Work and Energy — Mind Map

Physics gets fun the moment you meet Work. All that grinding through Newton's law equations? It melts away. A problem that took a page of force balancing now falls in three or four lines. And JEE loves this chapter because it lets them braid concepts together and ask something genuinely interesting.

One idea carries it all: the Work-Energy Theorem. Learn it well and it pays you back across the whole of mechanics.

But there's a catch — and it's a famous one. In physics, work is not what you think it is. Push a wall all afternoon. You'll sweat, your arms will ache, you'll feel like you did plenty. The work you did on that wall? Zero. Wait, what?

Let's map it out.

What "work" actually means

Here's the whole idea in one branch. A force only does work when it moves something.

• Work — done when a force causes a displacement.
  • No force, no work. Nothing pushes, nothing happens.
  • The force must produce displacement. Stand still against the wall and nothing moves — so zero work, no matter how tired you get.
  • The angle matters. If the force is at $90°$ to the displacement, the work is zero. (Carry a bag straight across a room: gravity pulls down, you move sideways, gravity does no work on the bag.)

That last point is the trap. The wall didn't move — that's the zero. And even when things do move, a force perpendicular to the motion still does nothing.

The formula behind it:

$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

Look at $\cos\theta$ and you can read the whole story. At $\theta = 90°$, $\cos\theta = 0$, so $W = 0$. That single cosine is doing all the work of explaining the wall.

More about work

Now branch out into the properties JEE actually tests.

• Work is a scalar. It has no direction. It comes from a dot product of two vectors, but the result is just a number (with sign).
• The sign tells a story. Work can be positive, negative, or zero — and the angle decides which.
  • $\theta < 90°$ → force helps the motion → positive work.
  • $\theta = 90°$ → force sideways → zero work.
  • $\theta > 90°$ → force fights the motion → negative work (friction, for instance, dragging on you).
• Conservative vs non-conservative forces. So you like going in circles? Then this one's for you.
  • A conservative force does zero net work when you start and finish at the same point. The path doesn't matter, only the endpoints. Examples: gravity, the Coulomb (electrostatic) force, the spring force.
  • A non-conservative force keeps draining energy no matter where you go — even on a closed loop. The classic culprit: friction. Walk in a circle back to your chair and friction has still stolen energy the whole way.

Quick mental test for "circular path": it doesn't have to be a literal circle. It means you ended where you started. Conservative forces don't care how you got back. Non-conservative ones charge you for every step.

Energy and the formulas

To do work, you need energy — and work is just energy changing hands. That's the Work-Energy Theorem in one sentence: the net work done on a body equals its change in kinetic energy.

$W_{net} = \Delta KE = \tfrac12 m v^2 - \tfrac12 m u^2, \qquad KE = \tfrac12 m v^2$

Now the formula branch — which one you reach for depends on a single question: is the force constant?

• Constant force → use $W = Fd\cos\theta$ directly. Done.
  • Example: gravity near the ground. Treat $mg$ as constant (it holds right up until the Gravitation chapter), so the simple formula works.
• Variable force → the simple formula breaks. You have to integrate: $W = \int \vec{F}\cdot d\vec{x}$
  • Example: a spring. The force grows as you stretch it, $F = kx$, so it's never constant. Integrate and you get the stored energy $\,U = \tfrac12 k x^2$.

That fork — constant or not — decides your whole approach. Spot it first, every time.

Your turn. You carry a 5 kg suitcase at a steady height across a 10 m platform. How much work do you do against gravity on the suitcase? And why?

Check: Zero. Gravity pulls straight down ($mg$), but the displacement is horizontal, so $\theta = 90°$ and $W = mgd\cos 90° = 0$. Same flavour as pushing the wall — the force never lines up with the motion.

The mind map in one screen

• Work $= \vec{F}\cdot\vec{d} = Fd\cos\theta$ — a scalar.
  • No force → no work. No displacement → no work. $\theta = 90°$ → no work.
  • Sign set by the angle: helping (+), sideways (0), opposing (−).
• Forces
  • Conservative (gravity, Coulomb, spring): zero work on a closed path.
  • Non-conservative (friction): drains energy on any path.
• Formulas
  • Constant force → $W = Fd\cos\theta$.
  • Variable force (spring, $F = kx$) → $W = \int \vec{F}\cdot d\vec{x}$.
• Work-Energy Theorem: $W_{net} = \Delta KE$, with $KE = \tfrac12 m v^2$.

This is a heavy-scoring chapter for JEE — count on more than one question leaning on these ideas. Keep this map next to you, and the next time the wall pushes back, you'll know exactly why your work is zero.