We are neglecting non-conservative forces, so we write the energy conservation formula relating the particle at the highest point (initial) and the lowest point in the swing (final) as Lastly, in step 5, we set the sum of energies at the highest point (initial) of the swing to the lowest point (final) of the swing to ultimately solve for the final speed. In step 4, we choose a reference point for zero gravitational potential energy to be at the lowest vertical point the particle achieves, which is mid-swing. This increase in kinetic energy equals the decrease in the gravitational potential energy, which we can calculate from the geometry. Because the particle starts from rest, the increase in the kinetic energy is just the kinetic energy at the lowest point. Therefore, the mechanical energy of the system is conserved, as represented by Equation 8.13, 0 = Δ ( K + U ) 0 = Δ ( K + U ). We neglect air resistance in the problem, and no work is done by the string tension, which is perpendicular to the arc of the motion. Second, only the gravitational force is acting on the particle, which is conservative (step 3). Using our problem-solving strategy, the first step is to define that we are interested in the particle-Earth system. It is shown when released from rest, along with some distances used in analyzing the motion. Now, we write this equation without the middle step and define the sum of the kinetic and potential energies, K + U = E K + U = E to be the mechanical energy of the particle.įigure 8.7 A particle hung from a string constitutes a simple pendulum. We then replaced the work done by the conservative forces by the change in the potential energy of the particle, combining it with the change in the particle’s kinetic energy to get Equation 8.2. Returning to our development of Equation 8.2, recall that we first separated all the forces acting on a particle into conservative and non-conservative types, and wrote the work done by each type of force as a separate term in the work-energy theorem. We first consider a system with a single particle or object. Thus, there is no physical law of conservation of water. A conserved quantity, in the scientific sense, can be transformed, but not strictly created or destroyed. However, in scientific usage, a conserved quantity for a system stays constant, changes by a definite amount that is transferred to other systems, and/or is converted into other forms of that quantity. Bring these atoms together to form a molecule and you create water dissociate the atoms in such a molecule and you destroy water. Water is composed of molecules consisting of two atoms of hydrogen and one of oxygen. (The same comment is also true about the scientific and everyday uses of the word ‘work.’) In everyday usage, you could conserve water by not using it, or by using less of it, or by re-using it. The terms ‘conserved quantity’ and ‘conservation law’ have specific, scientific meanings in physics, which are different from the everyday meanings associated with the use of these words. The last section of this chapter provides a preview. As you continue to examine other topics in physics, in later chapters of this book, you will see how this conservation law is generalized to encompass other types of energy and energy transfers. ![]() This will lead us to a discussion of the important principle of the conservation of mechanical energy. In this section, we elaborate and extend the result we derived in Potential Energy of a System, where we re-wrote the work-energy theorem in terms of the change in the kinetic and potential energies of a particle.
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