Physics · Module 3
Energy & Work
Dynamics told us how forces cause acceleration. Energy gives us a scalar shortcut — a single number that encodes the capacity of a system to do work, conserved across every interaction in the universe.
§ 1
Work
Work is done when a force moves its point of application through a displacement. It is a scalar — the dot product of force and displacement vectors.
DEFINITION OF WORK
$$W = \mathbf{F} \cdot \mathbf{s} = Fs\cos\theta$$
$\theta$ is the angle between the force vector and the displacement vector. Only the component of force along the displacement does work.
✅
Positive work
$\theta < 90°$ — force has a component along displacement. Energy transferred into the object.
🚫
Zero work
$\theta = 90°$ — force is perpendicular to motion. Carrying a bag horizontally: no work done on bag.
🔻
Negative work
$\theta > 90°$ — force opposes displacement. Friction does negative work, removing energy from the system.
The unit of work is the joule: $1\ \text{J} = 1\ \text{N m} = 1\ \text{kg m}^2\text{ s}^{-2}$. It is also the unit of energy — work and energy are the same physical quantity.
Wwork doneJ
Fforce magnitudeN
sdisplacementm
θangle between F and srad
§ 2
Kinetic energy
Kinetic energy is the energy an object possesses by virtue of its motion. The derivation follows directly from the work done by a net force accelerating a mass from rest.
Derivation from $F = ma$
$W = Fs$
Work done by net force over displacement $s$ (force parallel to motion, $\theta=0$)
$W = (ma)s$
Newton's second law: $F = ma$
$W = m\,\dfrac{v^2 - u^2}{2s} \cdot s$
From SUVAT: $v^2 = u^2 + 2as$, so $a = \dfrac{v^2 - u^2}{2s}$
$$\boxed{W = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mu^2}$$
Cancel $s$. The work done equals the change in the quantity $\frac{1}{2}mv^2$, which we name kinetic energy.
KINETIC ENERGY
$$E_k = \tfrac{1}{2}mv^2$$
Quadratic in $v$ — doubling speed quadruples kinetic energy. This is why high-speed collisions are so destructive.
§ 3
Potential energy
Potential energy is stored energy associated with position in a force field. When a conservative force (gravity, spring) does work, it converts potential energy to kinetic energy and vice versa — no energy is lost.
Gravitational potential energy
$W_\text{against gravity} = F \cdot h = mgh$
Lifting an object height $h$ requires force $mg$ upward over distance $h$
$$\boxed{E_p = mgh}$$
This work is stored as gravitational potential energy. Reference level ($h=0$) is arbitrary.
Only height differences matter. The path taken to reach a height is irrelevant — only the vertical displacement $\Delta h$ determines the change in gravitational PE. This is what it means for gravity to be a conservative force.
Elastic potential energy (springs)
HOOKE'S LAW + SPRING PE
$$F = -kx$$
Hooke's law: restoring force proportional to extension $x$. $k$ is the spring constant (N m⁻¹).
$$E_s = \tfrac{1}{2}kx^2$$
Spring PE: area under the $F$–$x$ graph (triangle with base $x$, height $kx$).
§ 4
The work–energy theorem
The result from §2 is so important it has a name. It is the bridge between forces (dynamics) and energy.
WORK–ENERGY THEOREM
$$W_\text{net} = \Delta E_k = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_i^2$$
The total work done by all forces on an object equals the change in its kinetic energy. No exceptions.
This is useful when you know forces and want to find speed (or vice versa) without knowing time. Compare with SUVAT which requires time.
🏋️
Net work positive
Object speeds up. $E_k$ increases. The applied forces collectively push the object along its path.
⛽
Net work zero
Speed unchanged. Normal force on circular motion: always perpendicular, never does work.
🛑
Net work negative
Object slows down. Brakes do negative work: $W = -\Delta E_k$. All that KE becomes heat.
§ 5
Conservation of energy
In a closed system with only conservative forces, total mechanical energy is constant. Energy is never created or destroyed — only converted between forms.
CONSERVATION OF MECHANICAL ENERGY
$$E_k + E_p = \text{constant}$$
$$\tfrac{1}{2}mv_1^2 + mgh_1 = \tfrac{1}{2}mv_2^2 + mgh_2$$
Valid when no non-conservative forces (friction, drag) act. The subscripts 1 and 2 refer to any two points in the motion.
With non-conservative forces (friction)
ENERGY WITH DISSIPATION
$$E_{k,i} + E_{p,i} = E_{k,f} + E_{p,f} + W_\text{friction}$$
Friction work $W_\text{friction} = f_k \cdot d$ is the energy converted to thermal energy (heat). Total energy including heat is still conserved.
Energy conservation is deeper than Newton's laws. It holds in quantum mechanics, special relativity, and every domain of physics — a consequence of the time-translation symmetry of the laws of nature (Noether's theorem).
Energy bar chart — pendulum at various positions
§ 6
Power
Power is the rate of doing work — how quickly energy is transferred. Same total work done more quickly requires more power.
POWER
$$P = \frac{W}{t} = \frac{\Delta E}{t}$$
Average power: work per unit time. Unit: watt (W = J s⁻¹).
$$P = Fv$$
Instantaneous power: force times velocity (when F is parallel to v). Crucial for engines and motors.
Derivation of $P = Fv$
$P = \dfrac{W}{t} = \dfrac{Fs}{t}$
Definition of power; work $W = Fs$ for constant force along displacement
$$\boxed{P = F \cdot \frac{s}{t} = Fv}$$
By definition, $v = s/t$
Terminal velocity from $P = Fv$: A car engine delivers constant power $P$. The driving force is $F = P/v$. As speed increases, driving force decreases. Terminal velocity is where $F_\text{drive} = F_\text{drag}$ — the engine can no longer accelerate the car.
§ 7
Springs and elastic energy
A spring obeys Hooke's law until its elastic limit. The restoring force is always directed back toward equilibrium — this creates oscillatory motion (covered fully in Module 7: SHM).
SPRING ENERGY SUMMARY
$$F = -kx \quad \Rightarrow \quad E_s = \tfrac{1}{2}kx^2$$
Force is the negative gradient of potential energy: $F = -\dfrac{dE_s}{dx}$. This is the general relationship between force and PE.
$$v_\text{max} = x_0\sqrt{\frac{k}{m}}$$
Maximum speed at equilibrium, from $\frac{1}{2}kx_0^2 = \frac{1}{2}mv^2$ (all PE → KE at centre).
The relationship $F = -dE_p/dx$ is profound: it means every potential energy landscape is also a force map. Where PE is steep, force is large. Where PE is flat, force is zero. This geometry of energy underlies all of theoretical physics.
§ 8
Live simulation
Three scenarios — each one making a different aspect of energy conservation visible. Watch energy flow between forms in real time.