Add Math Pendulum
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~Contents~ Introduction 1 Aim
-------------------------------------------------------------------------------------
2
Task Specification 3
-------------------------------------------
Problem Solving 4 – 15
-------------------------------------------
Further Exploration ------------------------------------------16 – 19 Conclusion
------------------------------------------- 20
Acknowledgements ------------------------------------------21
~Introduction~ Time is a component of a measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects. Time has been a major subject of religion, philosophy, and science, but defining time in a non-controversial manner applicable to all fields of study has consistently eluded the greatest scholars. There are many ways to measuring time; some ancient methods include using a sundial, or an hourglass. A pendulum can also be used as a basis for measuring time. A simple pendulum consists of an object, usually a pendulum bob, suspended by a string from a fixed point. When displaces, and then released, the bob will swing back and forth in a vertical plane under the influence of gravity. This periodic motion can be used as a basis to determine time. In this project work, I will focus on using the simple pendulum to measure time, with the help of some mathematical knowledge.
Photo 1: A sundial is an ancient method for measuring time
Photo 2: Another way to measure time is using an hourglass
2 1
~Aim~ The aims of carrying out this project work are to: i)
Develop mathematical knowledge in a increases students’ interest and confidence;
way
ii)
Apply mathematics to everyday situations and to begin to understand the part that mathematics play in the world we live;
iii)
Improve thinking skills and mathematical communication;
iv)
Assist students to develop positive attitude and personalities, intrinsic mathematical values such as accuracy, confidence, and systematic reasoning;
v)
Stimulate learning and enhance effective learning
promote
which
effective
2 1
~Task Specification~ Identifying and stating all required information
Figure 1
The simple pendulum shown in Figure 1 is set in motion by releasing the object through a small angle of displacement, (10 to 15 ) from the vertical. Procedure: 1. A simple pendulum is set up as shown in Figure 1 by attaching an object to a string of length 60cm. 2. The pendulum is set in motion and the time taken, t s is measured for 20 complete oscillations.
2 1
3. The period, T s, that is the time taken for one complete oscillation is calculated. 4. Steps (1) to (3) are repeated using at least 10 different lengths of strings with the minimum length of 5 cm. 5. The readings are recorded in a suitable table. 6.
A graph of period (T s) is plotted against length ( graph obtained is commented.
). The
~Problem Solving~ Results from the conducted experiment:
2 1
Length of Number of pendulum complete , l (cm) oscillation (x) 60 20 55 20 50 20 45 20 40 20 30 20 20 20 15 30 10 30 5 30
Time taken for number of complete oscillation, t (s) 1st 2nd Average Attempt Attempt 32 32 32.0 30 31 30.5 29 29 29.0 27 27 27.0 26 26 26.0 23 23 23.0 19 19 19.0 25 25 25.0 21 21 21.0 17 17 17.0
Period, T (s)
1.600 1.525 1.450 1.350 1.300 1.150 0.950 0.833 0.700 0.567
The Relationship between Period, T and length, l The relationship between period and length is shown by the given formula:
The gravitational acceleration, g is not a variable, it is a constant. Therefore the variables in the following formula are T and . (a)Suggest at least two pairs of variables for the horizontal and vertical axes to obtain a linear relation. For each pair, plot the graphs to draw the lines of best fit manually and by using ICT.
2 1
To obtain a linear equation, the equation above can be squared to form a linear equation.
The two pairs of variables are: (i) T and ( T as the y-axis and as the x-axis) (ii) and ( as the y-axis and as the x-axis)
In order to draw the graphs of these two pairs of variables, we need to find the vales of and . Period, T (s) 1.600 1.525 1.450 1.350 1.300 1.150 0.950 0.833 0.700 0.567
2.560 2.326 2.103 1.823 1.690 1.323 0.903 0.694 0.490 0.322
Length of pendulum, (cm) 60 55 50 45 40 30 20 15 10 5
(cm) 7.746 7.416 7.071 6.708 6.325 5.477 4.472 3.873 3.162 2.236
2 1
Graph of
against
Graph of
against
2 1
Drawing the graphs manually i. Graph of T against
2 1
ii.
