define and use pressure

Cambridge International AS & A‑Level Physics (9702) – Concise Syllabus Notes

0. How to Use These Notes

• Each main heading corresponds to a syllabus topic (AS 1‑11, A‑level 12‑25).
• Key definitions and formulas are highlighted in bold or boxed equations.
• AO (Assessment Objective) tags indicate which objectives are addressed:

  • AO1 – Knowledge & understanding
  • AO2 – Application of knowledge
  • AO3 – Experimental skills & analysis

• Worked examples illustrate typical exam questions; practice questions are provided at the end of each section.
• Uncertainty propagation and error analysis are included where relevant (AO3).

1. Quantities, Units, Vectors & Mathematical Tools (AO1, AO2)

1.1 SI Units & Dimensional Check

1.2 Vector Operations (AO1)

• Resultant: \(\mathbf{R}= \sum \mathbf{F}_i\)
• Dot product: \(\mathbf{A}\!\cdot\!\mathbf{B}=AB\cos\theta\) (scalar)
• Cross product: \(\mathbf{A}\!\times\!\mathbf{B}=AB\sin\theta\,\hat{\mathbf{n}}\) (vector, \(\hat{\mathbf{n}}\) ⟂ plane of \(\mathbf{A},\mathbf{B}\))
• Resolution into components: \(\mathbf{F}=Fx\hat{\mathbf{i}}+Fy\hat{\mathbf{j}}+F_z\hat{\mathbf{k}}\)

1.3 Calculus Refresher (AO2)

• Derivative: \(\displaystyle \frac{dy}{dx}\) – rate of change (used for acceleration, pressure gradient).
• Integral: \(\displaystyle \int f(x)\,dx\) – area under curve (used for work, hydrostatic pressure).
• Typical forms:

  • \(\displaystyle \int v\,dt = s\) (displacement)
  • \(\displaystyle \int F\,dx = W\) (work)
  • \(\displaystyle \int \rho g\,dh = \rho g h\) (hydrostatic pressure)

2. Kinematics (AO1, AO2)

• Equations of motion for constant acceleration:

  \[

  v = u + at,\qquad

  s = ut + \tfrac12 at^2,\qquad

  v^2 = u^2 + 2as

  \]

• Graphical interpretation (s‑t, v‑t, a‑t) – slope = velocity, area = displacement.

Practice

Calculate the stopping distance of a car travelling at 20 m s⁻¹ that brakes uniformly to rest in 3 s.

3. Dynamics (Newton’s Laws) (AO1, AO2)

• First law: An object remains at rest or in uniform motion unless acted on by a net external force.
• Second law: \(\displaystyle \mathbf{F}_{\text{net}} = m\mathbf{a}\) (vector form).
• Third law: For every action there is an equal and opposite reaction.

Example

A 2.0 kg block is pulled horizontally with a 10 N force; kinetic friction is 2 N. Find the acceleration.

4. Forces – Density & Pressure (AO1, AO2, AO3)

4.1 Definition of Pressure

Pressure is the magnitude of the component of a force that acts normal (perpendicular) to a surface, divided by the area over which it acts.

\[

\boxed{p = \frac{F_{\perp}}{A} = \frac{\mathbf{F}\!\cdot\!\hat{\mathbf{n}}}{A}}

\]

• \(\mathbf{F}\) – total force vector acting on the surface.
• \(\hat{\mathbf{n}}\) – outward unit normal to the surface.
• Only the normal component contributes to pressure; tangential components produce friction, not pressure.

4.2 Units & Dimensional Check (AO1)

Always verify that the final unit reduces to pascals (Pa).

4.3 Hydrostatic Pressure (AO1, AO2)

In a fluid at rest the pressure increases with depth because the weight of the fluid above exerts a force on the layers below.

Derivation (constant density)

1. Consider a horizontal fluid slab of thickness \(\Delta h\), area \(A\), density \(\rho\).
2. Weight of the slab: \(\Delta W = \rho g A \Delta h\).
3. Force balance on the slab:

   \[

   p(h+\Delta h)A - p(h)A = \Delta W

   \]

4. Divide by \(A\) and let \(\Delta h\to0\):

   \[

   \frac{dp}{dh}= \rho g

   \]

5. Integrate from the free surface (\(p_0\)) to depth \(h\):

   \[

   \boxed{p = p_0 + \rho g h}

   \]

If the fluid density varies with height (e.g. atmosphere), integrate \(\displaystyle p = p0 + \int0^{h}\rho(h')g\,dh'\).

