Experiment 7 – Period of oscillation pendulums of different lengths complete (in air)

Aim of experiment:

Main aim: To find out how the period of oscillation differ when the length of the pendulum varies.

Sub aim: To find out if the length of pendulum affects the period of oscillation proportionally.

Hypothesis:

Main hypothesis: As stated in my summary of research (continued), the longer the pendulum, the wider the arc of the oscillation, the longer the distance traveled by the pendulum. The period of oscillation should thus, be longer as the time taken to cover the distance will increase. If this is true, the period of oscillation for the pendulum with length of 5cm should be the shortest, while the period of oscillation for the pendulum with length of 15cm should be the longest.

Sub hypothesis: Based on the equation: T = 2pÖ(L/G), the length of the pendulum should not affect the period of oscillation proportionally.

Materials Needed:

- 1 pendulum (5cm)

- 1 pendulum (10cm)

- 1 pendulum (15cm)

- 1 stopwatch (up to milliseconds)

Experiment Set-up:

Experiment Procedure:

1. Raise the 15cm long pendulum at least 15cm from the ground.

2. Clamp it to a support.

3. Raise the weight, parallel to the ground and make sure the string is straight.

4. Release the weight and start the stopwatch simultaneously.

5. Stop the stopwatch after 5 oscillations are completed.

6. Repeat the experiment 4 times more for more accurate results.

7. Repeat steps 1-6 with the 10cm and 5cm long pendulums.

Independent Variable: Length of the pendulum.

Constant Variables:

1. Weight of pendulum.

2. Height to which pendulum is raised to before release.

Dependent Variable: Time the pendulum took to complete 5 oscillations. / Period of oscillation.

Results:

Length of pendulum/cm	Time taken for 5 oscillations/s	*Period of oscillation/s
5	2.81
	2.69
	2.98
	3.06
	2.84
Average time taken for 5 oscillations/s	2.876	0.5752
10	3.95
	3.97
	3.93
	3.95
	3.95
Average time taken for 5 oscillations/s	3.95	0.79
15	4.69
	4.69
	4.8
	4.48
	4.82
Average time taken for 5 oscillations/s	4.696	0.9392

* Calculated by dividing average time taken to complete 5 oscillations by 5.

Data Analysis:

The longest pendulum (15cm) took the longest time to complete 5 oscillations, 4.696s, resulting in the largest period of oscillation of 0.9392s. The shortest pendulum (5cm) took the shortest time to complete 5 oscillations, 2.876s, resulting in the shortest period of oscillation of 0.5752s. The 10cm long pendulum had a period of oscillation of 0.79s. Although the period of oscillations are not equivalent to the values derived from the equation (0.4486, 0.6344, 0.7769), the values still increase as the length of the pendulum increases. However, the period of oscillation of the pendulums of varying lengths do not increase proportionally. For example, the 10cm long pendulum does not have a period of oscillation that is twice as long as that of the 5cm long. Likewise, the period of oscillation of the 15cm long pendulum does not triple that of the 5cm long.

Conclusion:

The longer the pendulum, the longer the period of oscillation. Also, the length of the pendulum does not affect the period of oscillation proportionally. This conclusion is valid only with the following assumptions:

1. The pendulum was raised to the same height before release, and there was no external force acting on it.

2. Human error in starting and stopping the stopwatch is minimal.

Summary of Research (Continued)

Pendulum
- The longer the pendulum, the wider the arc of the oscillation, the longer the distance travelled by the pendulum. The period of oscillation should thus, be longer as the time taken to cover the distance will increase.
- Formula to calculate period of oscillation:



Where:
T = period of oscillation
L = Length of pendulum in metres
G = acceleration of gravity (m/s)
|o| = amplitude (angle)


http://en.wikipedia.org/wiki/Harmonic_oscillator


T² is proportional to L:


L 
T^{2 = constant}


http://www.educationalelectronicsusa.com/p/shm-II.htm

Experiment Refinement

To find out more about the movement of a pendulum, I have decided to further my research to other variables affecting the oscillations of a pendulum:

1. Viscosity (refined experiment procedure; 1cSt, 10cSt, 20cSt, 30cSt)

2. Length of pendulum (5cm, 10cm, 15cm)

I will be modifying the method of measuring the time per oscillation. Previously, I chose to time only one period of oscillation, which resulted in inaccurate results. Thus, to obtain more accurate results, I will be measuring the total time taken for the pendulum to complete 5 oscillations and then divide the total time by 5 to obtain one period of oscillation. 

Firstly, I will be conducting an experiment with different lengths of pendulum in air. This will act as a control before I move on to conducting the experiment in liquids of different viscosities. From this, I will also be able to find out if the length of the pendulum affects the period of oscillation proportionally. (E.g. x2 length of pendulum = x2 period of oscillation?)

