Work, Energy and Power

Work, Energy and Power

Info: Board Coverage AQA Paper 1 | Edexcel CP1, CP2 | OCR (A) Paper 1 | CIE P2

Explore the simulation above to develop intuition for this topic.

Intuition

Energy is the ability to do work — and work is energy transferred by a force: Think of pushing a box across a floor. You do work on the box because you apply a force and the box moves. The energy you expend becomes kinetic energy of the box (if it speeds up) or heat (if there’s friction). Energy isn’t “used up” — it’s transformed from one form to another.

Why it matters: Work-energy principles explain why roller coasters work (potential → kinetic → potential), why brakes get hot (kinetic → heat), and why power stations exist (chemical/nuclear → electrical). Understanding energy transformations is the key to understanding all physical processes.

The key insight: Power is the rate of energy transfer, not the amount. A 100W bulb and a 1000W heater both convert electrical energy, but the heater does it 10 times faster. This is why a small fire can be more dangerous than a large one — it’s not the total energy but how quickly it’s delivered.

1. Work Done by a Force

Definition. The work done by a constant force $\mathbf{F}$ when its point of application Moves through displacement $\mathbf{s}$ is:

$W = \mathbf{F} \cdot \mathbf{s} = Fs\cos\theta$

Where $\theta$ is the angle between $\mathbf{F}$ and $\mathbf{s}$.

Derivation. We decompose $\mathbf{F}$ into components parallel and perpendicular to $\mathbf{s}$. Only the parallel component $F\cos\theta$ does work (the perpendicular component Produces no displacement in its direction). Work is defined as force multiplied by the displacement In the direction of the force, hence $W = (F\cos\theta) \cdot s = Fs\cos\theta$.

Units. Work has the unit joule (J), where $1 \mathrm{ J} = 1 \mathrm{ N m} = 1 \mathrm{ kg m}^2\mathrm s^{-2}$.

Sign convention. Work is positive when the force has a component in the direction of motion, Negative when opposing motion.

Work done by a variable force. For a force $F(x)$ that varies with position:

$W = \int_{x_1}^{x_2} F(x)\,dx$

2. Kinetic Energy

Definition. The kinetic energy of a body of mass $m$ moving with velocity $v$ is:

$\boxed{E_k = \frac{1}{2}mv^2}$

Derivation from the work-energy theorem. Consider a body of mass $m$ accelerated from rest to Speed $v$ by a constant net force $F$ over distance $s$. The work done is:

$W = Fs = mas$

Using $v^2 = u^2 + 2as$ with $u = 0$: $s = v^2/(2a)$. Therefore:

$W = ma \cdot \frac{v^2}{2a} = \frac{1}{2}mv^2$

We define this work done as the kinetic energy gained. $\square$

General proof (variable force). From Newton’s second law: $F = m\frac{dv}{dt}$. The work done Is:

$W = \int_{s_1}^{s_2} F\,ds = \int_{s_1}^{s_2} m\frac{dv}{dt}\,ds = \int_{v_1}^{v_2} mv\,dv = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$

This is the work-energy theorem: the net work done on a body equals its change in kinetic Energy.

$\boxed{W_{\mathrm{net}} = \Delta E_k}$

3. Gravitational Potential Energy

Definition. The gravitational potential energy of a body of mass $m$ at height $h$ above a Reference level is:

$\boxed{E_p = mgh}$

Derivation. The work done against gravity in lifting a mass $m$ through a vertical height $h$ at Constant velocity (so $a = 0$ and applied force $= mg$):

$W = Fd = mg \times h = mgh$

This work is stored as gravitational potential energy. $\square$

Intuition. $E_p$ depends on the choice of reference level. Only changes in potential energy Are physically meaningful. The convention is to take $E_p = 0$ at the lowest point of the system.

4. Elastic Potential Energy

Definition. The elastic potential energy stored in a spring (or any elastic body obeying Hooke’s law) extended by $x$ from its natural length is:

$\boxed{E_e = \frac{1}{2}kx^2}$

Where $k$ is the spring constant (force per unit extension).

Derivation. By Hooke’s law, the force required to extend the spring by $x$ is $F = kx$. Since The force varies with extension, we integrate:

$W = \int_0^x F\,dx' = \int_0^x kx'\,dx' = \left[\frac{1}{2}kx'^2\right]_0^x = \frac{1}{2}kx^2$

This work is stored as elastic potential energy. $\square$

Graphical interpretation. On a force-extension graph, the elastic potential energy is the area Under the line $F = kx$Which is a triangle of base $x$ and height $kx$Giving $\frac{1}{2}kx^2$.

Springs in Series and Parallel

Parallel (same extension, forces add). Two springs with constants $k_1$ and $k_2$ share the same Extension $x$:

$E_e = \frac{1}{2}k_1 x^2 + \frac{1}{2}k_2 x^2 = \frac{1}{2}(k_1 + k_2)x^2 = \frac{1}{2}k_{\mathrm{eff}}x^2$

$\boxed{k_{\mathrm{parallel}} = k_1 + k_2}$

Proof from equilibrium. Both springs exert force on the mass: $F = k_1 x + k_2 x = (k_1 + k_2)x$So $k_{\mathrm{eff}} = k_1 + k_2$. $\square$

Series (same force, extensions add). Two springs with constants $k_1$ and $k_2$ experience the Same force $F$:

$F = k_1 x_1 = k_2 x_2$So $x_1 = F/k_1$ and $x_2 = F/k_2$. Total extension: $x = x_1 + x_2 = F(1/k_1 + 1/k_2)$.

