PhysicsChapter 5: Work and Kinetic Energy
Conservative & Non-Conservative Fields
Explanation level
5.2 Conservative & Non-Conservative Fields
Overview
Earlier in this chapter, work was described as energy transferred when a force acts through a displacement. This topic asks whether that work depends only on the starting and ending points or also on the route. That distinction separates conservative fields from non-conservative fields.
A field of force, or force field, is the region around a body where it can influence another body through an associated force. Electric, viscous and gravitational fields are textbook examples.
Learning Outcomes
- Define force, conservative and non-conservative fields.
- Explain path independence and path dependence.
- Apply the closed-path test.
- Differentiate the two types of fields.
- Calculate simple path and closed-loop work.
- Relate conservative work to potential energy.
- Classify the textbook examples correctly.
Definition
Force Field
Formal definition: A force field is the region around a body in which it can influence other bodies through an associated force. In easy words, it is an area where another object can feel a force without direct contact. For example, an object near Earth lies in Earth's gravitational field.
Conservative Field
Formal definition: A field is conservative if work done on a particle moving between two points is independent of the path and work done around every closed path is zero. In easy words, any route between the same start and finish gives the same work; a complete round trip gives zero net work by that force.
Non-Conservative Field
Formal definition: A non-conservative field is one in which work depends on the path followed, or work done along a closed path is not zero. In easy words, the route matters. A longer rough path usually requires more work against friction.
Key Terms
| Term | Symbol | Meaning | SI unit |
|---|---|---|---|
| Work | W | Energy transferred by a force through displacement | joule (J) |
| Conservative work | W_c | Work done by a conservative force | joule (J) |
| Non-conservative work | W_nc | Work done by a non-conservative force | joule (J) |
| Potential energy | U | Energy associated with position or configuration | joule (J) |
| Closed path | ∮ | A route ending at its starting point | No unit |
| Frictional force | f | Force opposing relative motion | newton (N) |
| Path length | s | Distance measured along the route | metre (m) |
Intermediate Level
At this level, classify a field by comparing work along different paths and by checking the net work around a closed route. The initial and final points and the physical conditions must remain the same during the comparison.
Path Independence
- W
- work done by the field (J)
- A
- initial point
- B
- final point
When to use: Compare two paths with the same endpoints in a conservative field.
Common mistake: Do not compare routes having different endpoints.
Closed-Path Test
A closed path begins and ends at the same point. For a conservative force, outward and return work cancel.
- ∮
- integral around a closed path
- F_c
- conservative force (N)
- dr
- small displacement (m)
When to use: Use for a complete closed trip in a conservative field.
When not to use: Do not assume zero closed-path work for friction or drag.
Potential Energy
- W_c
- conservative work (J)
- ΔU
- Uf - Ui (J)
- U_i
- initial potential energy (J)
- U_f
- final potential energy (J)
When to use: Use when potential energy is defined for a conservative interaction.
Common mistake: Keep the minus sign: positive conservative work means potential energy decreases.
Friction and Path Length
- W_f
- work done by friction (J)
- f
- friction magnitude (N)
- s
- actual path length (m)
When to use: Constant friction directly opposing motion.
Common mistake: Use actual path length, not straight-line separation.
Conservative and Non-Conservative Fields Compared
| Feature | Conservative | Non-conservative |
|---|---|---|
| Work from A to B | Independent of path | Depends on path |
| Closed-path work | Zero | Not necessarily zero |
| Potential energy | Can be defined | No single potential-energy function describes the force alone |
| Mechanical energy | Can remain constant if only conservative forces work | Usually changes when non-conservative work occurs |
| Examples | Gravitational and electric fields | Frictional field, viscous drag, air resistance |
Further Explanation
Connection Between the Two Conservative Properties
Take path I from A to B and return along path II. The route is closed.
Therefore, zero closed-path work implies path-independent work between fixed endpoints.
Positive, Negative and Zero Cases
| Case | Condition | Meaning |
|---|---|---|
| Positive conservative work | W_c > 0; ΔU < 0 | Potential energy decreases. |
| Negative conservative work | W_c < 0; ΔU > 0 | Potential energy increases. |
| Zero conservative work | ΔU = 0 | Initial and final potential-energy states are equal. |
| Closed conservative path | Initial point = final point | Wc = 0. |
| Friction opposing motion | θ = 180° | W_f is negative and depends on distance travelled. |
A non-conservative force does not destroy energy. Friction may reduce mechanical energy while increasing thermal energy; total energy remains conserved when all forms are included.
