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A good exercise is to solve the motion on an inclined plane at angle ##\theta##. The usual way by looking at tangential motion. Then, convert back to Cartesian coordinates.Atomillo said:
Equations of motion of a point sliding on a line of arbitrary shape
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SUMMARY
The discussion centers on deriving the equations of motion for a point sliding along a curve defined by a function f(x) in a two-dimensional space. The user initially attempts to apply conservation of energy, leading to the velocity equation v(x) = √(2*(E - mgf(x))/m). However, confusion arises when trying to relate this to time, resulting in an incorrect interpretation of motion as free fall. Key insights reveal that in two-dimensional motion, both x and y components must be considered, and the user must utilize energy conservation correctly to derive the velocity as a function of time.
PREREQUISITES- Understanding of conservation of mechanical energy in physics
- Familiarity with calculus, particularly integration and derivatives
- Knowledge of two-dimensional motion and vector components
- Basic understanding of the Brachistochrone problem in physics
- Study the Brachistochrone curve and its implications for motion along curves
- Learn how to derive velocity as a function of time in two-dimensional motion
- Explore the relationship between kinetic energy and potential energy in two dimensions
- Investigate the calculus of variations for solving complex motion problems
Students of physics, particularly those studying mechanics, engineers working on motion analysis, and anyone interested in understanding the dynamics of particles on curved paths.
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