Worked Example – Pressure at 5 m depth in water

1. Given: \(\rho{\text{water}} = 1.0\times10^{3}\,\text{kg m}^{-3}\), \(g = 9.81\,\text{m s}^{-2}\), \(h = 5.0\,\text{m}\), \(p0 = 1.01\times10^{5}\,\text{Pa}\).
2. \(\Delta p = \rho g h = (1.0\times10^{3})(9.81)(5.0)=4.91\times10^{4}\,\text{Pa}\).
3. Total pressure: \(p = p_0 + \Delta p = 1.01\times10^{5}+4.91\times10^{4}=1.50\times10^{5}\,\text{Pa}=1.50\;\text{bar}\).

4.4 Buoyancy – Archimedes’ Principle (AO1, AO2)

The upward force on a body wholly or partially immersed in a fluid equals the weight of the fluid displaced.

\[

\boxed{F{\text{up}} = \rho{\text{fluid}}\,g\,V_{\text{disp}}}

\]

• If \(F_{\text{up}} >\) weight of the object → it rises.
• If \(F_{\text{up}} <\) weight → it sinks.
• If equal → neutral buoyancy.

Worked Example – Will a wooden block float?

1. Block dimensions: \(0.10\times0.10\times0.20\;\text{m}\) → \(V = 2.0\times10^{-3}\,\text{m}^3\).
2. \(\rho{\text{wood}} = 600\,\text{kg m}^{-3}\), \(\rho{\text{water}} = 1000\,\text{kg m}^{-3}\).
3. Weight: \(W = \rho_{\text{wood}} g V = 600(9.81)(2.0\times10^{-3}) = 11.8\;\text{N}\).
4. Maximum upthrust (fully submerged): \(F{\text{up}} = \rho{\text{water}} g V = 1000(9.81)(2.0\times10^{-3}) = 19.6\;\text{N}\).
5. Since \(F{\text{up}} > W\), the block floats; only a fraction \(\dfrac{W}{\rho{\text{water}} g V}=0.60\) of its volume is submerged.

4.5 Pascal’s Principle – Transmission of Pressure (AO1, AO2)

In a fluid at rest, any change in pressure applied at one point is transmitted unchanged throughout the fluid.

\[

p1 = p2 \quad\Longrightarrow\quad \frac{F1}{A1}= \frac{F2}{A2}

\]

• Useful for hydraulic lifts, brakes, and pressure‑sensor calibrations.

Worked Example – Hydraulic piston

1. Small piston: \(A1 = 2.0\times10^{-4}\,\text{m}^2\), applied force \(F1 = 50\;\text{N}\).
2. Large piston: \(A_2 = 5.0\times10^{-3}\,\text{m}^2\).
3. Transmitted pressure: \(p = F1/A1 = 2.5\times10^{5}\,\text{Pa}\).
4. Force on large piston: \(F2 = pA2 = 1.25\times10^{3}\,\text{N}\).

4.6 Contact Pressure on Solids (AO1, AO2)

When a solid surface bears a load, the average pressure is

\[

\boxed{p = \frac{F}{A}}

\]

where \(F\) is the normal force and \(A\) the contact area. Real contacts are often non‑uniform; the maximum pressure may be higher than the average.

Uncertainty Propagation (AO3)

\[

\boxed{\frac{\Delta p}{p}= \sqrt{\left(\frac{\Delta F}{F}\right)^2+\left(\frac{\Delta A}{A}\right)^2}}

\]

Example: \(F = 100.0\pm0.2\;\text{N}\), \(A = 0.0200\pm0.0001\;\text{m}^2\).

• \(p = 5.00\times10^{3}\;\text{Pa}\).
• \(\displaystyle \frac{\Delta p}{p}= \sqrt{(0.2/100)^2+(0.0001/0.0200)^2}=2.2\times10^{-3}\).
• \(\Delta p = 5.00\times10^{3}\times2.2\times10^{-3}=11\;\text{Pa}\).
• Result: \(p = (5.00\pm0.01)\times10^{3}\;\text{Pa}\).

4.7 Common Misconceptions (AO1)

• Pressure is a force. It is force per unit area; the same force can give different pressures on different surfaces.
• All components of a force contribute to pressure. Only the component normal to the surface does.
• Pressure is always uniform. In fluids it varies with depth; in solids it can vary with contact geometry.
• Buoyancy equals the weight of the object. Buoyancy equals the weight of the displaced fluid.

4.8 Summary Checklist (AO1, AO2)

• Definition: \(p = \dfrac{F_{\perp}}{A}\) (Pa = N m⁻²).
• Hydrostatic pressure: \(p = p_0 + \rho g h\).
• Variable density: \(p = p0 + \int0^{h}\rho(h')g\,dh'\).
• Buoyancy: \(F{\text{up}} = \rho{\text{fluid}} g V_{\text{disp}}\).
• Pascal’s principle: pressure transmitted unchanged in a static fluid.
• Always resolve forces into normal and tangential components before applying pressure formulas.
• Check units and significant figures; propagate uncertainties when experimental data are used.
• Identify whether the problem assumes uniform pressure (e.g., contact pressure) or depth‑dependent pressure (hydrostatic).

