Topic 5.414 min read
Work-Energy Principle
The work-energy principle states that the net work done on an object equals its change in kinetic energy: Wnet = Kf · Ki = ΔK. For conservative forces, ΔK = -ΔU.
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Beginner Level
Work transfers energy. If all forces together make an object speed up, they do positive net work and its kinetic energy increases. If the object slows down, net work is negative and kinetic energy decreases.
- W_net
- net work (J)
- K_i
- initial kinetic energy (J)
- K_f
- final kinetic energy (J)
When to use: Use when total work or a change in speed is involved.
Recommended for Class 11
5.4 Work-Energy Principle
Overview
A force doing work transfers energy to or from an object. The work-energy principle connects the net work done by all forces with the resulting change in the object's kinetic energy.
Learning Outcomes
- State and explain the work-energy principle.
- Distinguish net work from work by one force.
- Use Wnet = Kf - Ki in numerical problems.
- Interpret positive, negative and zero net work.
- Relate kinetic and potential-energy changes for conservative forces.
Definition and Main Equation
The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy.
- W_net
- sum of work by all forces (J)
- K_i
- initial kinetic energy (J)
- K_f
- final kinetic energy (J)
When to use: Use when net work changes an object's speed.
Common mistake: Use net work, not automatically the work of one force.
Intermediate Level
Derivation
For a constant net force parallel to displacement:
Substitute Fnet = ma and the expression for d:
Meaning of the Sign
| Net work | Kinetic-energy change | Effect |
|---|---|---|
| W_net > 0 | ΔK > 0 | Speed increases |
| W_net < 0 | ΔK < 0 | Speed decreases |
| Wnet = 0 | ΔK = 0 | Speed remains constant; direction may change |
Connection with Potential Energy
Potential energy can be defined for conservative forces. Work done by a conservative force is the negative of the change in potential energy:
- W_c
- work by conservative force (J)
- ΔU
- potential-energy change (J)
If only conservative forces do work, Wnet = Wc:
When to use: Use when only conservative forces transfer energy.
An increase in kinetic energy is exactly balanced by an equal decrease in potential energy, so total mechanical energy K + U remains constant.
Units and Dimensions
The SI unit is the joule: 1 J = 1 N m = 1 kg m² s⁻².
Solved Numerical Examples
Example 1: Finding Net Work
A 4.0 kg body speeds up from 3.0 m s⁻¹ to 7.0 m s⁻¹. Find the net work.
- m = 4.0 kg
- vi = 3.0 m s⁻¹
- vf = 7.0 m s⁻¹
Wnet = 12 m(vf² - vi²)
Wnet = 12 (4.0)(7.0² - 3.0²)
Wnet = 2.0(49 - 9)
Wnet = 2.0(40)
Final answer: Wnet = 80 J
Example 2: Stopping Work
A 1000 kg car moving at 20 m s⁻¹ is brought to rest. Find the net work.
- m = 1000 kg
- vi = 20 m s⁻¹
- vf = 0
Wnet = 12 m(vf² - vi²)
Wnet = 12 (1000)(0² - 20²)
Wnet = 500(-400)
Final answer: Wnet = -2.0 × 10⁵ J
Example 3: Conservative Energy Change
A falling object loses 120 J of gravitational potential energy. Ignore air resistance. Find ΔK.
- ΔU = -120 J
ΔK = -ΔU
ΔK = -(-120 J)
Final answer: ΔK = +120 J
Real-Life Applications
- Vehicle brakes do negative work and reduce kinetic energy.
- A bat does positive work and raises a ball's kinetic energy.
- A roller coaster exchanges potential and kinetic energy.
- Gravity changes direction while doing zero work on a body in uniform circular motion.
Common Mistakes
- Using one force instead of total net work.
- Writing Ki - Kf instead of Kf - Ki.
- Thinking negative work means negative kinetic energy.
- Assuming zero net work means no forces act.
- Using ΔK = -ΔU when resistance also acts.
Short-Question Preparation
Q. State the work-energy principle.
A. The net work done on an object equals its kinetic-energy change: Wnet = ΔK = Kf - K_i.
Q. What does negative net work mean?
A. Kf < Ki, so the object loses kinetic energy and usually slows down.
Q. What does zero net work mean?
A. ΔK = 0, so kinetic energy and speed remain unchanged, although direction may change.
Q. How are ΔK and ΔU related for conservative forces?
A. When only conservative forces work, ΔK = -ΔU and K + U remains constant.
Long-Question Preparation
Q. Derive the work-energy principle.
A. 1. Write Wnet = Fnet d. 2. Use Fnet = ma. 3. Use vf² - vi² = 2ad. 4. Rearrange d = vf² - vi²2a. 5. Substitute both expressions. 6. Wnet = mavf² - vi²2a. 7. Cancel a. 8. Wnet = 12 m(vf² - vi²). 9. Expand. 10. Wnet = 12 mvf² - 12 mvi². 11. Therefore Wnet = Kf - Ki = ΔK.
Q. Explain its conservative-force form.
A. 1. For a conservative force, Wc = -ΔU. 2. The principle gives Wnet = ΔK. 3. If only conservative forces work, Wnet = Wc. 4. Therefore ΔK = -ΔU. 5. Hence ΔK + ΔU = 0. 6. Thus Ki + Ui = Kf + Uf.
Sticky Notes
- Wnet = ΔK.
- Use final minus initial.
- Positive work → K increases.
- Negative work → K decreases.
- Zero work → K unchanged.
- Conservative case: ΔK = -ΔU.
60-Second Revision
When to use: Only conservative forces.
Summary
Net work transfers kinetic energy, so W_net = ΔK. Its sign shows whether kinetic energy rises, falls or remains unchanged. For purely conservative interactions, ΔK = -ΔU and mechanical energy stays constant.
Ask-the-Tutor Modes
- Explain the theorem simply.
- Show the derivation one step per line.
- Help identify net work.
- Explain the sign of my answer.
- Check my numerical.
- Test me with an MCQ.
Exam Focus
Exam Focus
Important points
- Statement
- Derivation
- Sign interpretation
- Net work
- Conservative form
Important formulas
- Wnet = ΔK
- Wnet = 12 m(vf² - vi²)
- ΔK = -ΔU
Common exam traps
- Wrong subtraction order
- Ignoring a force
- Using conservation with resistance
Short question tips
- Explain the sign.
Long question structure
Statement → equations → substitution → simplification → interpretation.
Common Mistakes
Common Mistakes
- Mistake
Use net work.
- Mistake
Calculate final minus initial.
- Mistake
Negative work means a decrease in K, not negative K.
- Mistake
Apply ΔK = -ΔU only to conservative-force situations.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Wnet = ΔK
Positive work raises K.
Negative work lowers K.
Conservative case: ΔK = -ΔU.
Net work equals change in kinetic energy.
Wnet = Kf - Ki = 12 m(vf² - vi²)
- Definition
- Net work equals change in kinetic energy.
- Formula
- Wnet = Kf - Ki = 12 m(vf² - vi²)
- Example
- If K rises from 20 J to 65 J, net work is 45 J.
- Critical warning
- Do not use one force when several forces do work.
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Exam Practice
Important Short Question
Q1. State the work-energy principle.
Conceptual Question
Q2. Can forces act while net work is zero?
Long Question Preparation
Q3. Derive Wnet = ΔK.
Important Short Question
Q4. State the conservative-force form.