Lab on Centripetal Motion

(a) To study the relationship between frequency, f, and centripetal force, Fc, when radius and mass are constant.

(b) To study the relationship between frequency, f and radius, when force, Fc, and mass, m, are kept constant.

Calibration of spring:

Data Collection:

Mass Added (g)

Reading on Scale (cm)

0

18

20

16.8

40

15.6

60

14.6

80

13.5

100

12.4

120

11.2

140

10.2

160

9.1

180

8

Constants:

* Mass of spring

* Acceleration due to gravity

Data Processing:

First, the results need to be converted to SI units

Mass Added (kg)

Reading on Scale (m)

0

0.18

0.02

0.168

0.04

0.156

0.06

0.146

0.08

0.135

0.1

0.124

0.12

0.112

0.14

0.102

0.16

0.091

0.18

0.08

When there was no mass added to the scale, the reading on the scale was 0.18 meters. This means that the extension after adding each mass can be calculated.

Mass Added (kg)

Extension (m)

0

0

0.02

0.012

0.04

0.024

0.06

0.034

0.08

0.045

0.1

0.056

0.12

0.068

0.14

0.078

0.16

0.089

0.18

0.01

When the scale is calibrated, the aim is to find the relationship between the force and the extension. This means that the force resulting from the addition of each mass must be calculated in this step.

Mass Added (kg)

Force (N)

0

0

0.02

0.196

0.04

0.392

0.06

0.588

0.08

0.784

0.1

0.98

0.12

1.176

0.14

1.372

0.16

1.568

0.18

1.764

Therefore,

Force (N)

Extension (m)

0

0

0.196

0.012

0.392

0.024

0.588

0.034

0.784

0.045

0.98

0.056

1.176

0.068

1.372

0.078

1.568

0.089

1.764

0.1

Now, the string constant, k, must be calculated. Since , where F is the force and x is the extension, the string constant can be calculated.

If a graph were to be drawn of the extension against the force, then the slope would be the spring constant

The slope of the line was calculated to be 17.745, therefore, the string constant, k, is . The string constant is multiplied by the extension on the spring in order to find the force applied on the spring.

Part 1:

Data Collection:

Reading on Scale (cm)

Time for 20 Revolutions (s)

Radius (cm)

13

9.63

29.5

16

17.09

26.5

15

14.59

28

14

10.63

25.5

12

8.87

30.5

Constants:

* Radius is meant to be constant; although that is impossible (the radius in the readings has a range of 5 cm, which is quite large)

* Mass of spring and ball

Data Processing

Now, the extension must be calculated.

Reading on Scale (cm)

Extension (cm)

13

5

16

2

15

3

14

4

12

6

The results must now be changed into SI units

Extension (m)

Time for 20 Revolutions (s)

Radius (m)

0.05

9.63

0.295

0.02

17.09

0.265

0.03

14.59

0.28

0.04

10.63

0.255

0.06

8.87

0.305

In order to find the relationship between the force and the frequency, the results for the readings on the scale must be converted into force (N).

Extension (m)

Force (N)

0.05

0.88725

0.02

0.3549

0.03

0.53235

0.04

0.7098

0.06

1.0647

The readings were taken for the time for 20 revolutions, from this, the time for one revolution must be calculated.

Time for 20 Revolutions (s)

Time for one Revolution (s)

9.63

0.4815

17.09

0.8545

14.59

0.7295

10.63

0.5315

8.87

0.4435

Since then the inverse of the readings for time for one revolution must be calculated

Time for one Revolution (s)

Frequency s-1

0.4815

2.0768

0.8545

1.1703

0.7295

1.3708

0.5315

1.8815

0.4435

2.2548

Now, a graph of the force against the frequency can be plotted.

The second point seems to be an outlier, it will be excluded from all graphs. Another graph will be plotted of Frequency against Force

The slope of the linear trend line was calculated to be

However, another polynomial trend line was added, since it correlation is higher than that of the linear trend line, it should be explored further.

