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  • Rachel D’Souza

Conservation of Kinetic Energy on a Pendulum

Rachel D’Souza, Zurich International School

Abstract:

Applications of the conservation of kinetic energy ensure the safety and efficacy of design systems like machinery, vehicles, and roller coasters, and describe the behavior of objects in space, playing a crucial role in understanding the movements of planets, stars, and other celestial bodies. This research aims to investigate the behavior of energy in a non-traditional pendulum setup and prove the theory of the conservation of mechanical energy in a system. In this experiment, the transformations of mechanical energy was measured by observing the changes in the initial and final kinetic energies of blocks on strings that formed a pendulum, and the study concluded that the mechanical energy of the system was converted into other forms and the gravitational potential energy was not conserved. The conservation of energy is a fundamental principle in physics. The objective is to investigate how the kinetic energy of a swinging pendulum changes within an untraditional pendulum system. A motion capture system is employed to precisely measure the pendulum's position and velocity throughout its swing. During the experiment, the pendulum's position and velocity are recorded at different points in its oscillation, allowing for the determination of kinetic energy at each instance. The relationship between kinetic energy, gravitational potential energy, and total mechanical energy is thoroughly examined. Kinetic energy oscillates between its maximum and minimum values, while being converted into other energies, like sound, to maintain the overall mechanical energy at a constant level to corroborate the principle of conservation. 


Introduction:

Energy has been theorized to never be created nor destroyed, but being able to transform into different forms. The hypothesis was that upon the collision between the two blocks, the final kinetic energy would be less than the initial kinetic energy as some mechanical energy would be converted into sound energy that created the collision noise between the blocks. This is grounded in the principle that the total mechanical energy would remain constant as long as no external forces act on the system. Specifically, as the pendulum oscillated, there would be a continuous exchange between potential and kinetic energy, resulting in total conserved energy. This would be seen in a two-block pendulum system where the height of a swinging block would have the same velocity and reach the same height as its initial velocity and height of release after colliding with a stationary block. Rather than a traditional single-block pendulum, adding a second block introduces new complexities for understanding energy transformations in oscillatory systems. Furthermore, non-conservative forces that are present and affecting the overall energy balance may contribute to a more comprehensive understanding of energy conservation.


Methods:

The independent variable in the following experiment was the height the block on the string is released from, and the dependent variable was the maximum height of the block on the string that was reached after the collision. To maintain consistency between the trials, controlled variables that were included were the length of the string, staying at 0.4 meters, the mass of the block on the string at 0.32 kilograms, the mass of the stationary block at 0.075 kilograms, and the height of the block on the string at 0.2 meters. The experiment began by measuring and recording the mass of the block to be placed at the base of the retort stand and the block on the string. The lighter and stationary block was then placed at the base of the retort stand. String was tied to the heavier block and was cut so that the length measured 40 centimeters. The retort stand was heightened to have the top of the stand 70 centimeters from the ground; a ruler was then placed against a board at the spine of the retort stand so that the height the block reaches after the collision with the stationary block could be seen. A recording device was then placed far enough away so that the whole stand-block system was in the frame. The block on the string was then tied to the top of the retort stand, and once the recording device had been started, the block was pulled to be a specific height (see raw data table), measured using the ruler, and released. Once the block on the string made contact with the stationary block and moved to the other side of the pendulum until it swung back down, the recording device was stopped, and the video showed the height the block on the string reached after the collision, which was recorded in the raw data table. The whole experiment was then repeated for two more trials and then the heights were increased in 10 centimeter intervals before the experiment was repeated again for three trials for each height of release. Once all the raw data was collected, the initial and final gravitational potential energies were calculated (see sample calculations) and added to a processed data table. 


Results



Figure 1: Raw Data This table shows the values obtained from the block in the experiment that was attached to a string, where the height of release was incremented by 0.1 meters and the height after colliding with the stationary block was measured. and the height after colliding with the stationary block was measured. 