Graph of
against
2 1
(b)Estimating the gradients of both graphs From the graph T against , we can find that the equation is 0.188x+0.115.From the formula , we can determine the gradient if we compare the formula to the equation obtained. Therefore, the gradient of the graph is 0.188. The equation relating the period and the length is T = 0.188 From graph T2 against , we find that the equation is y = 0.041x + 0.112. From the formula , we can determine the gradient if we compare the formula to the equation obtained. Therefore, the gradient of the graph is 0.041. The equation relating the period and the length is T 2 = 0.041 + 0.112 (c)
Determining the value of the gravitational acceleration, using the gradient of each graphs.
2 1
(i) From the graph T against √l,
or
T = y-axis = x-axis So gradient =
To find the value of gravitational acceleration, g ms-2
Since 1116.663 is in cm s-2, we have to change it into m s-2 by dividing the value by (10 x 10 = 100). We get 11.17 m s-2. This value is bigger than the original value of 9.807 m s-2. Value of obtained from the experiment = 11.17 Value of actual value of = 9.807
2 1
Difference
= 11.17 = 1.363
9.807
Percentage of error = = 13.898%
Comment: The value obtained from the experiment is higher by 1.363 from the actual value of g, which is 9.807. The reason is the experiment not conducted in an enclosed system, therefore, many factors can influence the accuracy of the results obtained, such as air movement, and air resistance. (ii)
From the graph of
against
or
2 1
= y-axis = x-axis So gradient = To find the value of gravitational acceleration,
m s-2,
Since 986.96 is in cm s-2, we have to change it into m s-2 by dividing the value by (10 x 10 = 100). We get 9.87 m s -2. This value is a little bigger than the original value of 9.807 m s-2. Value of obtained from the experiment = 9.87 Value of actual value of = 9.807 Difference
= 9.87 = 0.063
9.807
Percentage of error = = 0.642% Comment: If compared with the first graph, the percentage of error is much smaller. The first value of obtained is slightly higher than the actual value of , but the second value of obtained is a bit higher than the actual value. This is probably caused by the presence of air resistance. Another probable cause of the deviation of the value from the actual value is that the pendulum bob did not oscillate in a plane but in a circle.
2 1
(d)Using the graph with the least percentage of error to determine the length of the string that will produce a complete oscillation in 1 second. The graph with the least percentage of error is the graph showing against . The relationship between period and length is:
When
= 1,
25.00cm is the length of string that will produce a complete oscillation in 1 second.
Using a Simple Pendulum to Calculate the Pulse Rate 1. A simple pendulum is made out of a 25.00cm string ( to ensure that 1 oscillation is equivalent to 1 second) 2. A friend is asked to count the number of oscillation
2 1
3. He is asked to give instructions when to start and stop (after 30 oscillations). 4. The counting of the pulse is started when he says start and stopped when he says stop after 30 oscillations. 5. The process is repeated 3 times to get the value, Paverage. 6. The pulse rate = Paverage x 2. Change in the Period, if the length of the string in increased by 4 times. When l is increased by 4 times, = Substitute into the equation,
Since
Then,
Therefore, when the length of the pendulum is increased by 4 times, the period increases by 2 times.
