Potential Energy Conservation Lesson 8 by Owen Borville 11.23.2025
In a single particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another.
ΔUab = Ub - Ua = -Wab
Since only differences of potential energy are physically meaningful, the zero of the potential energy function can be chosen at a convenient location.
ΔU = U(r) - U(r0)
The potential energies for Earth's constant gravity, near its surface, and for Hooke's law force are linear and quadratic functions of position, respectively.
Gravitational potential energy near the Earth's surface: U(y) = mgy + constant
Potential energy for an ideal spring: U(x) = 1/2kx^2 + constant
A conservative force is one in which the work done is independent of path and a force is conservative if the work done over any closed path is zero. A non-conservative force is one for which the work done depends on the path. For conservative forces, the infinitesimal work is an exact differential. This implies conditions on the derivatives of the force's components. The component of a conservative force, in a particular direction, equals the negative of the derivative of the potential energy for that force, with respect to a displacement in that direction. The work done by a conservative force over a closed path is Wclosed path = ∫Fcons * dr = 0
A conserved quantity is a physical property that stays constant regardless of the path taken. A form of the work-energy theorem says that the change in the mechanical energy of a particle equals the work done on it by non-conservative forces. If non-conservative forces do no work and there are no external forces, the mechanical energy of a particle stays constant. This is a statement of the conservation of mechanical energy and there is no change in the total mechanical energy.
For one-dimensional particle motion, in which the mechanical energy is constant and the potential energy is known, the particles position, as a function of time, can be found by evaluating an integral that is derived from the conservation of mechanical energy.
Conservative force in two dimensions: (dFx/dy) = (dFy/dx)
Conservative force is the negative derivative of potential energy: Fl = -dU/dl
For no non-conservative forces, conservation of energy is 0 = Wnc, ab = Δ(K + U)ab = ΔEab
Interpreting a one dimensional potential energy diagram allows you to obtain qualitative, and some quantitative, information about the motion of a particle. At a turning point, the potential energy equals the mechanical energy and the kinetic energy is zero, indicating that the direction of the velocity reverses there. The negative of the slope of the potential energy curve, for a particle, equals the one-dimensional component of the conservative force on the particle. At an equilibrium point, the slope is zero and is a stable (unstable) equilibrium for a potential energy minimum (maximum).
Energy can be transferred from one system to another and transformed or converted from one type into another. Some of the basic types of energy are kinetic, potential, thermal, and electromagnetic. Renewable energy sources are those that are replenished by ongoing natural processes, over human time scales, such as wind, water, geothermal, and solar power. Nonrenewal energy sources are those that are depleted by consumption, over human time scales, such as fossil fuel and nuclear power.
In a single particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another.
ΔUab = Ub - Ua = -Wab
Since only differences of potential energy are physically meaningful, the zero of the potential energy function can be chosen at a convenient location.
ΔU = U(r) - U(r0)
The potential energies for Earth's constant gravity, near its surface, and for Hooke's law force are linear and quadratic functions of position, respectively.
Gravitational potential energy near the Earth's surface: U(y) = mgy + constant
Potential energy for an ideal spring: U(x) = 1/2kx^2 + constant
A conservative force is one in which the work done is independent of path and a force is conservative if the work done over any closed path is zero. A non-conservative force is one for which the work done depends on the path. For conservative forces, the infinitesimal work is an exact differential. This implies conditions on the derivatives of the force's components. The component of a conservative force, in a particular direction, equals the negative of the derivative of the potential energy for that force, with respect to a displacement in that direction. The work done by a conservative force over a closed path is Wclosed path = ∫Fcons * dr = 0
A conserved quantity is a physical property that stays constant regardless of the path taken. A form of the work-energy theorem says that the change in the mechanical energy of a particle equals the work done on it by non-conservative forces. If non-conservative forces do no work and there are no external forces, the mechanical energy of a particle stays constant. This is a statement of the conservation of mechanical energy and there is no change in the total mechanical energy.
For one-dimensional particle motion, in which the mechanical energy is constant and the potential energy is known, the particles position, as a function of time, can be found by evaluating an integral that is derived from the conservation of mechanical energy.
Conservative force in two dimensions: (dFx/dy) = (dFy/dx)
Conservative force is the negative derivative of potential energy: Fl = -dU/dl
For no non-conservative forces, conservation of energy is 0 = Wnc, ab = Δ(K + U)ab = ΔEab
Interpreting a one dimensional potential energy diagram allows you to obtain qualitative, and some quantitative, information about the motion of a particle. At a turning point, the potential energy equals the mechanical energy and the kinetic energy is zero, indicating that the direction of the velocity reverses there. The negative of the slope of the potential energy curve, for a particle, equals the one-dimensional component of the conservative force on the particle. At an equilibrium point, the slope is zero and is a stable (unstable) equilibrium for a potential energy minimum (maximum).
Energy can be transferred from one system to another and transformed or converted from one type into another. Some of the basic types of energy are kinetic, potential, thermal, and electromagnetic. Renewable energy sources are those that are replenished by ongoing natural processes, over human time scales, such as wind, water, geothermal, and solar power. Nonrenewal energy sources are those that are depleted by consumption, over human time scales, such as fossil fuel and nuclear power.