5. Work, Energy & Power (AO1, AO2)

• Work: \(W = \mathbf{F}\!\cdot\!\mathbf{s} = Fs\cos\theta\) (J).
• Kinetic Energy: \(K = \tfrac12 mv^2\).
• Potential Energy (gravity): \(U = mgh\).
• Conservation of Mechanical Energy (no non‑conservative forces): \(Ki+Ui = Kf+Uf\).
• Power: \(P = \dfrac{dW}{dt} = Fv = \dfrac{E}{t}\).

Example – Lifting a 5 kg mass 2 m at constant speed

Force required = weight = \(mg = 5\times9.81 = 49.1\;\text{N}\).

Work = \(F s = 49.1\times2 = 98.2\;\text{J}\).

If the lift takes 4 s, power = \(98.2/4 = 24.5\;\text{W}\).

6. Deformation of Solids (AO1, AO2)

• Stress: \(\sigma = \dfrac{F}{A}\) (Pa). Tensile or compressive.
• Strain: \(\varepsilon = \dfrac{\Delta L}{L}\) (dimensionless).
• Young’s Modulus: \(E = \dfrac{\sigma}{\varepsilon}\) (Pa).
• Shear stress \(\tau = F_{\text{parallel}}/A\); shear strain \(\gamma = \Delta x / L\); shear modulus \(G = \tau/\gamma\).
• Bulk modulus \(K = -\dfrac{\Delta p}{\Delta V/V}\) – relates pressure change to volume change (useful for liquids).

Worked Example – Extension of a steel wire

Wire: \(L = 1.0\;\text{m}\), \(A = 1.0\times10^{-6}\;\text{m}^2\), \(E = 2.0\times10^{11}\;\text{Pa}\).

Force \(F = 500\;\text{N}\).

Stress \(\sigma = F/A = 5.0\times10^{8}\;\text{Pa}\).

Strain \(\varepsilon = \sigma/E = 2.5\times10^{-3}\).

Extension \(\Delta L = \varepsilon L = 2.5\;\text{mm}\).

7. Waves (AO1, AO2)

• Wave speed: \(v = f\lambda\).
• Longitudinal vs transverse.
• Reflection, refraction, diffraction.
• Standing waves – nodes and antinodes; resonance condition \(L = n\lambda/2\) (fixed‑fixed) or \(L = (2n-1)\lambda/4\) (fixed‑free).

Example – Fundamental frequency of a 1.2 m string under tension 80 N (mass per unit length \(\mu = 0.005\;\text{kg m}^{-1}\))

Wave speed \(v = \sqrt{T/\mu} = \sqrt{80/0.005}=126.5\;\text{m s}^{-1}\).

Fundamental frequency \(f_1 = v/(2L) = 126.5/(2\times1.2)=52.7\;\text{Hz}\).

8. Superposition & Interference (AO1, AO2)

• Resultant displacement = algebraic sum of individual displacements (linear superposition).
• Constructive interference: \(\Delta\phi = 2n\pi\) → amplitude doubles.
• Destructive interference: \(\Delta\phi = (2n+1)\pi\) → cancellation.
• Path‑difference condition for bright fringes (double‑slit): \(d\sin\theta = n\lambda\).

9. Electricity – Charge & Electric Fields (AO1, AO2)

• Charge: \(q\) (C). Conservation of charge.
• Electric field: \(\mathbf{E} = \dfrac{\mathbf{F}}{q}\) (N C⁻¹). For a point charge: \(E = k\frac{Q}{r^2}\).
• Potential difference: \(V = W/q\) (V). Relation \(E = -\nabla V\).

Example – Field at 5 cm from a 2 µC point charge

\(E = kQ/r^2 = (9.0\times10^9)(2\times10^{-6})/(0.05)^2 = 7.2\times10^{6}\;\text{N C}^{-1}\).

10. DC Circuits (AO1, AO2, AO3)

• Ohm’s law: \(V = IR\).
• Series: \(R{\text{eq}} = \sum Ri\), \(I\) same, \(V\) divides.
• Parallel: \(\dfrac{1}{R{\text{eq}}}= \sum \dfrac{1}{Ri}\), \(V\) same, \(I\) divides.
• Power: \(P = VI = I^2R = V^2/R\).

Mini‑Experiment (AO3) – Verifying Ohm’s law

1. Set up a circuit with a variable resistor, ammeter and voltmeter.
2. Record \(V\) and \(I\) for at least five resistance settings.
3. Plot \(V\) against \(I\); the gradient gives the resistance.
4. Calculate uncertainties in \(V\) and \(I\) (instrument least count) and propagate to obtain \(\Delta R\).