~Further Exploration~ 2 1
Determining the new period if the pendulum is set in motion on the moon Gravitational field strength of the moon, gmoon = 1.622ms-2 Period of pendulum on Earth, TEarth =1s Gravitational acceleration of the Earth, gEarth = 9.807 ms-2 Period of pendulum on the moon, TMoon = TMoon Technique of solving: Using simultaneous equation
Solve (1) and (2) simultaneously
Investigating whether a pendulum will continuously swing in air
2 1
A pendulum cannot swing continuously swing in air due to damping. Damping is defined as loss of energy to the surroundings, usually in the form of heat energy. In air, there is air resistance, which induces friction upon the swinging pendulum bob. This causes the energy in the pendulum bob to be loss in the form of heat energy, hence damping occurs. This results in the pendulum swinging slower and slower. For the pendulum to become a perpetual machine (swing continuously), it has to swing in complete vacuum. Conditions for the pendulum to swing continuously: - No hinge friction - In a vacuum - No internal friction Comparing the time taken for the pendulum to stop in water with the time taken for the pendulum to stop in air. Time taken for the pendulum to complete one oscillation in water: T1 = 4s T2 = 5s T3 = 4s TAverage = 4.33s Time taken for the pendulum to complete one oscillation in air: T1 = 1.55s T2 = 1.32s T3 = 1.62s TAverage = 1.50s Based on the values above, the time taken for the pendulum to make a complete oscillation in air is shorter, if compared to the time taken for the same pendulum to make a complete oscillation in water.
2 1
This is due to the water resistance and the buoyant force of the water acting on the pendulum bob, thus making the pendulum bob swing slower, as the water resistance is stronger to overcome, compared to air resistance. Comparing the motion of the pendulum in air, water and in a vacuum In Air
In water
In Vacuum
2 1
Comparing and contrasting the graphs
2 1
~Conclusion~ A pendulum can be used to measure time, as shown and proven in the experiment I conducted. The period, time taken for a complete oscillation can be calculated using the formula below:
The period of the pendulum swinging in air and in water varies, as the water resistance is stronger, compared to air resistance. The pendulum is not affected by gravity, but due to internal resistance, hinge resistance and air resistance, it is not able to oscillate continuously, i.e becoming a perpetual machine. As shown in the experiment, the pendulum is quite a reliable instrument to calculate time, but is not used, as the process of calculating time using a pendulum is very complex.
2 1
~Acknowledgements~ I would like to express my gratitude and thanks to my teacher, Mr Annuar Ali for her wonderful guidance for me to be able to complete this project work, to my parents for their continuous support to me throughout this experiment, to my friends for their help, and to all those who contributed directly or indirectly towards the completion of this project work. Throughout this project, I acquired many valuable skills, and hope that in the years to come, those skills will be put to good use.
2 1
-------------------------------------------------------------------------------------
2
Task Specification 3
-------------------------------------------
Problem Solving 4 – 15
-------------------------------------------
Further Exploration ------------------------------------------16 – 19 Conclusion
------------------------------------------- 20
Acknowledgements ------------------------------------------21
~Introduction~ Time is a component of a measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects. Time has been a major subject of religion, philosophy, and science, but defining time in a non-controversial manner applicable to all fields of study has consistently eluded the greatest scholars. There are many ways to measuring time; some ancient methods include using a sundial, or an hourglass. A pendulum can also be used as a basis for measuring time. A simple pendulum consists of an object, usually a pendulum bob, suspended by a string from a fixed point. When displaces, and then released, the bob will swing back and forth in a vertical plane under the influence of gravity. This periodic motion can be used as a basis to determine time. In this project work, I will focus on using the simple pendulum to measure time, with the help of some mathematical knowledge.
Photo 1: A sundial is an ancient method for measuring time
Photo 2: Another way to measure time is using an hourglass
2 1
~Aim~ The aims of carrying out this project work are to: i)
Develop mathematical knowledge in a increases students’ interest and confidence;
way
ii)
Apply mathematics to everyday situations and to begin to understand the part that mathematics play in the world we live;
iii)
Improve thinking skills and mathematical communication;
iv)
Assist students to develop positive attitude and personalities, intrinsic mathematical values such as accuracy, confidence, and systematic reasoning;
v)
Stimulate learning and enhance effective learning
promote
which
effective
2 1
~Task Specification~ Identifying and stating all required information
Figure 1
The simple pendulum shown in Figure 1 is set in motion by releasing the object through a small angle of displacement, (10 to 15 ) from the vertical. Procedure: 1. A simple pendulum is set up as shown in Figure 1 by attaching an object to a string of length 60cm. 2. The pendulum is set in motion and the time taken, t s is measured for 20 complete oscillations.
