Work, Energy & Power
Physics Study Notes – VITEEE / IIT‑JEE
(Comprehensive, exam‑oriented, with derivations, key formulae, tricky points & a hard problem)
1. Work
| Symbol | Definition | Units | Sign Convention |
|---|---|---|---|
| (W) | Energy transferred by a force ( \vec F ) acting over a displacement ( \vec s ) | Joule (J) | (W>0) if ( \vec F) does work, (W<0) if it opposes the motion |
1.1 Work done by a constant force
[ W = \vec{F}\cdot \vec{s} = Fs\cos\theta ]
1.2 Work done by a variable force
[ W = \int_{s_i}^{s_f} \vec{F}(s)\cdot d\vec{s} ] Important: The integral is path‑dependent unless ( \vec{F} ) is a conservative force.
2. Work–Energy Theorem
2.1 Statement
[ W_{\text{total}} = \Delta K = K_f - K_i ] where (K=\tfrac12 mv^2) is the kinetic energy.
2.2 Derivation (Newton’s 2nd Law)
[ \begin{aligned} \vec{F} &= m\vec{a} = m\frac{d\vec{v}}{dt} \ \vec{F}\cdot d\vec{s} &= m\frac{d\vec{v}}{dt}\cdot \vec{v},dt = m,\frac{d}{dt}!\left(\frac{v^2}{2}\right)dt \ \Rightarrow W_{\text{total}} &= \int_{t_i}^{t_f} m,\frac{d}{dt}!\left(\frac{v^2}{2}\right)dt = \frac12 m v_f^2-\frac12 m v_i^2 \end{aligned} ]
Tricky Point – The theorem holds for any forces, but the work you compute must include all forces (conservative + non‑conservative).
3. Kinetic Energy (KE)
[ K = \frac12 m v^2 ]
- Units: J
- Scalar quantity – orientation of velocity is irrelevant.
- Rate of change: ( \displaystyle \frac{dK}{dt} = \vec{F}\cdot\vec{v} ) (power supplied by the net force).
4. Potential Energy (PE)
4.1 Conservative Force and PE
A force ( \vec{F} ) is conservative if the work done between two points is path‑independent.
[
W_{\text{cons}} = -\Delta U
]
where ( U ) is the potential energy.
4.2 Derivation of PE from a Conservative Force
[ U(\vec{r}) = -\int_{\vec r_0}^{\vec r} \vec{F}{\text{cons}}\cdot d\vec{s} ] For a spherically symmetric force ( \vec{F}(r) = -\frac{GmM}{r^2}\hat r ): [ U(r)= -\int{\infty}^{r} -\frac{GmM}{r'^2},dr' = -\frac{GmM}{r} ]
4.3 Common Potential Energies
| System | PE (relative to chosen reference) | Expression | Reference Point |
|---|---|---|---|
| Particle in uniform gravitational field (near Earth) | (U_g = mgh) | (h) above ground | (h=0) at ground |
| Mass–spring (Hooke’s law) | (U_s = \frac12 k x^2) | (x) measured from equilibrium | (x=0) at equilibrium |
| Electrostatic point charges | (U_e = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}) | (r) separation | (U=0) at (r\to\infty) |
| Gravitational two‑body system | (U_g = -\frac{GmM}{r}) | (r) separation | (U=0) as (r\to\infty) |