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Under what condition does an applied force do ZERO work on an object?

When force is perpendicular to displacement (θ = 90°). Work W = F · d · cos(θ). When θ = 90°, cos(90°) = 0, resulting in zero work.

What is the instantaneous power delivered to a car of mass m traveling at constant velocity v against resistive force f?

f · v. Instantaneous power is given by P = F · v. To maintain constant velocity, engine force equals resistive force f.

Physics · Class 11-12 · JEE / NEET / AP Physics

Work, Energy, and Power

Work, Energy, and Powerform the foundational pillar of mechanics. Work is done when an applied force causes a displacement, transforming mechanical energy betweenkineticandpotentialstates.

11. Visual Overview & Intuitive Discovery: Work, Energy, and Power

Imagine standing at the highest crest of a colossal 100-meter roller coaster. For a brief second, you are motionless in the sky. Behind this exhilarating sensation lies one of the most elegant universal truths in the cosmos: energy can neither be created nor destroyed—it merely morphs from silent gravitational potential into a thundering kinetic roar.

Work, Energy, and Power Framework: Work Formula, Potential Energy Peak, Kinetic Energy Maximum, Conservation of Energy, Power Rating — Work, Energy, and Power Framework
Energy transformation on a roller coaster track showing potential energy at the peak converting into maximum kinetic energy at the base, along with mechanical power calculation.
1Work Formula — W = F · d · cos(θ), measuring the dot product of force and displacement vectors
2Potential Energy Peak — PE = mgh is maximized at maximum height where velocity is minimal
3Kinetic Energy Maximum — KE = 1/2 m v² is highest at the lowest elevation of the trajectory
4Conservation of Energy — E_total = PE + KE remains constant when non-conservative friction is neglected
5Power Rating — P = W / t = F · v, the rate of doing work measured in Watts (Joules per second)

Interactive Study Prompts (Click to Explore)

Clickable

AI Deep Dive 🪄

22. Physical Mechanism Flowchart: Work, Energy, and Power

Work, Energy, and Power Step-by-Step Flowchart

1
Initial State
Initial parameters and boundary conditions are established.
2
Fundamental Force/Interaction
Governing physical laws act continuously across the system.
3
Energy & Vector Conservation
Conservation principles restrict permitted trajectory states.
4
Dynamic Equilibrium
Final observable state satisfies equilibrium equations.

Follow the numbered pipeline above. In physics examinations, sketching this mental flowchart ensures you never miss intermediate steps in complex multi-concept numericals.

33. Rigorous Mathematical Proofs & Governing Equations

Derivation — Derivation of Governing Formula for Work, Energy, and Power

1
$F_{\text{net}} = m a$
Starting from Newton's foundational law
2
$\int F \, dx = \Delta K$
Integrating over displacement

The Work-Energy Theorem

W_{\text{net}} = \Delta K = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2

$W_{\text{net}}$ = Net work done by all acting forces (conservative + non-conservative)
$\Delta K$ = Change in kinetic energy of the body
m = Mass of the object
$v_f, v_i$ = Final and initial velocities

Conservative vs. Non-Conservative Forces

Feature	Conservative Force (e.g., Gravity, Spring)	Non-Conservative Force (e.g., Friction, Drag)
Work over Closed Loop	Zero (∮ F · dr = 0)	Non-zero (energy lost as heat/sound)
Path Dependence	Completely independent of path taken	Directly depends on the total path length
Potential Energy Association	Has an associated Potential Energy: F = -dU/dr	No potential energy function exists

Work over Closed Loop
Conservative Force (e.g., Gravity, Spring)
Zero (∮ F · dr = 0)
Non-Conservative Force (e.g., Friction, Drag)
Non-zero (energy lost as heat/sound)
Path Dependence
Conservative Force (e.g., Gravity, Spring)
Completely independent of path taken
Non-Conservative Force (e.g., Friction, Drag)
Directly depends on the total path length
Potential Energy Association
Conservative Force (e.g., Gravity, Spring)
Has an associated Potential Energy: F = -dU/dr
Non-Conservative Force (e.g., Friction, Drag)
No potential energy function exists

44. Real-World Engineering: Regenerative Braking in Electric Vehicles

In conventional gasoline cars, stepping on the brakes forces friction pads against spinning metal rotors, permanently throwing away precious kinetic energy as useless heat into the atmosphere. In modern electric vehicles (like Tesla and Formula E cars), the electric motor reverses its role during deceleration. It operates as an electric generator, turning the car's kinetic energy back into electrical potential energy stored in the lithium-ion battery pack—recovering up to 70% of energy that would otherwise be wasted!

Common mistake: Misconception 1: "Holding a heavy 50 kg barbell above your head for 10 minutes does enormous mechanical work." In physics, displacement is strictly zero (

d = 0

). Therefore, the mechanical work done on the barbell is identically ZERO (

W = F \cdot 0 = 0

). Your physiological muscles feel exhausted because microscopic muscle fibers continuously contract and relax, burning chemical ATP internally, but zero mechanical energy is transferred to the barbell!

Exam tip: Exam Pro-Tip: Whenever a problem involves tension in a pendulum or normal force on a curved track, NEVER attempt to calculate work done by normal force or string tension! Because these constraint forces are perpetually perpendicular to the instantaneous velocity vector at every point of the curved path (

\theta = 90^\circ

), their work is ALWAYS exactly zero.

55. Solved Standard & Advanced Exam Problems

Exam question · JEE Main & AP Physics Standard

A 2 kg block slides down an incline from a height of 5 m. At the bottom, its measured speed is 8 m/s. What is the work done by the frictional force during the slide? (Take g = 9.8 m/s²)

A.-34 Joules
B.-98 Joules
C.+64 Joules
D.-64 Joules

Exam question · JEE Advanced / AP Physics C

A variable force

F(x) = (3x^2 - 2x + 5)

Newtons acts on an object of mass 2 kg moving along the x-axis from

x = 0

m to

x = 3

m. If the object was initially at rest, what is its velocity at

x = 3

A.3.0 m/s
B.5.74 m/s
C.33.0 m/s
D.4.5 m/s

Remember

Work is the scalar product of force and displacement: W = F · d cos(θ).
The net work done on a particle equals the change in its kinetic energy (Work-Energy Theorem).
Conservative forces conserve total mechanical energy and depend solely on initial and final positions.
Power is the time rate at which energy is transferred: P = dW/dt = F · v.
Minimum speed required to complete a vertical loop of radius R under gravity is √(5gR) at the lowest point.

Check yourself

Question 1
Under what condition does an applied force do ZERO work on an object?
Question 2
What is the instantaneous power delivered to a car of mass m traveling at constant velocity v against resistive force f?