FC.P from gradient of graph × 100 FC.P from mass of weight 0.4905 - 0.5460N × 100 = 0.4905

=

=11.3% Comment On Graph and Results 1. The line of best fit almost past through the origin as it was displaced downwards from the origin by only 0.003 units. This could be due to a systematic error. Also, the centripetal force derived from the gradient of the graph varied from the actual centripetal force provided by the weights by 11.3% which is fairly accurate. Hence, with minimal error, the graph shows that the square of the orbital period is directly proportional to the radius of circular orbit as expected. 2. The range of coordinates were above and below the line of best fit indicating that there was a variation of random errors in the coordinates. 3. The uncertainties from the apparatus were negligible as the error bars are not even visible from the graph. Hence, the errors were mainly errors caused by errors in the procedure.

Relationship between Period and Centripetal Force

F C . P = 4 mπ 2 R ×

1 T2

, Of the form y = mx + c

Where y = Centripetal Force m , gradient = 4mπ 2 R x = Inverse Square of Period,

1 T2

c=0

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Jacob Yambasu Physics write up

Graph of Centripetal Force Against the Inverse Square of the Orbital Period

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Jacob Yambasu Physics write up

y 1.4

1.2

1

0.8

0.6

entripalFoc/N 0.4 C

0.2 x 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Inverse Square of Orbital Period, T-2 /s-2

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2.6

2.8

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Jacob Yambasu Physics write up

To judge the success of the experiment, the mass of object obtained using the gradient of the graph will be compared with the actual measured mass of object ; The gradient of the best fit line calculated using the graphing software. The gradient was 0.4806 N s2 Therefore as Gradient = 4mπ 2 R Gradient 4π 2 R 0.4806 Mass, m = 4π 2 (0.646)

Mass, m =

Mass , m from gradient of graph = 0.0188 And Mass, m from measured mass = 0.01860 Hence percentage different of Derived mass and actual measured mass = FC.P from measured mass of weight - FC.P from gradient of graph × 100 = FC.P from mass of weight 0.01860 - 0.0188 × 100 = 0.01860

=1.075269% =1.1% Comment On Graph and Results •

•

•

The line of best fit past 0.0790 units below the origin. This could be due to a systematic error from experimental procedures. Also, the mass of the object in orbit derived from the gradient of the graph varied from the actual mass of the object in orbit by 1.1=% which is very accurate. Hence, with minimal error, the graph shows that the centripetal force is inversely proportional to the square of the orbital period. The line past directly through the first 3 coordinates but past above and below the last 2 coordinates. This shows that random errors were reduced when the centripetal force is lower as seen from the first 3 coordinates and the random error was increased when the centripetal force is higher as evident from the last 2 coordinates. The uncertainties from the apparatus were negligible as the error bars are not even visible from the graph. Hence, the errors were mainly errors causes by errors in the procedure

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Jacob Yambasu Physics write up Conclusion The findings of the experiment were as follows; a) The relationship between the orbital period and the centripetal force is that the square of the orbital period is inversely proportional to the centripetal force ( T 2 ∞

1 F C .P

)

as a straight line passing close to the origin was observed when F C . P was plotted 1 against 2 . T b) The relationship between the orbital period and the radius of the circular orbits is that the square of the orbital period is directly proportional to the radius of the circular orbit (T 2 ∞R ) as a straight line passing through the origin was observed when R was plotted against T 2 . These relationships are valid as they correctly relate to the known equation relating these physical quantities as shown below;

F

C.P

4mπ 2 R = T2

Possible sources of weakness and error in experiment 1. a. The experimenter might have applied some amount of force to the rotating system thereby by adding on to the weights to produce the centripetal force. This may have caused a small error in the experiment. b. The rotating system was difficult to keep horizontal. Hence, at certain times during the experiment the rotating system was not rotating in a circular plane but was rotating in a conical motion. Hence, this could have lead to a systematic error. 2. As the experiment progressed, the experimenters hand became more tired and hence could not rotate the rotating system well. 3. As seen from the experiment B, when the weights became too heavy, timing the revolutions became difficult and thereby increased the degree of error in the time and the final results. Similarly in experiment A, timing the revolutions became difficult when the orbital radius became too short and thereby increased the degree of error in the time and the final results.( although it was not clearly evident from the graph that these errors were significant). Also when the orbital radius became too long, it became difficult too perform the horizontal circular rotation.

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Jacob Yambasu Physics write up

Improvements to Experimental Design and Procedures 1. a. For perfect horizontal circular motion, a machine can be used to replace the human experimenter. An example of one such machine is shown below ; 2

b. If no other improved apparatus for the rotating system is available, an extended portion of time should be allocated (example 30 minutes) to mastering the circular rotation so that during the actual experiment the rotating system will remain horizontal and thereby reduces systematic error of poor performance by the experimenter. 2. To avoid poor rotating of the rotating system caused by experimenters hand becoming tired, after every measurement the partners should switch roles (i.e. the roles of timing the revolutions and rotating the rotating system). 3.

2

www.sidney.k12.oh.us/schools/tenneb/Labs/CentripetalForce.pdf

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Jacob Yambasu Physics write up a. The mass of the weight should be 100 grams or less. This would ensure that the rotating system rotates at a reasonable velocity that will enable the timing of the revolutions with minimal error. Also the range of measurements for the orbital radius should be between 30cm and 70 cm as this would reduce the errors of the orbital radius being too long (leading to difficulty in rotation) and of the orbital radius being too short (leading to difficulty in timing). b. Alternatively, a data logging apparatus can be used to record the time for the revolutions. This would improve the accuracy as it removes the error of the slow reaction time of the experimenter in stopping the stop clock. References www.sidney.k12.oh.us/schools/tenneb/Labs/CentripetalForce.pdf

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