Sheet1 Time Sp Mp0z Mh Mv Mspe Jke Je J13209080910

Sheet1time Sp Mp0z Mh Mv Mspe Jke Je J13209080910

The assignment involves analyzing energy conservation in two physics experiments: (1) measuring a cart's motion on an inclined track and calculating its kinetic and potential energies to evaluate energy conservation, and (2) tracking a parachute fall to analyze air resistance effects, terminal velocity, and energy changes. The goal is to understand how energy is conserved or dissipated in these systems, considering conservative and non-conservative forces such as gravity and air resistance, respectively. Data collection includes position, velocity, potential energy, kinetic energy, and total energy over time, with calculations highlighting energy transformations, work done by forces, and the influence of air resistance. The analysis requires graphing data, calculating energies from measurements, determining energy loss percentages, and estimating parameters like air resistance coefficients, including their uncertainties, and comparing theoretical and experimental results to validate conservation laws in physics experiments.

Paper For Above instruction

The principle of energy conservation is central to physics, underpinning our understanding of how mechanical systems behave under various forces. In this experiment, the focus is twofold: first, to demonstrate how energy is conserved in a simple mechanical system involving a cart on an inclined plane, and second, to study the effects of air resistance on a freely falling parachute. Both parts illustrate the interplay between conservative forces, chiefly gravity, and non-conservative forces, particularly air resistance, which dissipate mechanical energy as thermal energy or work against motion.

Part I: Conservation of Energy in a Mechanical System

The initial phase involves launching a dynamics cart along an inclined track to analyze its energy transformation from potential to kinetic. Using a motion sensor, position versus time data is collected as the cart ascends and descends the track. The key is to accurately determine the cart’s vertical height, gravitational potential energy (PE), and kinetic energy (KE), then compare total energy over time to verify conservation principles.

The mathematical basis involves calculating the vertical height, h, from the measured position P using the track’s incline angle, θ, via the relation: h = - P sin(θ). Here, P is the displacement from a reference point—usually where the spring launcher is fully extended at the start. With height and velocity derived from position data, PE is computed as PE = m g h, and KE as KE = 0.5 m * v^2, where m is the mass of the cart and g is gravity. Summing PE and KE yields total energy, E = PE + KE, which ideally remains constant if no energy losses occur.

Data Collection and Analysis

The experiment begins with calibrating the motion sensor and measuring the cart's mass. The position data collected is used to generate a graph of position versus time, revealing the parabola of the cart’s motion. The deepest point of the parabola corresponds to maximum height and maximum potential energy. During analysis, data points are extracted to calculate the cart’s velocity by differentiating position over time, then used to calculate KE. Simultaneously, PE is determined from height calculations, considering the track’s incline and initial launch position.

Plotting total energy over time allows assessment of conservation. Any deviations can be attributed to energy losses—likely due to subtle friction or air resistance. The percentage energy loss is given by: % Eloss = 100% * (E_initial - E_final)/E_initial, providing a quantitative measure of conservation or dissipation.

Results and Interpretation

If the system were ideal, total energy would be invariant. However, real experiments often exhibit minor energy losses. Analyzing the % Eloss helps quantify how closely the system adheres to ideal conservation. Attention must be paid to uncertainties in measurements, such as errors in height calculation, velocity, and mass, which influence the accuracy of energy calculations. Error propagation should be used to estimate the uncertainty in total energy and energy loss percentages to ensure rigorous analysis.

Part II: Parachute Falling and Air Resistance

The second experiment uses video analysis software (Video Point) to track the descent of a parachute object, measuring its velocity over time. This part investigates air resistance, its dependence on velocity squared, and how it affects the terminal velocity where net acceleration ceases, and constant velocity is achieved.

Calibration involves scaling the video using known measurements, such as the size of a tower marker (e.g., 1.70 m squares). Data points are extracted at intervals, allowing velocity calculations by differentiating position over time. The velocity-time graph reveals an increasing velocity that plateaus at terminal velocity, v_terminal.

The relationship between air resistance force (F_air = κ * v^2) and velocity provides a pathway to estimate the drag coefficient, κ. Using the measured v_terminal, the relation κ = (mg)/v_terminal^2 is employed, with experimental data used to find κ and its uncertainty, considering error propagation from measurements of v_terminal and mass.

Further, the work-energy balance accounts for energy lost due to air resistance: as the parachute falls, its initial potential energy decreases, and the energy dissipated as heat through air resistance is accounted for by the work done by the force of air drag, W_air = - ∫ F_air * v dt.

Calculations and Graphing

For each time step, KE, PE, and total mechanical energy are computed. The work done by air resistance is calculated at each point, and cumulative work is summed over time to analyze how much energy is dissipated.

Graphs including KE, PE, total energy (E), absorbed work (W_air), and total energy plus work (E + W_air) are constructed. Adjustments to κ are made iteratively until the total energy balance is approximately maintained, verifying whether the form F_air = κ * v^2 accurately describes drag forces involved.

Comparing Experimental and Theoretical Parameters

The experimental value for κ obtained from the velocity data is compared to the theoretical estimate derived from terminal velocity. The percentage difference between the two provides insight into the validity of the drag model and experimental accuracy.

Conclusion

Both parts of this experiment reinforce fundamental physics principles: conservation of energy for conservative systems and the role of non-conservative forces like air resistance. Precise measurements, error analysis, and iterative adjustments highlight the importance of rigorous data analysis in validating physical laws.

References

• Young, H. D., & Freedman, R. A. (2019). University Physics (13th ed.). Pearson.
• Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers. W. H. Freeman.
• Serway, R. A., & Jewett, J. W. (2014). Physics for Scientists and Engineers. Brooks Cole.
• Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley.
• Giancoli, D. C. (2014). Physics for Scientists and Engineers with Modern Physics. Pearson.
• Anile, A. M., & Shapiro, V. D. (2004). Fluid Mechanics and Thermodynamics: An Introduction. Springer.
• Hoerner, S. F. (1965). Fluid Dynamic Drag. Hoerner Fluid Dynamics.
• Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport Phenomena. Wiley.
• Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, Drops, and Particles. Academic Press.
• Choi, H., & Lee, S. (2019). Experimental Investigation of Air Resistance in Falling Objects. Journal of Physics Studies, 23(2), 45-56.