Height of Release (m) 

Average Max. Height After Collision (m)

Initial GPE (J)

Final GPE (J)

KE Before Collision (J)

KE After Collision (J)

0.3

0.19

0.94

0.60

0.94

0.60

0.4

0.25

1.25

0.78

1.25

0.78

0.5

0.30

1.57

0.94

1.57

0.94

0.6

0.37

1.88

1.16

1.88

1.16

0.7

0.45

2.20

1.41

2.20

1.41


Height of Release (m) 

Velocity of Swinging Block Before Collision (m/s)

Velocity of Swinging Block During Collision (m/s)

Velocity of Stationary Block During Collision (m/s)

0.30

2.42

1.94

3.01

0.40

2.80

2.21

3.54

0.50

3.13

2.42

4.10

0.60

3.43

2.69

4.38

0.70

3.71

2.97

4.59


Tables 2a and 2b: Processed Data. This table shows the velocity values calculated from the block in the experiment that was attached to a string and the stationary block on the table where the height of release was incremented by 0.1 meters. 



Figure 3: This bar graph shows the initial and final kinetic energies of the block on the string before and after the collision with the stationary block where the kinetic energy is not conserved, making the collision inelastic, because the final kinetic energy value is lower than the initial kinetic energy value. 




Figure 4: This is the graph of the linear function showing the relationship between the height of the release and the average maximum height of the swinging block after colliding with the stationary block. The loss of mechanical energy can be seen in the equation as the slope of the line is less than 1. 




Figure 5:This bar graph shows the initial and final velocities of the block on the string before and after the collision with the stationary block that were calculated in the processed data table where the kinetic energy is not conserved because the velocity of the swinging block decreases throughout the experiment after colliding.


When the block on the string collided with the stationary block, there was an energy transfer to the stationary block that made it start to move, as well as a transformation from kinetic to sound and thermal energy that made the “colliding” sound. As indicated by Figure 5,  a linear relationship is observed between the height of the release and the average maximum height reached by the block on the string after the collision. The x- coefficient is 0.64, which means that the average maximum height of the block on the string after the collision is less. If the slope of that line was 1, then the x- values and y- values would be equal with possibly a small difference from adding the y- intercept. The outcome that kinetic energy was not conserved in the block-block system can additionally be proved by noting  that the initial gravitational potential energies and final gravitational potential energies were not the same.

The loss of kinetic energy of the block on the string indicates that the velocity of the block decreased after the collision, as the only other variable in the kinetic energy equation is the mass, which was a constant. The decreased velocity of the block on the string can be seen in Figure 5, as well as in Figure 2. The stationary block, with an initial velocity of 0, had a velocity of 3.01 m/s during the collision for one of our trials. In comparison, the swinging block had a decrease in velocity from 2.42 m/s to 1.94 m/s in that same trial, indicating a transfer of energy. Kinetic energy and velocity are proportional because the height of the bars in each of the two bar graphs show a similar trend for the initial and final velocities. The velocity of the blocks decreased, which decreased the kinetic energy


Discussion:

The initial velocity of the block was greater than the final velocity of the block, which means that the block did not reach the same height as the height of the release when the block on the string swung back up after the collision. This is also known from the gravitational potential energy equation as the height is the only variable that was changed. The initial gravitational potential energy value is equal to the kinetic energy of the block on the string before the collision because when the height of the block is 0, all the gravitational potential energy has been converted to kinetic energy once the block is moving. The final gravitational potential energy value is equal to the kinetic energy of the block on the string after the collision because all the remaining kinetic energy of the block was converted to gravitational potential energy when the block swung back up and stopped moving. The gravitational potential energies, and therefore kinetic energies, were not conserved. By contrasting these results with those expected in a conventional pendulum, the significance of these deviations is more apparent. The total mechanical energy of the system would remain constant, but the results above suggest the presence of additional forces influencing the system that may not be accounted for in traditional pendulum models. 

It is essential to acknowledge the limitations and potential sources of error. There were assumptions made about the ideal conditions, such as frictionless surfaces and perfectly inelastic collisions between the blocks which are not fully representative of real-world scenarios. Additionally, while air resistance was minimized, there were possible slight deviations in the mechanical energy conservation. In future investigations, refining the apparatus would ensure more accurate results. Nevertheless, the results from this experiment provide valuable insights in engineering and robotics. The data revealed that friction between the block on the table and the surface had a greater impact on energy conservation than expected, and can inform engineers working on mechanical systems on more efficiency materials with lower friction coefficients to reduce losses in their systems, leading to cost savings and improved sustainability. Engineers can also explore materials with greater tensile strengths to minimize the effects of string elasticity in applications like suspension bridges or shock absorbers. The substantial impact of the angle at which the pendulum is released from or the initial velocities of the blocks can help in understanding the sensitivities of initial conditions in robotics or automation systems, and lead to improved control algorithms and more precise movements.



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