2 1
3. The period, T s, that is the time taken for one complete oscillation is calculated. 4. Steps (1) to (3) are repeated using at least 10 different lengths of strings with the minimum length of 5 cm. 5. The readings are recorded in a suitable table. 6.
A graph of period (T s) is plotted against length ( graph obtained is commented.
). The
~Problem Solving~ Results from the conducted experiment:
2 1
Length of Number of pendulum complete , l (cm) oscillation (x) 60 20 55 20 50 20 45 20 40 20 30 20 20 20 15 30 10 30 5 30
Time taken for number of complete oscillation, t (s) 1st 2nd Average Attempt Attempt 32 32 32.0 30 31 30.5 29 29 29.0 27 27 27.0 26 26 26.0 23 23 23.0 19 19 19.0 25 25 25.0 21 21 21.0 17 17 17.0
Period, T (s)
1.600 1.525 1.450 1.350 1.300 1.150 0.950 0.833 0.700 0.567
The Relationship between Period, T and length, l The relationship between period and length is shown by the given formula:
The gravitational acceleration, g is not a variable, it is a constant. Therefore the variables in the following formula are T and . (a)Suggest at least two pairs of variables for the horizontal and vertical axes to obtain a linear relation. For each pair, plot the graphs to draw the lines of best fit manually and by using ICT.
2 1
To obtain a linear equation, the equation above can be squared to form a linear equation.
The two pairs of variables are: (i) T and ( T as the y-axis and as the x-axis) (ii) and ( as the y-axis and as the x-axis)
In order to draw the graphs of these two pairs of variables, we need to find the vales of and . Period, T (s) 1.600 1.525 1.450 1.350 1.300 1.150 0.950 0.833 0.700 0.567
2.560 2.326 2.103 1.823 1.690 1.323 0.903 0.694 0.490 0.322
Length of pendulum, (cm) 60 55 50 45 40 30 20 15 10 5
(cm) 7.746 7.416 7.071 6.708 6.325 5.477 4.472 3.873 3.162 2.236
2 1
Graph of
against
Graph of
against
2 1
Drawing the graphs manually i. Graph of T against
2 1
ii.
Graph of
against
2 1
(b)Estimating the gradients of both graphs From the graph T against , we can find that the equation is 0.188x+0.115.From the formula , we can determine the gradient if we compare the formula to the equation obtained. Therefore, the gradient of the graph is 0.188. The equation relating the period and the length is T = 0.188 From graph T2 against , we find that the equation is y = 0.041x + 0.112. From the formula , we can determine the gradient if we compare the formula to the equation obtained. Therefore, the gradient of the graph is 0.041. The equation relating the period and the length is T 2 = 0.041 + 0.112 (c)
Determining the value of the gravitational acceleration, using the gradient of each graphs.
2 1
(i) From the graph T against √l,
or
T = y-axis = x-axis So gradient =
To find the value of gravitational acceleration, g ms-2
Since 1116.663 is in cm s-2, we have to change it into m s-2 by dividing the value by (10 x 10 = 100). We get 11.17 m s-2. This value is bigger than the original value of 9.807 m s-2. Value of obtained from the experiment = 11.17 Value of actual value of = 9.807
2 1
Difference
= 11.17 = 1.363
9.807
Percentage of error = = 13.898%
Comment: The value obtained from the experiment is higher by 1.363 from the actual value of g, which is 9.807. The reason is the experiment not conducted in an enclosed system, therefore, many factors can influence the accuracy of the results obtained, such as air movement, and air resistance. (ii)
From the graph of
against
or
2 1
= y-axis = x-axis So gradient = To find the value of gravitational acceleration,
m s-2,
Since 986.96 is in cm s-2, we have to change it into m s-2 by dividing the value by (10 x 10 = 100). We get 9.87 m s -2. This value is a little bigger than the original value of 9.807 m s-2. Value of obtained from the experiment = 9.87 Value of actual value of = 9.807 Difference
= 9.87 = 0.063
9.807
Percentage of error = = 0.642% Comment: If compared with the first graph, the percentage of error is much smaller. The first value of obtained is slightly higher than the actual value of , but the second value of obtained is a bit higher than the actual value. This is probably caused by the presence of air resistance. Another probable cause of the deviation of the value from the actual value is that the pendulum bob did not oscillate in a plane but in a circle.