Next, I will then conduct the experiment with the 10cm long pendulum in the liquids of different viscosities. I will be able to find out if the viscosity of the liquid affects the period of oscillation proportionally. (E.g. x2 viscosity of liquid = x2 period of oscillation?)

Experiment 6 – Number of oscillations pendulum completes in liquids of different viscosities

Aim of experiment: To find out how number of oscillation the pendulum completes differ when it is placed in liquids of different viscosities.

Hypothesis:

As stated in my summary of research, viscosity is a measure of the resistance of a liquid to flow, or in other words, the friction against flow. Thus, the more viscous the liquid, the more friction it should have against flow. Therefore, the number of oscillations the pendulum completes in water (viscosity 1) should be the highest number as the pendulum will lose less energy through friction; while the number of oscillations the pendulum completes in honey water (viscosity 30) should be the lowest number as more energy will be lost to overcome friction, resulting in the least number of oscillations.

Materials Needed:

- 1 litre of water (viscosity 1)

- 1 litre of honey water (viscosity 10)

- 1 litre of honey water (viscosity 20)

- 1 litre of honey water (viscosity 30)

- 1 pendulum

- 1 measuring beaker (1 litre)

Experiment Procedure:

1. Fill the measuring beaker with water till it reaches the 1 litre mark.

2. Allow the honey to reach the temperature of the air-conditioned room (20ºC).

3. Use the thermometer to check the temperature.

4. Position the chopsticks above the beaker such that the pendulum (string included) is fully submerged.

5. Raise the pendulum till it is just touching the surface of the water.

6. Release the pendulum.

7. Count the number of complete oscillations the pendulum makes.

8. Repeat the experiment 4 times more for more accurate results.

9. Repeat steps 1-8 with honey water (viscosity 10,20,30) instead of water.

Independent Variable: Viscosity of liquids the pendulum moves through

Constant Variables:

1. Amount of mixture poured into beaker (1 litre).

2. Length of string of pendulum.

3. Weight of pendulum.

4. Temperature of liquids tested.

5. Height to which pendulum is raised to before release.

Dependent Variable: Number of complete oscillations made by the pendulum before coming to a stop.

Results:

Liquid	Viscosity	No. of Oscillations
Water (20ºC)	1	5
		6
		5
		5
		6
	Average no. of oscillations	5.4
Honey Water (20ºC)	10	4
23% Honey		5
77% Water		5
		5
		4
	Average no. of oscillations	4.6
Honey Water (20ºC)	20	3
49% Honey		3
51% Water		4
		4
		4
	Average no. of oscillations	3.6
Honey Water (20ºC)	30	2
74% Honey		2
26% Water		2
		3
		2
	Average no. of oscillations	2.2

Data Analysis:

The honey water with viscosity of about 30 resulted in the shortest number of oscillations while water with viscosity of about 1 resulted in the most number of oscillations completed by the pendulum. The number of oscillations completed decreased as the viscosity of the liquids increased.

Conclusion:

The more viscous the liquid, the less number of oscillations the pendulum completes before coming to a stop. This conclusion is valid only with the following assumption:

1. The pendulum was raised to the same height before release, and there was no external force acting on it.

Experiment 5 – Time per oscillation of pendulum in liquids of different viscosities.

Aim of experiment: To find out how the time per oscillation of a pendulum differ when it is placed in liquids of different viscosities.

Hypothesis:

As stated in my summary of research, particularly with reference to Stoke’s Great Law of Drag:

Where

r = the radius of a sphere in cm

d1 = the density of the sphere in g/cm3

d2 = the density of the fluid in g/cm3

g = the local gravitational acceleration in cm/S2

c = the Viscosity of the fluid in Poise

v = the terminal velocity in cm/S

I can derive that the more viscous the fluid, the smaller the terminal velocity. Thus, I can form an hypothesis for my experiment (how does viscosity affect the movement of a pendulum in terms of speed?): The higher the viscosity of the liquid, the slower the pendulum will travel, therefore, the longer the time taken for one oscillation.

Materials Needed:

- 1 litre of water (viscosity 1)

- 1 litre of honey water (viscosity 10)

- 1 litre of honey water (viscosity 20)

- 1 litre of honey water (viscosity 30)

- 1 pendulum

- 1 measuring beaker (1 litre)

- 1 stopwatch

Experiment Procedure:

1. Fill the measuring beaker with water till it reaches the 1 litre mark.

2. Allow the honey to reach the temperature of the air-conditioned room (20ºC).

3. Use the thermometer to check the temperature.

4. Position the chopsticks above the beaker such that the pendulum (string included) is fully submerged.

5. Raise the pendulum till it is just touching the surface of the water.

6. Release the pendulum and start the stopwatch at the same time.

7. Stop the stopwatch as soon as one oscillation is completed.

8. Repeat the experiment 4 times more for more accurate results.

9. Repeat steps 1-8 with honey water (viscosity 10,20,30) instead of water.

Independent Variable: Viscosity of liquids the pendulum moves through.