$\boxed{\frac{1}{k_{\mathrm{series}}} = \frac{1}{k_1} + \frac{1}{k_2}}$

Intuition. Parallel springs are stiffer (each contributes to resisting displacement). Series Springs are less stiff (each stretches under the same load, giving more total compliance). This is Analogous to electrical circuits: parallel resistors decrease, series resistors increase.

5. Conservation of Energy

Principle of Conservation of Energy. Energy cannot be created or destroyed, only transferred From one form to another.

$\sum E_{\mathrm{initial}} = \sum E_{\mathrm{final}}$

For a conservative system (no dissipative forces like friction):

$E_k + E_p + E_e = \mathrm{constant}$

Proof for a mass falling under gravity. A mass $m$ falls from height $h_1$ (at rest) to height $h_2$ (speed $v$). By the work-energy theorem:

$W_{\mathrm{gravity}} = \Delta E_k$

But $W_{\mathrm{gravity}} = mg(h_1 - h_2) = -\Delta E_p$. Therefore:

$-\Delta E_p = \Delta E_k \implies \Delta E_k + \Delta E_p = 0$

$E_k + E_p = \mathrm{constant} \quad \square$

With non-conservative forces:

$W_{\mathrm{nc}} = \Delta E_k + \Delta E_p$

Where $W_{\mathrm{nc}}$ is the work done by non-conservative forces (e.g., friction does negative Work, reducing total mechanical energy).

Energy Dissipation by Friction

When friction acts, mechanical energy is not conserved — it decreases monotonically as energy is Transferred to thermal energy. For a block sliding to rest on a rough horizontal surface:

$E_k(\mathrm{initial}) = E_k(\mathrm{final}) + W_f$

Where $W_f = \mu mg d$ is the work done against friction over distance $d$.

Setting $E_k(\mathrm{final}) = 0$ (block comes to rest):

$\frac{1}{2}mv^2 = \mu mgd$

$\boxed{d = \frac{v^2}{2\mu g}}$

Intuition. The stopping distance is proportional to $v^2$ (not $v$), which is why driving at Twice the speed requires four times the braking distance. This quadratic dependence is a direct Consequence of energy conservation: kinetic energy scales as $v^2$So removing that energy requires Friction to act over a distance proportional to $v^2$.

Enrichment: Noether’s Theorem

Emmy Noether proved in 1915 that every conservation law corresponds to a symmetry of nature. Conservation of energy corresponds to time-translation symmetry — the laws of physics do not Change over time. If you perform an experiment today or tomorrow, you expect the same result. This Deep connection between symmetry and conservation is one of the most profound results in physics.

6. Power

Definition. Power is the rate at which work is done (or energy is transferred):

$\boxed{P = \frac{dW}{dt} = \frac{\Delta E}{\Delta t}}$

Units. 1 watt (W) $= 1$ J s$^{-1}$.

Power and Velocity

For a force $F$ moving a body at velocity $v$:

$P = \frac{dW}{dt} = \frac{F \, ds}{dt} = Fv$

More generally, for a force at angle $\theta$ to the velocity:

$\boxed{P = Fv\cos\theta}$

Average Power for Constant Acceleration

When a constant force accelerates a body from $u$ to $v$ over distance $s$:

$P_{\mathrm{avg}} = \frac{W}{t} = \frac{Fs}{t} = F \cdot \frac{s}{t} = F\bar{v}$

Since the average velocity is $\bar{v} = (u + v)/2$:

$\boxed{P_{\mathrm{avg}} = \frac{F(u + v)}{2}}$

This is useful because it relates power to the initial and final velocities directly, without Needing to know the distance or time separately.

Definition. The efficiency of an energy transfer is:

$\eta = \frac{\mathrm{useful energy output}}{\mathrm{total energy input}} \times 100\%$

Or equivalently for power:

$\eta = \frac{P_{\mathrm{out}}}{P_{\mathrm{in}}} \times 100\%$

Efficiency is always between 0% and 100%. In practice, some energy is always dissipated ( as Thermal energy due to friction or resistance).

8. Energy as the Universal Currency

Energy is the most unifying concept in physics. It connects mechanics, thermodynamics, Electromagnetism, waves, quantum physics, and relativity. The conservation of energy is the one Principle that holds across all domains:

• In mechanics, $E_k + E_p = \mathrm{const}$.
• In circuits, $E = VIt = I^2Rt$ is dissipated as heat.
• In waves, intensity (energy per unit area per unit time) decreases with distance.
• In nuclear physics, $E = \Delta mc^2$ converts mass to energy.
• In thermodynamics, the first law $\Delta U = Q + W$ is a statement of energy conservation.

Problem Set

Common Pitfalls

1. Misidentifying the system boundary when applying conservation laws. Define what is included before writing equations.

2. Using the wrong equation from the data sheet. Take time to read the full equation, including conditions and variable definitions.

3. Forgetting to include units in final answers, especially when working with derived units like $\text{N}\,\text{kg}^{-1}\,\text{m}^2$.

4. Confusing scalar and vector quantities. Always check whether direction matters for the quantity in question.

Summary

The key principles covered in this topic are linked in the sub-pages above. Focus on understanding the definitions, applying the formulas or frameworks, and evaluating strengths and limitations of each approach.

Worked Examples

Worked examples demonstrating the application of key concepts are covered in the detailed sub-pages linked above.

Cross-References

• Dynamics — The work-energy theorem is derived from Newton’s second law, connecting force, displacement, and kinetic energy.
• Momentum — Impulse relates force applied over time to change in momentum, complementary to the work-energy approach.
• Oscillations — Energy conservation in SHM uses elastic potential energy from springs, which obeys Hooke’s law.
• Gravitational Fields — Gravitational potential energy near the surface is the small-height approximation of the full gravitational potential.