Lab Report 6

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Jan 9, 2024

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Phys 207 Lab CD7 Report #6: Simple Harmonic M±ion Lindsey Rosario Introduction For the laboratory experiment, we investigated the simple harmonic motion of a mass hanging from a spring, using a motion detector to collect data. Our primary objective was to confirm some of the fundamental relationships that govern the simple harmonic motion of a mass on a spring. During the experiment, we utilized a Logger Pro file that we downloaded and used to collect and visualize the data. We could remove graphs by selecting them and deleting them. The mass was attached to the spring and allowed to oscillate freely up and down. We used the motion detector to collect data on the mass's motion, which was then displayed on the laptop in real-time graphs of the mass's position and velocity. Procedure Review your station's items and make sure they comprise of the following: ● A meter stick ● A 12-inch ruler ● A 5-mass set ● A LabPro interface ● A spring assortment ● A motion detector Once you're done with organizing your station, continue the lab report and proceed with the next steps. Suspend 1.000 kg in the air through a spring so that the mass is not moving.
Lab 6: Simple Harmonic M±ion . Presuming the mass is at rest, sketch the free-body diagram of the load. Classify each of the forces. Use the sketch in relation to the sum forces to determine the spring constant k. Demonstrate how you arrived at your assessment. Next, reset your station and hang 1.000 kg on a spring again. This time use Logger Pro to lay out position vs. time, velocity vs. time, and acceleration vs. time. Amass a set of results while the load is in motion. Construct experiments to observe how the period depends on mass and amplitude. Your report should include with the following graphs: ● Period as a function of mass ● Period as a function of the square root of mass ● Period as a function of amplitude Amass a series of position vs. time, velocity vs. time, and acceleration vs. time data for an oscillating mass of 1.000 kg. Inquire conservation of energy by using your collected data to determine and differentiate the total energy of the system at 2 points in the mass’s trail. With the first point being at the highest placement of mass and the second one being placed where the mass is at the equilibrium position. Construct an experiment to assess how long it will take for the amplitude to dwindle to 2/3 of the starting amplitude? Complete the experiment and report your data. After all is completed, reset your station and organize your items in an orderly fashion. Make sure nothing is missing and leave the station as it is. Data and Calculations Spring Constant: -23N/m Energy at highest point: 0.0414J Energy at lowest point: 0.0414J Constant Mass and Amplitude Graphs: 2
Lab 6: Simple Harmonic M±ion 3
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Lab 6: Simple Harmonic M±ion Change in mass with constant amplitude Graphs: 4
Lab 6: Simple Harmonic M±ion Change in amplitude with constant mass Graphs: 5
Lab 6: Simple Harmonic M±ion 6
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Lab 6: Simple Harmonic M±ion Questions 1. Describe how the mass moves relative to the equilibrium position. a. During this experiment, the mass attached to the spring moves in simple harmonic motion relative to the equilibrium position. As the mass is displaced from its equilibrium position, the spring exerts a force that acts to restore the mass back to the equilibrium position. The resulting motion of the mass is periodic and oscillatory, where the mass moves back and forth about the equilibrium position with a constant amplitude and frequency. 2. Calculate the maximum velocity from the position vs. time graph. Show your calculations. Compare with the value from the velocity time graph. a. By observing the steepest peaks of the graph, and through Excel, gaining the trendline for those steep turns, as well as observing the velocity data from the LogPro, we can say the maximum velocity is 0.1667m/s. 3. At what position is the velocity a maximum? 7
Lab 6: Simple Harmonic M±ion a. On the position vs. time graph, the velocity is maximum at the points where the curve is steepest, meaning where the slope of the curve is maximum (positive or negative). These points are located at the maximum or minimum displacement of the oscillating object, where it changes direction. At these points its magnitude is maximum. 4. Calculate the minimum velocity from the position vs. time graph. Show your calculations. Compare with the value from the velocity time graph. a. The minimum velocity during harmonic motion will be zero. This is due to the short time during each end of the oscillation period before the object quickly reverses and moves in the opposite direction due to spring force. 5. At what position is the velocity a minimum? a. On the position vs. time graph, the velocity is minimum (i.e., zero) at the points where the curve is at its highest or lowest position and momentarily stops before changing direction. These points are known as the turning points or the equilibrium positions of the object in simple harmonic motion. At these points, the object's kinetic energy is zero, and all of its energy is in the form of potential energy. 6. Calculate the maximum acceleration from the velocity vs. time graph. Show your calculations. Compare with the value from the acceleration vs. time graph. a. By observing the LogPro data, we can see that the maximum acceleration value is 0.998m/s^2. 7. At what position is the acceleration a maximum? a. On the position vs. time graph, the acceleration is maximum at the points where the curve changes direction most quickly, which corresponds to the 8
