WORK, ENERGY & POWER in One Shot: All Concepts & PYQs Covered | JEE Main & Advanced

Main Ideas / Lessons Conveyed (Work, Energy, Power — JEE Focus)

Work (definition & computation)

• Work is tied to how a force changes a particle’s kinetic energy.

• For an infinitesimal displacement (ds): [ \delta W=\vec F\cdot d\vec s = F\,ds\cos\theta ]

• For a force over a path/interval:

  • Net work is found by integration: [ W_{\text{net}}=\int \vec F\cdot d\vec s ]

• Special case (zero work):

  • If displacement is perpendicular to force ((\theta=90^\circ)), work is 0.
  • If a force is applied but there is no displacement of that point, work is 0.

Work by constant force

If the force is constant and makes angle (\theta) with displacement (\vec d): [ W=\vec F\cdot \vec d = Fd\cos\theta ] Common values:

• (\theta=0^\circ \Rightarrow W=Fd)
• (\theta=180^\circ \Rightarrow W=-Fd)
• (\theta=90^\circ \Rightarrow W=0)

Component-based perspective

Work can also be understood as:

• ((\text{component of force along displacement})\times d)

Work by common forces (mechanics toolkit)

• Gravity: depends on vertical displacement; the sign comes from the relative direction of motion.
• Normal reaction: typically perpendicular to displacement on smooth contact, so work is often 0 (though geometry/frame can matter).
• Tension: usually zero work when the mass moves with the string direction (tension is perpendicular to instantaneous displacement in typical constraint setups).
• Friction: work is negative when friction opposes displacement: [ W_f=-f_k\,d=-\mu_k N d ] (On horizontal surfaces, (N\approx mg).)

Vector / dot-product emphasis

• For variable directions or vector expressions, use dot product:

  • If (\vec F=\langle F_x,F_y,F_z\rangle) and (d\vec r=\langle dx,dy,dz\rangle), then: [ dW = F_x\,dx + F_y\,dy + F_z\,dz ]

• The lecture warns against incorrectly mixing components—use (\vec F\cdot d\vec r).

Path independence (constant / conservative force idea)

• For a constant force, work depends only on displacement (not on the path).
• The lecture hints that conservative forces (like spring/gravity under suitable conditions) also lead to path-independent work.

Work–Energy Theorem (core JEE link)

• Central theorem: [ W_{\text{by all forces}}=\Delta K = K_f-K_i ]

• Used to avoid solving full dynamics when you need only initial/final speeds (or energies).

• “Many arrows” approach:
  • Compute work done by each force individually and add them.
  • Equivalent net form: [ W=\int \vec F_{\text{net}}\cdot d\vec r ]

---

Methodology / Step-by-Step Instructions Explicitly Taught

A) Standard procedure to solve Work–Energy problems

1. Choose system and state frame implicitly
   • Identify which object/particle you’re applying work to.
2. Identify all forces acting
   • Examples: gravity (mg), normal (N), friction (f_k), tension (T), spring (kx), applied external force.

3. For each force, choose the correct work form

   • Constant force: [ W=F d\cos\theta ]

   • Variable force: [ W=\int \vec F\cdot d\vec s ]

4. Determine signs

   • Same direction as displacement (\Rightarrow) positive work
   • Opposite direction (\Rightarrow) negative work
   • Perpendicular (\Rightarrow) zero work

5. Add works

   • [ W_{\text{total}}=\sum W_i ]

6. Apply Work–Energy Theorem

   • [ W_{\text{total}}=K_f-K_i ]

7. Solve for the asked quantity

   • Often final speed (v), final kinetic energy, maximum compression/elongation, etc.

B) Work with multiple forces (example-block workflow)

• Example patterns highlighted:

  • Applied force: (W_F=Fd) if (\theta=0)
  • Gravity: often (W_{mg}=0) if displacement is horizontal
  • Normal: (W_N=0) if normal is perpendicular to displacement
  • Friction: [ W_f=-\mu N d ]

• Then: [ W_{\text{net}}=W_F+W_f ]

• Use: [ W_{\text{net}}=\Delta K ]

C) Variable force work (integration workflow)

• If (\vec F) depends on position:

  • [ dW=\vec F\cdot d\vec r ]

  • Expand: [ dW=F_x\,dx+F_y\,dy+F_z\,dz ]

• Pick limits from initial and final coordinates.

• Integrate component-wise.

D) Spring force work (high-value formula + method)

• Spring force: [ F_s=-kx ] (sign depends on how displacement is defined)

• General result for work done by spring: [ W_s=-\frac12 k\left(x_f^2-x_i^2\right) ]

• Key point: it works whether (x) is compression or elongation because the expression uses squares.

• Equivalent viewpoint: [ \text{Energy stored}=\frac12 kx^2 ]

E) “Force–displacement dot product in vector form” (advanced/vector JEE)

• For forces in (i,j,k) form: [ \vec F\cdot d\vec r ]

• If direction/magnitude varies: [ W=\int \left(F_x\,dx+F_y\,dy+F_z\,dz\right) ]

F) Frame/observer caution for normal & pseudo forces (lift frame example)

• Work values can be frame-dependent because displacement components change.
• Work–Energy theorem can still be applied in any frame if pseudo forces are included appropriately.
• In an accelerating lift:
  • Add pseudo force (m\vec a) to satisfy Newton’s 2nd law in the non-inertial frame.
  • Compute works in that frame and use (\Delta K) for that frame’s kinetic energy.

---

High-Frequency JEE Patterns Mentioned (PYQ-style Coverage)

• Compute work done by:
  • gravity during vertical displacement
  • friction during sliding distance on rough surfaces
  • normal is often zero in standard setups (but verify geometry/frame)
  • tension is typically zero when constraint geometry makes tension perpendicular to instantaneous displacement
• Work–Energy theorem shortcuts:
  • Directly find final speed when initial speed is known and only displacement/height changes are involved.
• Maximum compression/elongation of spring:
  • Use work–energy with spring work and include gravitational/friction work if present.
  • Include external forcing only if it exists.

---

Speakers / Sources Featured

• Speaker: Main instructor/teacher referred to as “Sir/Guruji/brother” (not clearly named).
• External sources: No specific books/videos are reliably identified by name beyond general references like “PYQ” and “SKSC book” (mentioned, but not fully identifiable from the text).