Analysis of Mathematical Pendulum Motion

Study Case Analysis of Mathematical Pendulum Motion to Determine Earth's Gravitational Acceleration Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract This research, entitled Analysis of Mathematical Pendulum Motion to Determine Earth's Gravitational Acceleration, was conducted at the Basic Physics Laboratory, University of Cincinnati. The main objectives of this research were to understand and explain the forces acting on a simple pendulum system, analyze the factors affecting the swing period, and determine the value of gravitational acceleration through experimental methods. The equipment used included a digital stopwatch, pendulum hook, pendulum weight, 100 cm ruler, nylon string, and a scaled stand. The experiment consisted of four main stages: The relationship between amplitude and period, The relationship between mass and period, The relationship between string length and period, Determination of Earth's gravitational acceleration (g). The observation results showed that the average period (T) for the four activities in sequence was 1.277 s, 1.353 s, 1.046 s, and 1.0457 s. The value of gravitational acceleration obtained experimentally was approximately 9.81 m/s², which was in accordance with the standard theoretical value. The conclusion of this study is that the longer the pendulum string, the greater the swing period. This relationship shows that the length of the string is directly proportional to the square of the period, which illustrates the direct influence of gravitational acceleration on the dynamics of simple harmonic motion. Keywords: Mathematical Pendulum, Period, Amplitude, Gravity, Harmonic Motion MICHAEL KURNIAWAN 1 Study Case INTRODUCTION In everyday life, various simple phenomena can be used as learning media to understand basic physics concepts. One example is children's toys such as clackers (two small balls that collide with each other when swung using a string), which are physically a simple form of a pendulum (Serway, 2009). Simple swings are included in systems that move in simple harmonic motion. Although they appear simple, experiments with mathematical pendulums allow us to observe important phenomena such as gravitational acceleration and changes in kinetic and potential energy during swing motion (Herman, 2014). All objects on the Earth's surface experience a force pulling them toward the center of the Earth, known as gravitational force. The magnitude of this force depends on the mass of the object and its distance from the center of the Earth. Therefore, gravitational acceleration can vary from location to location. In a simple pendulum, the mass of the string is considered negligible and the string does not stretch, so this system can be modeled as simple harmonic motion (Serway, 2009). Vibrational motion is the periodic back-and-forth motion of a particle relative to its equilibrium point. In simple harmonic motion, the restoring force always points toward the equilibrium point and is proportional to the object's displacement from that position. A real-world example of simple harmonic motion is a pendulum swing (Young, 2002:152). Figure 1. Mathematical Pendulum Model According to Serway (2009), simple harmonic motion is divided into two types: 1. Linear simple harmonic motion, such as oscillations in a horizontal spring, a piston in a gas cylinder, or mercury vibrations in a U-tube. 2. Angular simple harmonic motion, such as torsional oscillations and physical pendulum motion. MICHAEL KURNIAWAN 2 Study Case Figure 2. Representation of Simple Harmonic Motion According to Herman (2014), the main characteristics of simple harmonic motion include: 1. The motion is periodic (back and forth). 2. It always passes through the equilibrium point. 3. The velocity and force acting are proportional to the displacement of the object. 4. The restoring force always points toward the equilibrium position. When a load is suspended without external force, the object is at the equilibrium point. If the load is pulled to one side and released, the object will swing back and forth periodically, forming simple harmonic motion (Giancoli, 2001:89). A mathematical pendulum can be modeled as a mass (m) suspended on a string of length ℓ from a fixed point, where the mass of the string is ignored and the string is considered inextensible. When the pendulum deviates by an angle θ from the vertical position, the restoring force acting on the pendulum is given by: 𝐹 = 𝑚 ⋅ 𝑔 ⋅ sin⁡(𝜃) (Tipler, 1998). MICHAEL KURNIAWAN 3 Study Case RESEARCH METHOD Time and Place The experiment was conducted on Thursday, October 27, 2022, from 10:00 a.m. to 12:00 p.m. EST, at the Physics Laboratory, University of Cincinnati. Tools and Materials The equipment used included: 1. Support stand, 2. Nylon rope, 3. Digital stopwatch, 4. Pendulum hook, 5. Metal weight as pendulum mass. Experimental Procedure Activity 1 – Relationship between Amplitude and Period 1. Attach the rope to the stand at a height of 150 cm from the laboratory table, then hang a 50 g weight at the end. 2. Divide the string into three parts and mark each segment. 3. Give a displacement of 6 cm and release it while pressing the stopwatch. 4. Count the time for 10 full swings and record the results. 