Topic 5.4.113 min read
Work-Energy Theorem in Resistive Medium
In a resistive medium, applied work is divided between the object's increase in kinetic energy and the work done against resistance: W = ΔK + W_r.
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Beginner Level
When an object moves through air, water or a rough surface, resistance opposes its motion. The applied force must both change the object's kinetic energy and overcome this resistance.
- W
- work by applied force (J)
- ΔK
- gain in kinetic energy (J)
- W_r
- work against resistance (J)
When to use: Use when friction, drag or another resistive force acts.
Recommended for Class 11
5.4.1 Work-Energy Theorem in Resistive Medium
Overview
A resistive force opposes a moving object or resists the start of motion. Friction, air resistance and viscous drag are common examples. A parachute protects a skydiver because air resistance grows and can balance weight, limiting further acceleration.
Learning Outcomes
- Define a resistive force.
- Explain the division of applied work.
- Derive Wr = W - ΔK.
- Solve problems involving friction or drag.
- Explain where energy transferred against resistance goes.
- Compare the energy method with Newton's laws.
Key Terms
| Term | Meaning | Examples |
|---|---|---|
| Resistive force | Force opposing motion or attempted motion | Friction, drag, viscous force |
| Applied work W | Energy supplied by an external applied force | Push or pull through a distance |
| ΔK | Change in kinetic energy | Increase or decrease in speed |
| W_r | Work done against resistance | Energy transferred to surroundings |
Intermediate Level
Energy Accounting
When an applied force moves an object against resistance, the applied work has two destinations. One part changes kinetic energy. The other part is work done against resistance.
- W
- work by applied force (J)
- ΔK
- kinetic-energy change (J)
- W_r
- work against resistance (J)
Common mistake: Do not count W_r as an additional gain in kinetic energy.
Rearrange to find the work against resistance:
For constant applied force F through distance d in its direction:
For constant resistive-force magnitude f through distance d:
When to use: Use when F and f are constant and parallel to the motion.
Physical Meaning
Resistance takes mechanical energy from the moving system and transfers it into thermal energy, sound, deformation or motion of the surrounding fluid. Energy is transformed rather than destroyed.
| Situation | Result |
|---|---|
| W > W_r | ΔK is positive; speed increases |
| W = Wr | ΔK = 0; speed remains constant |
| Applied force removed | Resistance does negative net work; kinetic energy decreases |
| Parachutist at terminal speed | Air resistance balances weight; net work and ΔK are zero |
Why the Energy Method Helps
Both Newton's laws and the work-energy theorem can solve motion problems. When forces vary with position or the route is complicated, the energy method can be simpler because it compares initial and final states without finding every intermediate acceleration.
Units and Dimensions
Every term is measured in joules, so the equation is dimensionally consistent.
Solved Numerical Examples
Example 1: Work Against Friction
An applied force does 500 J of work while a box gains 320 J of kinetic energy. Find the work done against friction.
- W = 500 J
- ΔK = 320 J
Wr = W - ΔK
Wr = 500 - 320
Final answer: Wr = 180 J
Example 2: Constant Forces
A 60 N force pulls a body 10 m while a 20 N resistive force opposes it. The body starts with 40 J of kinetic energy. Find its final kinetic energy.
- F = 60 N
- f = 20 N
- d = 10 m
- Ki = 40 J
W = Fd = 60(10) = 600 J
Wr = fd = 20(10) = 200 J
ΔK = W - Wr
ΔK = 600 - 200 = 400 J
Kf = Ki + ΔK
Kf = 40 + 400
Final answer: Kf = 440 J
Example 3: Constant Speed
A crate moves at constant speed while 150 J of work is done against friction. Find the applied work.
- Wr = 150 J
- constant speed, so ΔK = 0
W = ΔK + Wr
W = 0 + 150
Final answer: W = 150 J
Real-Life Applications
- Parachutes use air resistance to reduce falling speed.
- Vehicle brakes use friction to remove kinetic energy.
- Lubricants reduce energy transferred against friction in machines.
- Streamlined vehicles reduce drag and required input energy.
- Cyclists must continually supply work against air and rolling resistance.
Common Mistakes
- Confusing work against resistance with work done by resistance.
- Assuming all applied work becomes kinetic energy.
- Using Wr = W + ΔK instead of W - ΔK.
- Forgetting that constant speed gives ΔK = 0.
- Saying resistive forces destroy energy.
Short-Question Preparation
Q. What is a resistive force?
A. A resistive force opposes motion or attempted motion. Friction, air drag and viscous force are examples.
Q. State the work-energy relation in a resistive medium.
A. Applied work equals the kinetic-energy gain plus work against resistance: W = ΔK + W_r.
Q. Where does energy used against resistance go?
A. It is transferred mainly into thermal energy and may also appear as sound, deformation or fluid motion.
Q. Why can the energy method be convenient?
A. It connects initial and final energy states and can avoid calculating a changing acceleration at every point.
Long-Question Preparation
Q. Explain the theorem in a resistive medium.
A. 1. Define a resistive force and give examples. 2. Let an applied force do work W. 3. One part changes kinetic energy by ΔK. 4. The remaining part is work Wr against resistance. 5. Therefore W = ΔK + Wr. 6. Rearrange: Wr = W - ΔK. 7. For constant forces, W = Fd and Wr = fd. 8. Hence ΔK = (F - f)d. 9. Explain that W_r becomes other energy forms.
Sticky Notes
- Resistance opposes motion.
- W = ΔK + Wr.
- Wr = W - ΔK.
- Constant speed → ΔK = 0.
- Work by resistance on object = -W_r.
- Energy becomes heat, sound or deformation.
60-Second Revision
When to use: Constant parallel forces.
Summary
With resistance present, applied work is shared between changing kinetic energy and overcoming resistance. The relation W = ΔK + W_r provides a clear energy account and explains why real motion produces thermal and other non-mechanical energy.
Ask-the-Tutor Modes
- Explain resistance simply.
- Draw the energy accounting in words.
- Explain the sign convention.
- Check my friction calculation.
- Compare Newton's laws with the energy method.
- Test me with an MCQ.
Exam Focus
Exam Focus
Important points
- Definition of resistance
- W = ΔK + Wr
- Sign convention
- Energy transformations
- Constant-speed case
Important formulas
- W = ΔK + Wr
- Wr = W - ΔK
- ΔK = (F - f)d
Common exam traps
- Wrong sign
- Assuming all work becomes K
- Calling energy lost or destroyed
Short question tips
- Name the transformed energy.
Long question structure
Definition → energy accounting → equations → sign convention → example.
Common Mistakes
Common Mistakes
- Mistake
Distinguish W_r from negative work by resistance.
- Mistake
Do not send all applied work into ΔK.
- Mistake
At constant speed, applied work equals work against resistance.
- Mistake
Energy is transformed, not destroyed.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
W = ΔK + Wr
Wr = W - ΔK
Resistance opposes motion.
Constant speed gives W = W_r.
Applied work in a resistive medium becomes kinetic-energy change plus work against resistance.
W = ΔK + Wr
- Definition
- Applied work in a resistive medium becomes kinetic-energy change plus work against resistance.
- Formula
- W = ΔK + Wr
- Example
- 500 J applied and 320 J gained as K gives W_r = 180 J.
- Critical warning
- Do not confuse work against resistance with its signed work on the object.
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Exam Practice
Important Short Question
Q1. Define a resistive force.
Important Short Question
Q2. State the energy relation with resistance.
Conceptual Question
Q3. Why is W_r not destroyed energy?
Long Question Preparation
Q4. Explain the work-energy theorem in a resistive medium.