Constant Variables:

1. Amount of mixture poured into beaker (1 litre).

2. Length of string of pendulum.

3. Weight of pendulum.

4. Temperature of liquids tested.

5. Height to which pendulum is raised to before release.

Dependent Variable: Time taken for the pendulum to complete one oscillation.

Results:

Liquid	Viscosity	Time/Osciliation (s)
Water (20ºC)	1	0.6
		0.68
		0.73
		0.72
		0.73
	Average time per oscillation	0.692
Honey Water (20ºC)	10
23% Honey		0.84
77% Water		0.87
		0.75
		0.84
		0.86
	Average time per oscillation	0.832
Honey Water (20ºC)	20	1.13
49% Honey		1.05
51% Water		1.19
		1.01
		1.16
	Average time per oscillation	1.108
Honey Water (20ºC)	30	1.38
74% Honey		1.42
26% Water		0.97
		1.3
		1.15
	Average time per oscillation	1.244

Data Analysis:

The honey water with viscosity of about 30 resulted in the longest time per oscillation of the pendulum, while water with viscosity of about 1 resulted in the shortest time. The time per oscillation increased as the viscosity of the liquids increased.

Conclusion:

The more viscous the liquid, the longer the time taken for the pendulum to complete one oscillation. This conclusion is valid only with the following assumptions:

1. The pendulum was raised to the same height before release, and there was no external force acting on it.

2. Human error of starting and stopping the stopwatch was minimal.

Pendulum

Materials used:

- 1 magnet ball (25g)

- 1 piece of string

- 1 pair of chopsticks

- Tape

Experiment 4 – Finding a mixture between honey and water with a viscosity of 30

Aim of experiment: To find a mixture with a viscosity of 30 by mixing honey and water in different percentages.

Hypothesis:

Using the equation derived from experiment 2b’s hypothesis,

The percentage of honey in the mixture

= (30 – 1.004) ÷ 0.39238

= 28.996 ÷ 0.39238

= 73.89775218%

= 74% (Round off to whole no.)

The percentage of water in mixture

= 100 – 74

= 26%

Therefore, if the viscosities of water and honey are accurate, the mixture should consist of 74% honey and 26% water to achieve a viscosity of 20.

Materials Needed:

- 1 Zahn cup set-up (From experiment 1)

- 1 measuring cup (100ml)

- 1 measuring cup (44ml)

- 1 stopwatch (times up to milliseconds)

- 1 bottle of honey

- 1 calculator

- 1 mixing spoon

- 1 thermomete

Experiment Procedure:

1. Measure 74ml of honey into the measuring cup (100ml).

2. Measure 26ml of water into the same measuring cup.

3. Mix it well.

4. Allow the honey to reach the temperature of the air-conditioned room (20ºC).

5. Use the thermometer to check the temperature of the mixture.

6. Pour the mixture into the Zahn Cup and start the stopwatch at the same time.

7. Stop the stopwatch when the continuous flow of honey water is broken.

8. Take down the timing up to milliseconds.

9. Repeat steps 1-8 4 times more for more accurate results.

10. Using the formula [1.1(time taken for water to flow from orifice – 29)], calculate the viscosity of the mixture based on the 5 timings.

11. Calculate the average viscosity of the 5 values.

*Note: If percentages result in a viscosity of more than 0.1 cSt off from the expected viscosity of 30 cSt, re-conduct the experiment and increase or decrease the percentage of honey by 1% each time till viscosity is close to 10 cSt.

Independent Variable: (only is hypothesis is false) Percentage of honey in mixture.

Constant Variables:

1. Amount of mixture poured into Zahn Cup.

2. Size of orifice of Zahn Cup.

3. Temperature of honey.

Dependent Variable: Time taken for the continuous flow of honey water through the orifice to stop.

Results:

Liquid		Time		Viscosity
Honey Water (20ºC)	56.43		30.173
74% Honey	56.33		30.063
26% Water	56.21		29.931
	56.51		30.261
	56.02		29.722
	Average Viscosity		30.03

Data Analysis:

A mixture of 74% honey and 26% water has a measured average viscosity of 30.03 cSt. This is very close to my intended 30 cSt.

Conclusion:

To obtain a liquid with a viscosity of 30 cSt, I have to create a mixture with 74% honey and 26% water. This conclusion is valid only with the following assumptions:

1. The mixture was mixed well and the honey was not clumped together.

2. Human error of starting and stopping the stopwatch was minimal.