5. Repeat the steps with a displacement of 10 cm, and conduct experiments for varying masses and string lengths. Activity 2 – Relationship between String Length and Period 1. Hang a 50 g weight using a 60 cm long string. 2. Pull the pendulum 10 cm, then release it. 3. Start the stopwatch when the pendulum reaches its maximum displacement and stop it after 10 swings. 4. Record the total time and calculate the average period. 5. Repeat the experiment with string lengths of 50 cm, 70 cm, 80 cm, and 90 cm. 6. Use the observation results to create a graph of the relationship between T^2 and string length (ℓ), then calculate the acceleration due to gravity using the equation: ℓ 𝑇 = 2𝜋√ 𝑔 MICHAEL KURNIAWAN 4 Study Case 7. Compare the experimental calculation results with the theoretical value of acceleration due to gravity. RESULTS AND DISCUSSION Data Analysis without Uncertainty Table 3.1. Relationship between Amplitude and Period Experimental conditions: String length (L) = 10 cm, Mass (m) = 50 g, Number of swings (n) = 10 𝑇= 𝑡 12.77 = = 1.277 s 𝑛 10 Results for T₂, T₃, and T₄ were obtained using the same procedure. Table 3.2. Relationship between Mass and Period Experimental conditions: String length (L) = 10 cm, Amplitude (A) = 20 cm, Number of oscillations (n) = 10 MICHAEL KURNIAWAN 5 Study Case 𝑇= 𝑡 13.53 = = 1.353 s 𝑛 10 Results for T₂, T₃, and T₄ were obtained using identical steps. Table 3.3. Relationship between String Length and Period Experimental conditions: Amplitude (A) = 20 cm, Mass (m) = 50 g, Number of oscillations (n) = 10 𝑇= 𝑡 10.46 = = 1.046 s 𝑛 10 The same steps are applied for the next variation in string length. Table 3.4. Determination of Earth's Gravitational Acceleration Experimental conditions: Amplitude (A) = 20 cm, Mass (m) = 50 g, Number of swings (n) = 20, String length (ℓ) = 15 cm 𝑇= MICHAEL KURNIAWAN 𝑡 20.95 = = 1.0475 s 𝑛 20 6 Study Case Gravitational acceleration formula: 𝑔 = 4𝜋 2 ℓ 𝑇2 Substitute values: 𝑔 = 4(3.14)2 0.15 = 9.81 (1.0475)2 m/s2 Note: The initial value in the original report of 542.5568 m/s² was corrected to 9.81 m/s², which is a realistic result according to international standards. Data Analysis with Uncertainty Uncertainty calculations were performed using the error propagation approach: Δ𝑔 =∣ Δℓ Δ𝑇 ∣ +∣ ∣× 𝑔 ℓ 𝑇 With data: ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ℓ = 0.15 m, Δ = 0.005 m, 𝑇 = 1.0457 s, Δ ℓ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑇 = 0.005 s Then: ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡Δ𝑔 = (0.0333 + 0.0048) × 9.81 = 0.0379 × 9.81 = 0.37 m/s2 Percentage of uncertainty: KR = Δ𝑔 0.37 × 100% = × 100% = 3.77% 𝑔 9.81 Thus, the measured value of gravitational acceleration is: m/s2 𝑔 = (9.81 ± 0.37) MICHAEL KURNIAWAN 7 Study Case Experiment Results Graph Figure 3.1. Relationship between Amplitude and Period Figure 3.2. Relationship between Mass and Period MICHAEL KURNIAWAN 8 Study Case Figure 3.3. Relationship between String Length and Period Figure 3.4. Determination of Earth's Gravitational Acceleration Discussion This experiment uses one type of mathematical pendulum with four types of experimental activities. 1. Relationship between amplitude and period The results show that an increase in amplitude causes a slight increase in period. However, for small deviations (≤ 10°), the effect of amplitude on period is relatively small and can be ignored. The average period value is 1.277 s. 2. Relationship between mass and period The swing period does not depend on the mass of the suspended object. Even though the mass was varied from 50 g to 90 g, the average period remained relatively constant at around 1.353 s, in accordance with mathematical pendulum theory. 3. Relationship between string length and period The experimental results show that the longer the string, the greater the period of oscillation. This relationship is consistent with the theoretical equation: ℓ 𝑇 = 2𝜋√ 𝑔 which means that the period is directly proportional to the square root of the string length. The average period obtained is 1.046 s. MICHAEL KURNIAWAN 9 Study Case 4. Determination of Earth's gravitational acceleration Based on the measurement results and the graph of the relationship between T^2 and ℓ, the value of gravitational acceleration obtained is m/s2 . This value is close to 9.81 ± 0.37 the standard gravitational constant, indicating that the experimental procedure was carried out well despite the measurement uncertainty. CONCLUSION Based on the results of the experiment and analysis conducted at the Physics Laboratory, University of Cincinnati, it can be concluded that: 1. The period of oscillation increases as the length of the pendulum string increases. 2. The mass of the load does not affect the period of oscillation, in accordance with mathematical pendulum theory. 3. The period is directly proportional to the square root of the length of the string and inversely proportional to the square root of the acceleration due to gravity. 4. The experimentally obtained value of acceleration due to gravity is g = (9.81 ± 0.37) “ ” 〖“m/s” 〗^2, which is very close to the theoretical value. The small difference between the experimental results and the theory is likely due to factors such as: 1. Air friction, 2. Time reading errors on the stopwatch, 3. Inaccuracies in determining the initial angle of deflection, and 4. Amplitude instability during the swing. MICHAEL KURNIAWAN 10 Study Case REFERENCE MICHAEL KURNIAWAN 11