Systems with a Single Particle or Object
We first consider a system with a single particle or object. Returning to our development of (Figure), 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 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 (Figure). Now, we write this equation without the middle step and define the sum of the kinetic and potential energies, [latex]K+U=E;[/latex] to be the mechanical energy of the particle.
This statement expresses the concept of energy conservation for a classical particle as long as there is no non-conservative work. Recall that a classical particle is just a point mass, is nonrelativistic, and obeys Newton’s laws of motion. In Relativity, we will see that conservation of energy still applies to a non-classical particle, but for that to happen, we have to make a slight adjustment to the definition of energy.
You are watching: University Physics Volume 1
It is sometimes convenient to separate the case where the work done by non-conservative forces is zero, either because no such forces are assumed present, or, like the normal force, they do zero work when the motion is parallel to the surface. Then
See more : The Return of Stone Ocean: A Comprehensive Exploration of JoJo’s Bizarre Adventure’s Latest Saga
In this case, the conservation of mechanical energy can be expressed as follows: The mechanical energy of a particle does not change if all the non-conservative forces that may act on it do no work. Understanding the concept of energy conservation is the important thing, not the particular equation you use to express it.
In these examples, we were able to use conservation of energy to calculate the speed of a particle just at particular points in its motion. But the method of analyzing particle motion, starting from energy conservation, is more powerful than that. More advanced treatments of the theory of mechanics allow you to calculate the full time dependence of a particle’s motion, for a given potential energy. In fact, it is often the case that a better model for particle motion is provided by the form of its kinetic and potential energies, rather than an equation for force acting on it. (This is especially true for the quantum mechanical description of particles like electrons or atoms.)
We can illustrate some of the simplest features of this energy-based approach by considering a particle in one-dimensional motion, with potential energy U(x) and no non-conservative interactions present. Figure and the definition of velocity require
See more : Inductor
Separate the variables x and t and integrate, from an initial time [latex]t=0[/latex] to an arbitrary time, to get
If you can do the integral in Figure, then you can solve for x as a function of t.
We will look at another more physically appropriate example of the use of Figure after we have explored some further implications that can be drawn from the functional form of a particle’s potential energy.
Source: https://en.congthucvatly.com
Category: Blog