Key Formula Summary
| Formula | Use | Units or condition |
|---|---|---|
| WA→B(I) = WA→B(II) | Path-independence test | W in J; same endpoints |
| ∮Fc · dr = 0 | Closed conservative path | F in N; dr in m |
| Wc = -ΔU | Conservative work and potential energy | All terms in J |
| Wf = -fs | Constant opposing friction | f in N; actual s in m |
| [W] = [ML²T⁻²] | Dimensions of work | Same as energy |
If U decreases, Wc becomes positive. If a rough path becomes longer while f stays constant, |Wf| increases. Changing the shape of a conservative closed path does not change its zero-work result.
Solved Numerical Examples
Example 1: Two Paths in a Gravitational Field
A 2.0 kg object is raised 3.0 m vertically or by a longer frictionless ramp. Find work done by gravity. Take g = 9.8 m s⁻².
- m = 2.0 kg
- h = 3.0 m
- g = 9.8 m s⁻²
- Same endpoints
Required: Work by gravity on each path
Formula: Wg = -mgh
Wg = -(2.0)(9.8)(3.0)
Wg = -58.8 kg m² s⁻²
Wg = -58.8 J
The height change is the same on the ramp.
Wg = -58.8 J
Final answer: For both paths, Wg = -58.8 J.
Example 2: Closed Conservative Trip
A conservative force does +24 J from A to B. Find its work from B to A and for the complete trip.
- WA→B = +24 J
- Closed trip
- Conservative force
Required: W{B→A} and Wclosed
Formula: Wclosed = WA→B + WB→A
0 = 24 J + WB→A
WB→A = -24 J
Wclosed = 24 J + (-24 J)
Wclosed = 0 J
Final answer: WB→A = -24 J; Wclosed = 0 J.
Example 3: Friction Along Different Paths
A 5.0 N frictional force opposes a box along paths of 4.0 m and 10.0 m. Compare the work.
- f = 5.0 N
- s₁ = 4.0 m
- s₂ = 10.0 m
Required: W{f1} and W{f2}
Formula: Wf = -fs
Wf1 = -(5.0)(4.0)
Wf1 = -20 J
Wf2 = -(5.0)(10.0)
Wf2 = -50 J
Final answer: The work values are -20 J and -50 J.
Real-Life Applications
- Hydroelectric power: gravitational energy depends on height difference, while turbine and pipe losses are non-conservative.
- Roller coasters: gravity connects speed and height; friction and drag reduce mechanical energy.
- Vehicle braking: non-conservative brake friction transforms mechanical energy mainly into heat.
- Falling and orbiting bodies: conservative gravitational work allows potential-energy methods.
- Motion through fluids: viscous drag and air resistance are treated as non-conservative effects.
Important Exam Facts
- State both conservative-field properties.
- Gravity and electric fields are conservative.
- Frictional, viscous-drag and air-resistance fields are non-conservative.
- Wc = -ΔU.
- Work has unit J and dimensions [ML²T⁻²].
- Closed-path work means the complete round trip.
- Non-conservative work may change mechanical energy but does not destroy total energy.
Common Mistakes
- Calling a force conservative because it gives zero work in one special motion.
- Writing that all conservative work is zero instead of only closed-path net work.
- Dropping the minus sign in W_c = -ΔU.
- Using endpoint separation instead of actual path length for friction.
- Calling frictional work positive when friction opposes motion.
- Saying friction destroys energy.
- Comparing paths having different endpoints.
Short-Question Preparation
Q. Define a force field and give examples.
A. A force field is the region around a body where it can influence another body through an associated force. The force can act without direct contact. Electric, viscous and gravitational fields are examples. A mass in Earth's gravitational field experiences gravitational force. A charge in an electric field experiences electric force.
Q. State the properties of a conservative field.
A. Work between the same two points is independent of path. Work around any closed path is zero. The first statement compares different routes with fixed endpoints. The second describes a complete trip back to the starting point. Gravitational and electric fields satisfy both properties.
Q. Define a non-conservative field and give examples.
A. Its work depends on the path, or its closed-path work is not zero. A longer route may give a different work value. Frictional field is an example. Friction, viscous drag and air resistance are non-conservative forces. Their work often transforms mechanical energy into heat.
Q. Why is gravity conservative?
A. Gravitational work between two positions depends only on the endpoints. Near Earth, it depends on height change. A return to the initial position gives zero change in gravitational potential energy. Therefore net gravitational work around the closed path is zero. Gravity is conservative.
Q. Why is friction non-conservative?
A. Friction acts along the actual contact path. For constant friction, W_f = -fs. A longer route has a larger s and greater work magnitude. Two routes between the same endpoints can therefore give different values. Friction is path dependent and non-conservative.