11. Particle Physics (AO1, AO2)

• Fundamental particles: quarks (u, d, s, c, b, t) and leptons (e, μ, τ and their neutrinos).
• Forces: strong, electromagnetic, weak, gravitational.
• Feynman diagrams – represent interactions.
• Key processes: β‑decay, positron emission, pair production.

Example – Energy released in β⁻ decay of \(\mathrm{^{14}C}\)

Q‑value ≈ 156 keV; convert to joules: \(E = 156\times10^{3}\times1.602\times10^{-19}=2.5\times10^{-14}\;\text{J}\).

12–25. A‑Level Extension Topics (Brief Overview) (AO1, AO2)

12. Motion in a Circle

• Centrepital acceleration: \(a_c = v^2/r = \omega^2 r\).
• Centripetal force: \(Fc = m ac\).

13. Gravitation

• Newton’s law: \(F = G\frac{m1 m2}{r^2}\).
• Gravitational field: \(g = GM/r^2\).
• Orbital speed: \(v = \sqrt{GM/r}\).

14. Temperature, Ideal Gases & Gas Laws

• Ideal‑gas equation: \(pV = nRT\).
• Combined gas law: \(\dfrac{p1V1}{T1}= \dfrac{p2V2}{T2}\).
• Mean kinetic energy: \(\tfrac32 k_B T\) per molecule.

15. Thermodynamics

• First law: \(\Delta U = Q - W\).
• Specific heat capacity: \(Q = mc\Delta T\).
• Latent heat: \(Q = mL\).
• Efficiency of a heat engine: \(\eta = 1 - \dfrac{TC}{TH}\).

16. Simple Harmonic Motion (SHM)

• Displacement: \(x = A\cos(\omega t + \phi)\).
• Acceleration: \(a = -\omega^2 x\).
• Energy: \(E = \tfrac12 kA^2 = \tfrac12 m\omega^2 A^2\).

17. Electric Fields & Potential

• Electric field between parallel plates: \(E = V/d\).
• Potential energy of a charge: \(U = qV\).

18. Magnetic Fields

• Force on a moving charge: \(\mathbf{F}=q\mathbf{v}\times\mathbf{B}\).
• Force on a current‑carrying conductor: \(F = BIL\sin\theta\).
• Biot–Savart law (qualitative).

19. Electromagnetic Induction

• Faraday’s law: \(\mathcal{E}= -\dfrac{d\Phi}{dt}\).
• Lenz’s law – direction of induced emf opposes change.
• Induced emf in a moving conductor: \(\mathcal{E}=Blv\) (when \(B\perp v\) and conductor length \(l\)).

20. AC Circuits

• RMS values: \(I{\text{rms}} = I{\max}/\sqrt{2}\), \(V{\text{rms}} = V{\max}/\sqrt{2}\).
• Impedance: \(Z = \sqrt{R^2+(XL-XC)^2}\).
• Power factor: \(\cos\phi = R/Z\).

21. Quantum & Nuclear Physics

• Photoelectric effect: \(hf = \phi + K_{\max}\).
• De Broglie wavelength: \(\lambda = h/p\).
• Radioactive decay law: \(N = N_0 e^{-\lambda t}\).
• Half‑life: \(t_{1/2}= \ln 2 / \lambda\).

22. Medical Physics

• X‑ray production: Bremsstrahlung and characteristic radiation.
• Diagnostic imaging – importance of contrast and dose.
• Ultrasound – reflection at tissue boundaries.

23. Astronomy

• Hubble’s law: \(v = H_0 d\).
• Black‑body radiation – Wien’s law and Stefan‑Boltzmann law.
• Kepler’s laws of planetary motion.

24. Practical Skills & Experimental Techniques (AO3)

• Planning: identify variables, choose appropriate apparatus, sketch circuit/diagram.
• Data collection: use of digital timers, force sensors, pressure transducers, and data loggers.
• Error analysis: random vs systematic errors, calculation of uncertainties, chi‑square goodness of fit.
• Typical mini‑investigations:

  • Measuring hydrostatic pressure with a U‑tube manometer.
  • Determining Young’s modulus of a metal wire using a static load.
  • Verifying the inverse‑square law for electric fields.
  • Investigating the relationship between frequency and period in a simple pendulum.

25. Summary of Core Formulas (Quick Reference) (AO1)

Practice Questions (Mixed AO1‑AO3)

1. Pressure & Buoyancy (AO2) – A solid sphere of radius 0.05 m and density 800 kg m⁻³ is gently released in a tank of oil (ρ = 900 kg m⁻³). Determine the fraction of the sphere’s volume that is submerged at equilibrium.
2. Hydrostatic Pressure (AO2) – Calculate the gauge pressure at the bottom of a 12 m deep swimming pool filled with fresh water (