Potential Energy, Force, and Equilibrium
For one-dimensional conservative motion, F_x is minus dU/dx; equilibrium occurs where dU/dx is zero, with stability determined by local curvature or nearby restoring direction.
Why this shows up in the exam
Reading U-x graphs · Locating turning points and equilibria · Deriving force from potential
Learn the idea
Force points down the potential-energy landscape. A potential graph behaves like a landscape: slope gives the opposite force, valleys are stable, and hills are unstable.
🧠 Memory hook: Force rolls downhill on U(x).
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- F_x = -dU/dx — force from a one-dimensional potential
- dU/dx = 0 — equilibrium condition
- d²U/dx² > 0 — stable equilibrium test
How to approach it
- 1Differentiate or read the slope
- 2Reverse its sign for force
- 3Inspect curvature and allowed energy region
Common slip-ups that cost marks
- •Using U itself as force
- •Forgetting the minus sign
- •Calling every zero-slope point stable
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A 2 kg body speeds up from 3 m/s to 7 m/s. What net work is done on it?
More from Work, Energy and Power
Work and its calculation
Work is the energy transferred by a force acting over a distance, and can be calculated using the dot product, area under a force-displacement graph, or for variable and constant forces.
Conservation of energy
The law of conservation of energy states that energy cannot be created or destroyed, only transformed, including cases with energy loss and efficiency considerations.
Work-energy theorem
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy, and applies to both constant and variable forces.
Conservative and non-conservative forces
Conservative forces, like gravity and spring force, conserve mechanical energy, while non-conservative forces, like friction, dissipate energy as heat.
Elastic potential energy
Elastic potential energy is the energy stored in a stretched or compressed spring, proportional to the square of its displacement.
Power
Power is the rate at which work is done or energy is transferred, and can be calculated as the product of force and velocity at any instant.