The formula for Centripetal force is:

This means that frequency squared is proportional to the centripetal force, so, a graph of force against frequency squared will be plotted.

Frequency s-1

Frequency2 s-2

2.0768

4.3133

1.1703

1.3695

1.3708

1.8791

1.8815

3.5399

2.2548

5.0841

For the sake of curiosity, the first reading will be excluded from the following graph to see if a better trend line is produced.

Yes, a better trend between the three points exists, but the absolute value of the y intercept is larger than the absolute value of the y intercept obtained from the previous graph.

It was deduced that, so, the slope will be calculated in each of the two graphs in order to see which is nearer.

If this was a perfect experiment, then the slope of one of the graphs would equal, unfortunately, this is an experiment with a very large error, and it is very likely that the slope on the graphs above deviates a lot from this theoretical slope.

Part B:

Data Collection:

Radius (cm)

Time for 20 Revolutions (s)

24

9.60

27

9.97

30

10.19

33

10.91

37

11.31

40

11.62

Constants:

* Force applied on spring

* Mass of spring and ball

Data Processing:

First, the time for one revolution must be calculated

Time for 20 Revolutions (s)

Time for One Revolution (s)

9.60

0.48

9.97

0.4985

10.19

0.5095

10.91

0.5455

11.31

0.5655

11.62

0.581

The results must now be changed to SI units

Radius (cm)

Radius (m)

24

0.24

27

0.27

30

0.30

33

0.33

37

0.37

40

0.40

The aim of the experiment is to study the relationship between frequency and radius, therefore, the times must be converted into frequencies by dividing by since

Time for One Revolution (s)

Frequency (s-1)

0.48

2.0833

0.4985

2.006

0.5095

1.9627

0.5455

1.8332

0.5655

1.7683

0.581

1.7212

Therefore,

Radius (m)

Frequency (s-1)

0.24

2.0833

0.27

2.006

0.30

1.9627

0.33

1.8332

0.37

1.7683

0.40

1.7212

A graph will now be plotted of the radius against the frequency

These trend line should be explored further, since there is obviously something additional that is still hidden.

The formula for centripetal force is:

If this is changed so that it is in terms of f, then

Therefore, f is proportional to the square root of

Radius (m)

(m-1)

(m-0.5)

0.24

4.1667

2.0412

0.27

3.7037

1.9245

0.30

3.0303

1.8257

0.33

3.3333

1.7408

0.37

2.7027

1.644

0.40

2.5

1.5811

A graph will now be plotted of f against the square root of

The slope of the trend line above is

If we were to rearrange the equation above in order to find the slope, then

Once again, the theoretical slope will not equal the slope of the trend line in the graph above; this is because the experiment has a lot of error, and large errors for that matter.

Conclusion:

In conclusion, it can be deduced from this experiment that the centripetal force is proportional to the frequency squared and that the frequency is related to the square root of the inverse radius.

Evaluation:

We tried to ensure the accuracy of our data by trying to have everything as accurate as possible. Firstly, we tried to draw straight lines on the scale of the spring in order to make accurate measurements. In the end we came up with a simple mechanism which we would move up and down to check where the reading is on the scale. Secondly, we tried to check our measurements before recording them. Finally, we decided to have only one person do the “experimentation” and the other to record the data.

We had several errors in our lab. Firstly, we are not sure that our measurements are 100% correct. Secondly, we are not exactly sure whether the spring was calibrated all the time and calibrated properly all the time. Thirdly, it was difficult to have a steady force while twirling the ball, so the force was not the same all the time. Fourth, we are not sure whether the reading on the scale was read correctly while twirling the ball. Finally, there was a human error while handling the stopwatch. Humans have a reaction time of around 0.25 seconds and also there might have been counting errors.

Many changes can be done to this lab in order to make it better, but this lab will always have a large error no matter what is done. What can be done to improve is include more than one trial for all the readings. In addition, if this lab was computerized and mechanized then the error would be small. But for now, what can be done is to precision of the data as there are a lot of random errors in the lab.