2 1
(d)Using the graph with the least percentage of error to determine the length of the string that will produce a complete oscillation in 1 second. The graph with the least percentage of error is the graph showing against . The relationship between period and length is:
When
= 1,
25.00cm is the length of string that will produce a complete oscillation in 1 second.
Using a Simple Pendulum to Calculate the Pulse Rate 1. A simple pendulum is made out of a 25.00cm string ( to ensure that 1 oscillation is equivalent to 1 second) 2. A friend is asked to count the number of oscillation
2 1
3. He is asked to give instructions when to start and stop (after 30 oscillations). 4. The counting of the pulse is started when he says start and stopped when he says stop after 30 oscillations. 5. The process is repeated 3 times to get the value, Paverage. 6. The pulse rate = Paverage x 2. Change in the Period, if the length of the string in increased by 4 times. When l is increased by 4 times, = Substitute into the equation,
Since
Then,
Therefore, when the length of the pendulum is increased by 4 times, the period increases by 2 times.
~Further Exploration~ 2 1
Determining the new period if the pendulum is set in motion on the moon Gravitational field strength of the moon, gmoon = 1.622ms-2 Period of pendulum on Earth, TEarth =1s Gravitational acceleration of the Earth, gEarth = 9.807 ms-2 Period of pendulum on the moon, TMoon = TMoon Technique of solving: Using simultaneous equation
Solve (1) and (2) simultaneously
Investigating whether a pendulum will continuously swing in air
2 1
A pendulum cannot swing continuously swing in air due to damping. Damping is defined as loss of energy to the surroundings, usually in the form of heat energy. In air, there is air resistance, which induces friction upon the swinging pendulum bob. This causes the energy in the pendulum bob to be loss in the form of heat energy, hence damping occurs. This results in the pendulum swinging slower and slower. For the pendulum to become a perpetual machine (swing continuously), it has to swing in complete vacuum. Conditions for the pendulum to swing continuously: - No hinge friction - In a vacuum - No internal friction Comparing the time taken for the pendulum to stop in water with the time taken for the pendulum to stop in air. Time taken for the pendulum to complete one oscillation in water: T1 = 4s T2 = 5s T3 = 4s TAverage = 4.33s Time taken for the pendulum to complete one oscillation in air: T1 = 1.55s T2 = 1.32s T3 = 1.62s TAverage = 1.50s Based on the values above, the time taken for the pendulum to make a complete oscillation in air is shorter, if compared to the time taken for the same pendulum to make a complete oscillation in water.
2 1
This is due to the water resistance and the buoyant force of the water acting on the pendulum bob, thus making the pendulum bob swing slower, as the water resistance is stronger to overcome, compared to air resistance. Comparing the motion of the pendulum in air, water and in a vacuum In Air
In water
In Vacuum
2 1
Comparing and contrasting the graphs
2 1
~Conclusion~ A pendulum can be used to measure time, as shown and proven in the experiment I conducted. The period, time taken for a complete oscillation can be calculated using the formula below:
The period of the pendulum swinging in air and in water varies, as the water resistance is stronger, compared to air resistance. The pendulum is not affected by gravity, but due to internal resistance, hinge resistance and air resistance, it is not able to oscillate continuously, i.e becoming a perpetual machine. As shown in the experiment, the pendulum is quite a reliable instrument to calculate time, but is not used, as the process of calculating time using a pendulum is very complex.
2 1
~Acknowledgements~ I would like to express my gratitude and thanks to my teacher, Mr Annuar Ali for her wonderful guidance for me to be able to complete this project work, to my parents for their continuous support to me throughout this experiment, to my friends for their help, and to all those who contributed directly or indirectly towards the completion of this project work. Throughout this project, I acquired many valuable skills, and hope that in the years to come, those skills will be put to good use.
2 1
