Pendulum motion Trung Bao Truong

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Dec 6, 2023

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Pendulum motion Trung Bao Truong W215647597 ABSTRACT: In this experiment we measure the period of a simple pendulum and use scientific methods to determine the relationships between the period of a pendulum and its length, its mass, and the amplitude of its motion. By recording data from the motion sensor, we change factors such as length, mass, and amplitude so that their relationship can be found. And based on the formulas to calculate the gravity of the earth. INTRODUCTION: A simple pendulum is made up of a mass known as the pendulum “bob”, which is suspended from a massless string of negligible thickness. When the pendulum bob is displaced from its equilibrium position and released, it oscillates back and forth. The period of the pendulum is defined as the time required for the pendulum bob to complete one full oscillation, moving from 1 position to the opposite 2 position, and then returning to the starting point (see Figure 2). We use a Motion Sensor in this experiment to accurately record the angular motion of the pendulum. We can precisely measure the period of the pendulum's oscillation thanks to the sensor's data acquisition capabilities and software tools (Capstone software). We also investigate the effects of three key variables: the length of the pendulum string (L), the mass of the pendulum bob (m), and the amplitude of its motion (A). To set up it we flow that step: 1. Constructing the Apparatus: - Attach the 90 cm Rod firmly to the Rod base. - Secure the Pendulum clamp onto the rod, positioning it near the upper end. - Fasten a string, approximately 1 meter long, to the Pendulum clamp, mimicking the configuration depicted in Figure 3. 2. String Configuration: - Thread the string through the opening in the brass cylinder. - Extend the string's ends to the inner and outer clips of the clamp, shaping a 'V' form as the string suspends. - Fine-tune the string's position to achieve a vertical distance of about 45 cm from the lower edge of the Pendulum clamp to the midpoint of the pendulum bob. 3. Placing the Motion Sensor: - Employ a table clamp and the 45 cm rod to anchor the motion sensor (as shown in Figure 1) in front of the pendulum. - Orient the motion sensor's brass-colored disk vertically and ensure it faces the

pendulum bob. Align the disk with the projected path of the pendulum's swing. - Configure the range switch on the motion sensor to the "top" setting. - Adjust the motion sensor's height to match that of the pendulum, ensuring they are in alignment. 4. Adjusting Distance and Alignment: - Arrange the rod base to position the motion sensor approximately 25 cm away from the pendulum bob. This spatial arrangement optimizes precise data acquisition. 5. Acquiring Data: - Establish a connection between the motion sensor and the data interface. - Facilitate data transmission and collection by connecting the data interface to the desktop computer. With this careful setup procedure executed, the experimental apparatus stands ready to record the simple pendulum's motion during oscillation. Change the length of the String and collect the required data to test the relationship between period and length. The length begins at 45 cm and gradually decreases to 40 cm, 35 cm, 30 cm, 25 cm, 20 cm, while maintaining the same amplitude (6 cm). We can you linear function in the software to find the linear (y = mL + b). In the following section, keep the length of the pendulum and the amplitude of the oscillation constant, but change the mass by changing the body material (Brass, AI, Plastic). Adjust the pendulum length for each of them so the distance is about 60cm. Calculate the period of an oscillation with an amplitude of approximately 6cm. In the final section, adjust the pendulum's length so that the distance is approximately 40cm. Measure and record the period of an oscillation with an amplitude of about 10cm. Repeat for 8 cm, 6 cm, 4 cm, and 2 cm amplitudes. Base on all of the data, we can know relationships between the period of a pendulum and its length, its mass, and the amplitude of its motion. DISCUSSION: In the section 1, when we use aluminum pendulum with a 6cm amplitude and a 40cm length to collect the data and compare it according to the formula y = Asin( 2 π T x + φ ), we get good results; this is how we make sure we're doing it right to get the most accurate data from the pendulum. The data from the computer shows that the Pendulum oscillation of the experiment and the calculation is almost identical (figure 6) . When we change the length in turn and keep the remaining factors unchanged (mass and amplitude), we get different results corresponding to each length. The period of the pendulum decreases as the length

decreases. The data is presented in the following table 1: Table 1 Figure 6. Pendulum Oscillation When using the linear curve fit tool from the graph tool pallet, we can see that the recorded period data when changing the length is almost a straight line and it changes with the length of the pendulum (figure 7). Figure 7. Period as function of length Based on this we can find the slope of the graph equal to 3.77 based on the formula y = mL + b In the next section, we decrease the amplitude of the oscillation. Measure the period for an oscillation with an amplitude of about 10 cm; the result of the period was 1.34. Repeat for an amplitude of 8 cm, 6 cm, 4 cm, 2 cm. We get the result that the change in period is negligible (table 2), or the period stays the same when the pendulum's amplitude changes.

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Table 2 In the final section, we decrease the mass of the pendulum bob by change the other bobs ( Brass, AI, Plastic) and keep the amplitude of the oscillation (6 cm) and the length of the pendulum constant (60 cm). Which different bobs we have period based on their mass (table 3). Based on the data, the pendulum's mass does not significantly affect the period of the pendulum, as the mass changes markedly (207.6 g - 66.9 g), but the period does not change significantly. Table 3 From the above data, we can see that: If we decrease the length of the pendulum, the period will decrease. If we decrease the amplitude of the oscillation, the period will stay about the same. If we decrease the mass of the pendulum bob, the period will stay about the same. In general, the period of a pendulum is independent of amplitude only if the angle is small (less than 20 0 ) and we calculate the angle of the amplitude on that is 5.74 0 (using Sin formular with length of the pendulum was about 60cm and amplitude was about 6cm) Base on theoretical formula for the period of pendulum is given by T = 2 π √ L g where g is acceleration due to gravity. By squatting both side of the equation, we can show that for a graph of T 2 vs L, the slope of the resulting straight line is given by Slope = 4 π 2 g With slope is 3.37 we have g = 10.47. that mean, if we take pendulum to the moon where gravity is less, the period will increase. CONCLUSION: Through this experiment and theoretical formula for the period of a pendulum, we can see that the period of a pendulum depends preeminent on two factors: the length of the pendulum and the acceleration due to gravity.  Pendulum Length (L): The length of the pendulum determines its period. This means that a longer pendulum takes longer to complete one full oscillation than a shorter pendulum.  Acceleration due to Gravity (g): The greater the gravitational acceleration, the shorter the period. Pendulums on planets with stronger gravitational forces have shorter periods than those on planets with weaker gravitational forces. The data taken from the experiment is good enough that we can see the relationship between the period of a pendulum and its length, its mass, and the amplitude of its motion. It gives almost the same result as the theoretical one. There are a number of factors that affect the accuracy of the results in the experiment, such as the accuracy of the amplitude of the oscillation or the length of the pendulum, or when the pendulum is moving, it may be deflected from the motion sensor leading to errors. deviation in the results. To improve, we need to measure

more accurately and meticulously and find ways to reduce the influence of the surrounding environment on the pendulum- like wind...

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