PHY5200 F06

Chapter 4: Energy

Reading

Taylor 4.3-4.5 (today) and 4.5-4.7 (Wednesday).

Recall

Time Dependent Potential Energy

It is possible to write a potential energy that depends on time, that is, U = U(r,t). The force is still given by F = -∇U, and will still satisf ∇×F = 0, but now the force can also depend on time, F = F(r,t). If the potential energy depends on time, then energy is not conserved. To see this, note that the differential change in U is

ΔU = (∂U/∂x)Δx + (∂U/∂y)Δy + (∂U/∂z)Δz + (∂U/∂t)Δt

and the differential change in kinetic energy is still

ΔT = dT/dt Δt = (mv⋅vdot)Δt = F⋅Δr.

Since F = -∇U, we can write ΔU = -F⋅Δr + (∂U/∂t)Δt. Adding ΔU and ΔT, we see that the force term cancels, leaving

ΔU + ΔT = Δ(T+U) = ΔE = (∂U/∂t)Δt

including the definition of mechanical energy E = T+U. Taking the limit as Δt goes to zero yields

∂E/∂t = ∂(T+U)/∂t = ∂U/∂t

The rate of change of mechanical energy equals the rate of change of the potential energy.

Energy for linear one-dimensional systems

What is meant by a linear one-dimensional system? The author means a system whose position is characterized by a single variable, and that moves in a straight line, as if on a track. For such systems, the requirements for a force to be conservative that (i) the work done must depend only on the position (let's call that x) and (ii) the work done must be independent of path are equivalent.

We can graph U as a function of x. Such a graph has the property that the force, F = -dU/dx, points downhill. There is a similarity with hilly terrain, or a roller coaster where the potential energy is given by mgh. Here the potential changes linearly with height, just like on a graph of U as a function of x.

The force is zero where the derivative dU/dx = 0, at the tops of hills or the bottom of valleys. These can be points of stable or unstable equilibrium: if the potential curves upward, then the object will tend to return to the equilibrium position if it is displaced slightly; if the potential curves downward, then the object will tend to move further away from the equilibrium position if displaced slightly.

Since the total (mechanical) energy is E = T + U, then a line drawn across the graph at U=E leads to a qualitative understanding of the motion. First, the object can be located only where U≤E. Locations where U=E are called turning points, since as these points are approached, the kinetic energy goes to zero, but the force doesn't, so the object then moves away from the point -- the motion turns at these points. If the object is in a region bounded by two turning points, then it moves back and forth between them. If it is in a region bounded by one turning point, then the other direction must be open to infinity, so if an object approaches the turning point, it will turn around and move away to infinity.