recall and understand that the efficiency of a system is the ratio of useful energy output from the system to the total energy input

Energy, Work, Power & Efficiency (Cambridge AS‑Level – Topic 5)

Law of conservation of energy: In a closed system the total energy remains constant.

\[

\sum E{\text{initial}}=\sum E{\text{final}}

\]

1. Work

• Definition: Work is the transfer of energy when a force causes a displacement. It is a scalar quantity.
• Constant force \(\mathbf{F}\) making an angle \(\theta\) with the displacement \(\mathbf{s}\):

  \[

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

  \]

• Sign convention

  • Positive work – component of the force is in the direction of motion (energy added to the system).
  • Negative work – component opposes the motion (energy removed from the system).

• Variable force: When the force varies with displacement the work equals the area under a force‑vs‑displacement graph:

  \[

  W = \int{si}^{s_f}F(s)\,\mathrm{d}s

  \]

• Work‑energy theorem:

  \[

  W{\text{net}} = \Delta K = Kf-K_i

  \]

  The net work done on an object equals the change in its kinetic energy.

Worked example – Variable force (spring)

A spring with force constant \(k = 200\ \text{N m}^{-1}\) is compressed by \(x = 0.15\ \text{m}\). Find the work done on the spring.

\[

W = \frac12 kx^{2}= \frac12 (200)(0.15)^{2}=2.25\ \text{J}

\]

2. Kinetic Energy (\(K\))

• Derived from the work‑energy theorem for a particle of mass \(m\) accelerated from rest to speed \(v\):

  \[

  K = \frac12 mv^{2}

  \]

• Scalar quantity – direction is contained in the velocity vector, not in the energy.
• Relation to linear momentum:

  \[

  K = \frac{p^{2}}{2m},\qquad p = mv

  \]

Worked example – Projectile

A 0.8 kg ball is thrown horizontally at \(12\ \text{m s}^{-1}\). Its kinetic energy at launch is

\[

K = \frac12 (0.8)(12)^{2}=57.6\ \text{J}

\]

3. Potential Energy

3.1 Gravitational potential energy (\(U_g\))

• Energy stored because of an object’s position in a uniform gravitational field.
• Reference level (zero of potential) may be chosen arbitrarily; only changes \(\Delta U_g\) are physically meaningful.
• For a vertical displacement \(h\):

  \[

  U_g = mgh

  \]

• Change when an object falls a height \(h\):

  \[

  \Delta U_g = -mgh

  \]

  (loss of potential energy appears as kinetic energy if friction is negligible).

3.2 Elastic (spring) potential energy (\(U_s\))

• For an ideal spring obeying Hooke’s law \(F = kx\):

  \[

  U_s = \frac12 kx^{2}

  \]

• The energy is stored in the deformation of the spring and is released when the spring returns to its natural length.

Worked example – Roller coaster (gravitational PE)

A 500 kg coaster car is at the top of a 30 m hill. Its gravitational potential energy relative to the base is

\[

U_g = (500)(9.81)(30)=1.47\times10^{5}\ \text{J}

\]

4. Mechanical Energy & Energy Transfers

• Total mechanical energy: \(E{\text{mech}} = K + Ug + U_s\).
• In the absence of non‑conservative forces (friction, air resistance) the total mechanical energy is conserved:

  \[

  \Delta K + \Delta U = 0

  \]

• When non‑conservative forces act, the work they do appears as a loss (usually as heat, sound, etc.).

5. Power

• Definition: Power is the rate at which energy is transferred or transformed.

  \[

  P = \frac{E}{t}

  \]

• Mechanical power (force acting through a displacement):

  \[

  P = \frac{W}{t}=Fv\cos\theta

  \]

• Electrical power:

  \[

  P = IV = I^{2}R = \frac{V^{2}}{R}

  \]

• Average AC power (rms values):

  \[

  \overline{P}=V{\text{rms}}I{\text{rms}}\cos\phi

  \]

  where \(\phi\) is the phase angle between voltage and current.

Worked example – Motor

A motor delivers a constant torque \(\tau = 2.5\ \text{N m}\) and rotates at \(\omega = 1500\ \text{rad s}^{-1}\). Mechanical power output:

\[

P = \tau\omega = (2.5)(1500)=3.75\times10^{3}\ \text{W}

\]

6. Efficiency of a System

• Definition: Efficiency (\(\eta\)) is the ratio of useful energy (or power) output to the total energy (or power) input.

  \[

  \eta = \frac{E{\text{useful}}}{E{\text{input}}}

  \qquad\text{or}\qquad

  \eta = \frac{P{\text{useful}}}{P{\text{input}}}

  \]

• Often expressed as a percentage:

  \[

  \eta_{\%}= \eta\times100\%

  \]

• For an ideal loss‑free device \(\eta = 1\) (100 %). Real devices always have \(\eta < 1\) because of unavoidable losses (heat, sound, friction, etc.).

Typical efficiencies

Step‑by‑step calculation of efficiency

1. Determine the total energy (or power) supplied: \(E{\text{input}}\) (or \(P{\text{input}}\)).
2. Identify the useful energy (or power) delivered to the intended output: \(E{\text{useful}}\) (or \(P{\text{useful}}\)).
3. Apply \(\displaystyle \eta = \frac{E{\text{useful}}}{E{\text{input}}}\) (or the power form).
4. Convert to a percentage if required.

Worked example – Electric heater

A 1500 W heater runs for 10 min. 80 % of the electrical energy heats the water. Find the efficiency.

• Energy input: \(E_{\text{in}} = Pt = 1500\ \text{W}\times 600\ \text{s}=9.0\times10^{5}\ \text{J}\).
• Useful energy: \(E{\text{useful}} = 0.80E{\text{in}} = 7.2\times10^{5}\ \text{J}\).
• \[

  \eta = \frac{7.2\times10^{5}}{9.0\times10^{5}} = 0.80 \; \Rightarrow\; 80\%

  \]

Common misconceptions

• “Efficiency can be 100 %.” Only an ideal, loss‑free system could achieve this; real devices always have some loss.
• “Output energy larger than input means a perpetual‑motion machine.” This would violate energy conservation; any apparent excess must come from an unaccounted source.
• “Efficiency is a unit of power.” Efficiency is dimensionless; power is measured in watts (W).

7. Practice Questions (AO1–AO2)

1. A car engine receives \(2.5\ \text{MJ}\) of chemical energy from fuel and delivers \(0.6\ \text{MJ}\) as kinetic energy to the car. Calculate the engine’s efficiency as a percentage.
2. A solar panel of area \(1.5\ \text{m}^{2}\) receives solar irradiance of \(800\ \text{W m}^{-2}\). If it produces \(120\ \text{W}\) of electrical power, determine its efficiency.
3. A pendulum of length \(0.45\ \text{m}\) is released from rest at its highest point. The theoretical gravitational potential energy is \(mgh\). The measured kinetic energy at the lowest point is only \(85\%\) of \(mgh\). What is the experimental efficiency of the energy conversion?
4. A constant force of \(30\ \text{N}\) pulls a crate 5 m up an incline that makes a \(30^{\circ}\) angle with the horizontal. Calculate (a) the work done on the crate, (b) the increase in its kinetic energy if it started from rest, assuming no friction.
5. A \(2\ \text{kg}\) block slides down a frictionless ramp from a height of \(3\ \text{m}\). Determine (a) its speed at the bottom, (b) the time taken if the ramp length is \(5\ \text{m}\), and (c) the average power delivered by gravity during the descent.

8. Suggested Diagram