Lab 6: Simple Harmonic M±ion maximum displacement of the oscillating object. At these points, the acceleration is in the opposite direction to the displacement, and its magnitude is equal to the product of the displacement and the square of the angular frequency. This is because the object experiences the maximum force when it is farthest from the equilibrium position, and this force produces the maximum acceleration. 8. Calculate the minimum acceleration from the velocity vs. time graph. Show your calculations. Compare with the value from the acceleration vs. time graph. a. The minimum acceleration is zero, because since the graph acts as a sin function does, there are points in the graph with a slope of zero that show zero acceleration. 9. At what position is the acceleration a minimum? a. On a position vs. time graph, the acceleration is minimum at the points where the curve is closest to the equilibrium position, where the displacement is zero. At these points, the object has zero acceleration because it experiences zero net force. As the object moves away from the equilibrium position, the force on it increases, causing the acceleration to increase, and as it moves toward the equilibrium position, the force decreases, causing the acceleration to decrease. Therefore, the minimum acceleration occurs when the object is at the equilibrium position. 10. Compare your position, velocity, and acceleration graphs with your predictions on page 1. Resolve any discrepancies. a. There are no discrepancies, my predictions were correct. When the mass is changed in a simple harmonic motion system, the period of oscillation will be affected. Specifically, increasing the mass will result in a longer period, while decreasing the mass will result in a shorter period. This is because the period of oscillation is inversely proportional to the square root of the mass 9
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Lab 6: Simple Harmonic M±ion attached to the spring. This relationship can be expressed as T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant. Thus, as the mass increases, the period will also increase. Moreover, when the initial amplitude is changed, the period of oscillation will not be affected. This is because the period of oscillation is only dependent on the mass and spring constant of the system, not the amplitude of the oscillation. Therefore, changing the amplitude will only affect the maximum displacement and velocity of the system, but not the period of oscillation. 11. Use the plots mentioned to validate the relations between the period and the mass or amplitude. a. In this experiment, there is a relationship between the period and amplitude of the mass's motion. The period is the time required for the mass to complete one full cycle of motion, while the amplitude is the maximum displacement of the mass from its equilibrium position. There is a direct relationship between the period and the amplitude of the motion. Specifically, as the amplitude of the motion increases, the period of the motion also increases. This means that the period is not solely determined by the properties of the mass and the spring, but is also dependent on the amplitude of the motion. The exact nature of this relationship can be described by a mathematical equation, which shows that the period of the motion is proportional to the square root of the mass-spring system's effective spring constant, which in turn is directly proportional to the square root of the amplitude. 12. What is the time constant, τ, of this damped oscillator? a. The exact value of the time constant depends on the specific damping mechanism and the properties of the system. In the case of a simple harmonic oscillator with damping, the time constant is given by: τ = 1 / (damping coefficient * angular frequency) 10
Lab 6: Simple Harmonic M±ion where the damping coefficient is a measure of the strength of the damping forces acting on the system, and the angular frequency is the frequency of the undamped oscillator. 13. Does the period change as the system loses energy? Explain. a. Yes, the period of the system can change as the system loses energy. As the system loses energy, the amplitude of the oscillations decreases, and the effective spring constant can also change due to various factors such as changes in the tension of the spring or the mass. This can lead to a change in the period of the oscillations. As the system loses energy due to damping, the amplitude of the oscillations decreases, and the period can increase. This is because the damping force reduces the energy available to the system, causing the motion to slow down and the period to increase. 14. The reduction in amplitude represents a loss of energy by the system. Where does the energy go? a. In a simple harmonic motion system, when the amplitude of the motion decreases, it represents a loss of energy by the system. The energy lost by the system is primarily dissipated as heat due to various forms of frictional forces, such as air resistance and frictional losses within the system. During the motion of the mass attached to the spring, the spring alternately stores and releases energy, but some of this energy is lost due to frictional forces acting on the system. As a result, the amplitude of the motion decreases over time, and the system loses energy to the surroundings in the form of heat. In a perfectly ideal system with no frictional forces, the energy of the system would be conserved, and the amplitude of the motion would remain constant over time. 11
Lab 6: Simple Harmonic M±ion Conclusion In conclusion, this lab experiment successfully demonstrated the relationship between mass, amplitude, and period of oscillation of a spring-mass system. By hanging a 1.000 kg mass from a spring and observing its motion through position vs. time, velocity vs. time, and acceleration vs. time graphs, we were able to analyze the behavior of the system and draw conclusions about its physical properties. We also investigated the conservation of energy in the system by comparing the total energy at two different points in the mass's trajectory. Through these experiments, we were able to gain a deeper understanding of the dynamics of simple harmonic motion and the behavior of spring-mass systems. References Lab Manual 12
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