Long-Question Preparation
Q. Compare conservative and non-conservative fields.
A. 1. Define a force field. 2. Define a conservative field. 3. Describe two paths between A and B. 4. Write WA→B(I) = WA→B(II). 5. Define a closed path. 6. State that conservative closed-path work is zero. 7. Write ∮Fc · dr = 0. 8. Connect conservative work with potential energy. 9. Write Wc = -ΔU. 10. Give gravitational and electric fields as examples. 11. Define a non-conservative field. 12. State that its work depends on path. 13. State that its closed-path work need not be zero. 14. Write W_f = -fs for constant opposing friction. 15. Explain how path length changes frictional work. 16. Give friction, viscous drag and air resistance as examples. 17. Describe the required two-path or closed-loop diagram. 18. Conclude that endpoints determine conservative work while route determines non-conservative work.
Q. Show that zero closed-path work implies path independence.
A. 1. Move from A to B along path I. 2. Return from B to A along path II. 3. The combined route is closed. 4. For a conservative force, Wclosed = 0. 5. WA→B(I) + WB→A(II) = 0. 6. Reversing a path reverses the work sign. 7. WB→A(II) = -WA→B(II). 8. Substitute this expression. 9. WA→B(I) - WA→B(II) = 0. 10. Move the second term to the other side. 11. W{A→B}^{(I)} = W_{A→B}^{(II)}. 12. Work is equal along both paths. 13. The result applies to any routes with the same endpoints. 14. Hence work is path independent. 15. Conversely, path-independent outward and return work cancel. 16. Thus both properties describe conservative behavior.
Sticky Notes
- Route does not matter → conservative.
- Closed loop: ∮Fc · dr = 0.
- W_c = -ΔU: keep the minus sign.
- Route matters → non-conservative.
- Friction uses actual path length: W_f = -fs.
- Friction transforms energy; it does not destroy it.
Quick Revision
| Item | Revision point |
|---|---|
| Definition | Conservative: path independent and zero closed-path work. Non-conservative: path dependent or non-zero closed-path work. |
| Formulas | Wc = -ΔU; ∮Fc · dr = 0; Wf = -fs |
| Units | Work and energy: J; force: N; distance: m |
| Dimensions | [ML²T⁻²] |
| Conservative examples | Gravitational and electric fields |
| Non-conservative examples | Frictional field, viscous drag, air resistance |
| Comparison | Endpoints matter versus complete route matters |
60-Second Revision
When to use: Same endpoints in a conservative field.
When to use: Closed conservative path.
When to use: Conservative work and potential energy.
When to use: Constant opposing friction.
- A force field is a region where an associated force acts.
- Conservative work is path independent.
- Conservative closed-path work is zero.
- Non-conservative work is path dependent.
- Gravity/electric force are conservative; friction/drag/air resistance are non-conservative.
Summary
A force field is a region where an associated force acts. Conservative work is path independent and zero around a closed path, with W_c = -ΔU. Non-conservative work is path dependent; friction, viscous drag and air resistance are examples. Endpoints determine conservative work, while the route affects non-conservative work.
Ask-the-Tutor Modes
- Explain path independence simply.
- Show the closed-path proof one step per line.
- Compare the two fields.
- Check my frictional-work numerical.
- Explain the sign in Wc = -ΔU.
- Test me with an MCQ.
- Explain why gravity is conservative and friction is not.
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Exam preparation
Exam Focus
Important points
- Exact definitions
- Both conservative properties
- Correct textbook examples
- Mechanical versus total energy
Common mistakes
- Do not say conservative means weak.
- Only closed-path conservative work is necessarily zero.
- Keep the minus sign in W_c = -ΔU.
- Use actual path length for friction.
- Friction transforms rather than destroys energy.
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Conservative & Non-Conservative Fields
Sticky Notes · Quick Revision
Core test
- Route does not matter → conservative.
- Route matters → non-conservative.
Closed path
∮Fc · dr = 0
In words Net work by a conservative force around a closed path is zero.
Energy relation
Wc = −ΔU
In words Positive conservative work means potential energy decreases.
Examples
- Conservative: gravitational and electric fields.
- Non-conservative: friction, viscous drag and air resistance.
Exam warning
- Only closed-path conservative work is necessarily zero.
- Friction transforms energy; it does not destroy it.
Ready to test yourself?
You've completed the learning section.
- 8 concept questions
- Easy → Medium → Hard
- Explanation after each answer
Exam Practice
Important Short Question
Q1. State the two properties of a conservative field.
Important Short Question
Q2. Define a non-conservative field and give examples.
Conceptual Question
Q3. Why is gravity conservative but friction non-conservative?
Long Question Preparation
Q4. Compare conservative and non-conservative fields with formulas and examples.
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