Conservative Forces and Path Independence
A force is conservative in a simply connected region when its line integral is path independent, equivalently its work around every closed path is zero and a single-valued potential exists.
Why this shows up in the exam
Comparing alternate paths · Gravity and ideal springs · Testing whether potential energy exists
Learn the idea
Conservative-force work depends only on endpoints. A conservative field stores recoverable energy, so different routes between the same points give the same work and a closed trip gives zero.
🧠 Memory hook: Same endpoints, same conservative work.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- W_AB = -Delta U — conservative-force work
- closed integral F . dr = 0 — closed-path test
- Delta U = U_B-U_A — potential-energy change
How to approach it
- 1Identify the force and domain
- 2Compare endpoints or test a closed path
- 3Use W=-Delta U with signs
Common slip-ups that cost marks
- •Calling every central-looking force conservative
- •Changing the sign in W=-Delta U
- •Applying path independence to friction
🌟 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.