doPhysics · Work, Energy and Power
Potential Energy and Conservation of Energy
Energy stored in an arrangement and the forces that can store it. Mechanical energy that stays constant, and friction that turns it into heat. Force as the slope of potential energy, and energy diagrams with their turning points and equilibria.
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Why this chapter
Why this chapter exists
Newton's laws follow a body instant by instant, and that is hard work when the forces change at every point. Energy offers a shortcut. Some forces store the energy they take and return it in full, and for them kinetic plus potential energy stays constant. Then a speed follows from a height alone, whatever the shape of the path. This chapter builds that shortcut and shows what happens when friction breaks it. Then a single graph of potential energy becomes a map of where a body can go and rest.
You will learn
By the end of this chapter you will
- Define potential energy through the work of a force, and choose its zero freely.
- Find the energy stored by gravity near the Earth and by a spring, including springs in series, in parallel and cut.
- Decide whether a force is conservative with the path test or the round-trip test, and explain why only such a force has a potential energy.
- Use conservation of mechanical energy to find speeds and heights on smooth paths of any shape.
- Account for friction with \(W_{nc} = \Delta E\), and follow the energy of a bungee jump through three stores.
- Find a force from its potential energy, \(F = -dU/dx\) in one dimension and \(\vec F = -\nabla U\) in three.
- Read an energy diagram for turning points, forbidden regions, bound and unbound motion, and stable, unstable and neutral equilibrium.
Before you begin
Before you begin
From the chapter on Work, Kinetic Energy and Power, you should know work, \(W = Fd\cos\theta\). You should know the work of a variable force as the area under its graph. You should know kinetic energy \(\tfrac12 mv^2\) and the work-energy theorem, Hooke's law \(F = kx\), and kinetic friction \(f = \mu_k N\). You should be able to differentiate a polynomial.
Lessons
1
Potential Energy
Energy stored in an arrangement. Its zero is a free choice, so only a change in it enters an answer. Near the Earth it is \(mgh\), in a spring it is \(\tfrac12 kx^2\), and it belongs to a pair of bodies, never to one.
Potential energy is stored in an arrangementGravitational potential energy and its zeroElastic potential energy of a springPotential energy belongs to the pair
2
Conservative and Non-Conservative Forces
The path test and the round-trip test that decide whether a force can store energy. Gravity and springs pass; friction and drag fail and turn energy into heat. Every central force passes, because arcs cost nothing.
The path test for a conservative forceOnly a conservative force has a potential energyCentral forces are conservative
3
Mechanical Energy Is Conserved
Kinetic plus potential energy stays constant while only conservative forces do work. The speed then follows from the heights alone, and the shape of a smooth track drops out. The system boundary is drawn first.
Total mechanical energy and its conservationThe shape of a smooth track drops outDraw the system boundary first
4
When Friction Acts
The work of the non-conservative forces equals the change in mechanical energy. Friction takes exactly f times the distance slid and turns it into heat, while the total energy is still conserved. A bungee jump shows three stores of energy in one account.
Work of non conservative forces changes mechanical energyFriction takes f times the distance slidThe energy account of a bungee jump
5
Force from Potential Energy
Going the other way. If the work of a force gives its potential energy, the slope of the potential energy gives the force, \(F = -dU/dx\). A flat \(U\) means no force, \(\tfrac12 kx^2\) returns Hooke's law, and in three dimensions the force is minus the gradient.
Force is minus the slope of potential energyA flat potential energy means no forceHookes law from the spring energyForce is minus the gradient in three dimensions
6
Energy Diagrams and Equilibrium
A graph of U against x with a horizontal line for the total energy answers the big questions before any equation is solved. Where can the particle go, where does it turn back, can it escape, and where can it rest, stable or not?
Reading an energy diagramTurning points and forbidden regionsBound and unbound motionStable unstable and neutral equilibrium
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- bound motion
- Motion trapped between two turning points, going back and forth for ever.
- central force
- A force that points along the line joining a body to a fixed centre, with a size that depends only on the distance. Every central force is conservative.
- conservative force
- A force whose work on a body moving between two points is the same on every path, or equivalently zero around every closed path. Only such a force has a potential energy.
- energy diagram
- A graph of potential energy against position, with a horizontal line at the total energy. The gap between the line and the curve is the kinetic energy.
- equilibrium
- A state in which the net force on a particle is zero. For a conservative force it occurs where \(dU/dx = 0\).
- forbidden region
- A region where the potential energy is greater than the total energy. A particle can never be there, because its kinetic energy would be negative.
- gradient
- The vector of the slopes of a function along the three axes, \(\nabla U = (\partial U/\partial x, \partial U/\partial y, \partial U/\partial z)\). It points where the function rises fastest; a conservative force is \(\vec F = -\nabla U\).
- heat
- Energy in the random motion of the molecules of a body, also called thermal energy. Friction turns mechanical energy into heat.
- mechanical energy
- The sum of the kinetic energy and the potential energy of a system, \(E = K + U\). It stays constant when no non-conservative force does work.
- neutral equilibrium
- An equilibrium in a region where the potential energy is constant. A displaced particle feels no force and stays where it is put.
- non-conservative force
- A force whose work depends on the path taken, such as kinetic friction, air drag or a push from outside. It has no potential energy.
- potential energy
- Energy stored in the arrangement of a system, such as a body raised above the Earth or a stretched spring. Its change is minus the work done by the force that stores it, \(\Delta U = -W\).
- reference level
- The position chosen to have zero potential energy. It may be placed anywhere, because only changes in potential energy enter any result.
- spring constant
- The force needed to stretch or compress a spring by one metre, \(k = F/x\), in N/m. A stiffer spring has a larger \(k\).
- springs in parallel
- Springs side by side between the same two bodies, so that each has the same stretch. The pair has \(k_p = k_1 + k_2\), and the stiffer spring stores more energy.
- springs in series
- Springs joined end to end, so that the same force acts through each; the pair has \(1/k_s = 1/k_1 + 1/k_2\), and the softer spring stores more energy.
- stable equilibrium
- An equilibrium at a minimum of potential energy, \(d^2U/dx^2 > 0\). A small displacement produces a force that brings the particle back.
- system
- The bodies chosen for an energy account, marked off by a boundary. A potential energy exists only when both interacting bodies are inside it.
- turning point
- A position where the potential energy equals the total energy. The kinetic energy there is zero, and the particle stops and reverses.
- unbound motion
- Motion with at most one turning point, so that the particle moves away and never returns.
- unstable equilibrium
- An equilibrium at a maximum of potential energy, \(d^2U/dx^2 < 0\). A small displacement produces a force that pushes the particle further away.