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What is the work-energy theorem?

Class 11 PhysicsNCERT chapter: Work, Energy and Powerइसे हिंदी में पढ़ें

Exam-ready answer

The work-energy theorem states that the change in kinetic energy of a particle is equal to the work done on it by the net force: K_f − K_i = W. Here kinetic energy K = ½mv². If the net force does positive work, the body speeds up; if it does negative work, the body slows down.

Explanation

Start with the equation of motion for constant acceleration: v² − u² = 2as, where u is the initial speed, v the final speed and s the displacement. Multiply both sides by m/2. This gives ½mv² − ½mu² = mas. By Newton's second law, ma is the net force F, so the right-hand side becomes Fs, which is the work done by the net force over the displacement s.

So ½mv² − ½mu² = Fs, or K_f − K_i = W. The quantity ½mv² is the kinetic energy of the body, a scalar measured in joules (J). The theorem also holds for a variable force, where the work is found by integrating F dx over the path. It is the integral form of Newton's second law and links force and displacement directly to the change in speed.

Example: a moving car brakes to a stop. The friction force acts opposite to the displacement, so it does negative work. That negative work equals the loss of the car's kinetic energy. Similarly, when a raindrop falls, gravity does positive work and air resistance does negative work; the net work done by both forces equals the change in the drop's kinetic energy.

Points to remember

  1. Statement: the change in kinetic energy of a particle equals the work done on it by the net force.
  2. Formula: K_f − K_i = W, where K = ½mv².
  3. Proof uses v² − u² = 2as multiplied by m/2, with F = ma.
  4. Work and kinetic energy are scalars; SI unit is the joule (1 J = 1 N m).
  5. Positive net work increases speed; negative net work (e.g. friction) decreases speed.

Exam tip

Show the three-line proof: v² − u² = 2as, multiply by m/2, replace ma by F. Then write K_f − K_i = W in a box and give the SI unit, joule.

Questions students also ask

What is the work done by a force?
Work done by a constant force is W = Fd cos θ, where F is the force, d the displacement and θ the angle between them. It is positive when θ is less than 90°, zero at 90° and negative when θ is more than 90°.
Does the theorem work for a variable force?
Yes. For a variable force, the work is the integral of F dx between the initial and final positions. Using F = m dv/dt and dx = v dt, this integral works out to ½mv² − ½mu², the change in kinetic energy.
Why does a body moving in a circle at constant speed keep its kinetic energy?
The centripetal force is always perpendicular to the velocity, so the angle between force and displacement is 90° and the work done is zero. By the work-energy theorem, zero work means no change in kinetic energy, so the speed stays constant.

Chapter labels help you find the topic in your textbook. Check your own edition: chapter names and order can change. These are Know.it explanations, not official NCERT solutions.