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Torque of a force about a point is the force times the perpendicular distance from the point (tau = F d). For equilibrium the sum of forces and the sum of torques both vanish. Levers: first class (fulcrum between effort and load, neck holding the head), second class (load between fulcrum and effort, calf raising), third class (effort between fulcrum and load, most muscles such as the biceps). Muscles usually attach close to the joint, so the muscle force must be much larger than the load: mechanical advantage below 1. Example: a forearm holding a 5 kg mass 30 cm from the elbow with the biceps attached 4 cm from the elbow needs a biceps force of about 5 x 9.8 x 30/4 = 370 N, ignoring the weight of the forearm.

The centre of mass of the standing body lies in the pelvis, about 55 percent of height. Stability requires the vertical line through the centre of mass to fall inside the base of support. Posture changes (bending forward, carrying loads) shift the centre of mass and increase muscle effort. The spine bears large compressive forces when lifting with a bent back because of the long lever arm of the trunk weight.

1. Define torque. 2. Which lever class is most common in the body? 3. Why is the biceps force much larger than the load it carries? 4. What condition ensures standing stability?

1. Force times the perpendicular distance from the pivot. 2. Third class. 3. The muscle attaches close to the joint so its lever arm is short compared with that of the load. 4. The vertical line through the centre of mass falls within the base of support.

Pressure P = F/A (pascal). Hydrostatic pressure: P = P0 + rho g h. Blood pressure varies with height relative to the heart (about 0.77 mmHg per cm). Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished. Archimedes: buoyant force equals the weight of fluid displaced. Continuity equation: A v = constant (flow rate Q = A v), so speed increases where the vessel narrows. Bernoulli's equation for an ideal fluid: P + rho v^2/2 + rho g h = constant; pressure falls where speed rises (a stenosis). Blood pressure is often given in mmHg (1 mmHg = 133 Pa).

Viscosity eta measures internal friction (blood about 3-4 times that of water). Laminar flow in a tube: Poiseuille's law Q = pi r^4 (P1 - P2)/(8 eta L). Vascular resistance R = 8 eta L/(pi r^4), so halving the radius increases resistance sixteen-fold, which is why small changes of arteriolar radius control blood flow and why atherosclerotic narrowing is so significant. Reynolds number Re = rho v D/eta: laminar flow below about 2000, turbulent above (murmurs, Korotkoff sounds). Blood is non-Newtonian: viscosity depends on haematocrit and shear rate.

Surface tension gamma is the force per unit length at a liquid surface. For a sphere (bubble or alveolus) the Laplace law gives pressure P = 2 gamma/r (4 gamma/r for a soap bubble with two surfaces); smaller alveoli would collapse into larger ones, which pulmonary surfactant prevents by lowering surface tension more in smaller alveoli. For a cylindrical vessel: wall tension T = P r (Laplace), so a dilated vessel or ventricle (larger radius) has higher wall tension for the same pressure: this explains aneurysm rupture risk and the increased work of a dilated heart.

Temperature is measured in kelvin or degrees Celsius; heat is energy transferred (joule; 1 calorie is 4.18 J). Specific heat c: Q = m c delta T. The body's metabolic heat production of about 100 W at rest must be lost to maintain 37 degrees Celsius. First law: energy is conserved (delta U = Q - W). Second law: heat flows spontaneously from hot to cold; entropy of an isolated system does not decrease. Heat loss mechanisms: conduction (contact), convection (air movement), radiation (Stefan-Boltzmann law: proportional to the fourth power of absolute temperature; the largest loss at rest in air) and evaporation of sweat (the main mechanism during exercise or in hot environments).

1. State the continuity equation. 2. By what factor does resistance change if the radius halves? 3. What does Laplace's law say about an aneurysm? 4. Name the four heat loss mechanisms. 5. Why are alveoli stable despite different sizes?

1. A v is constant along a tube of incompressible flow. 2. It increases sixteen times. 3. The wall tension increases with radius for a given pressure, so a dilated segment is under more tension and more likely to rupture. 4. Conduction, convection, radiation, evaporation. 5. Surfactant lowers surface tension more in smaller alveoli.

A wave has amplitude, frequency f, wavelength lambda and speed v = f lambda (sound in air about 340 m/s, in soft tissue about 1540 m/s). The audible range is about 20 Hz to 20 kHz. Intensity I is power per unit area (W/m^2). The threshold of hearing is 10^-12 W/m^2. The intensity level in decibels: beta = 10 log10(I/I0); each 10 dB is a factor of ten in intensity. Conversation is about 60 dB, and above about 85 dB prolonged exposure damages hearing. The ear: outer ear, middle ear (ossicles match the impedance of air to the fluid of the cochlea) and cochlea.

Ultrasound frequencies above 20 kHz; diagnostic use 2-15 MHz. A transducer (piezoelectric crystal) emits pulses and detects echoes; the time delay gives depth (d = v t/2) and the echo strength depends on the acoustic impedance mismatch at interfaces (Z = density times speed). A gel removes air between probe and skin. Higher frequency gives better resolution but less penetration because attenuation increases with frequency. Doppler effect: the frequency shift of echoes from moving blood gives velocity. Safety: no ionising radiation; heating and cavitation at high intensities.

Reflection: angle of incidence equals angle of reflection. Refraction (Snell's law): n1 sin(theta1) = n2 sin(theta2), with n = c/v. Total internal reflection occurs beyond the critical angle when passing from higher to lower index (optical fibres). Thin lens equation: 1/f = 1/do + 1/di; power in dioptres is 1/f in metres. Converging (convex) lenses have positive power, diverging (concave) negative power. Magnification m = -di/do.

The cornea gives about two thirds of the eye's power (about 43 D) and the lens adds accommodation (up to about 15-20 D in youth, declining with age: presbyopia). Myopia (short-sightedness): the image falls in front of the retina, corrected with a diverging lens. Hyperopia: image behind the retina, corrected with a converging lens. Astigmatism: uneven corneal curvature, corrected with a cylindrical lens. Compound microscope: objective and eyepiece lenses; total magnification is the product. Resolution is limited by diffraction (about lambda divided by twice the numerical aperture). Aberrations: spherical, chromatic and astigmatism, reduced by lens design.

1. Define the decibel scale. 2. How is depth calculated in ultrasound imaging? 3. Why does higher ultrasound frequency give better resolution but less penetration? 4. State Snell's law. 5. Which lens corrects myopia?

1. Sound level beta = 10 log10(I/I0); each 10 dB is a tenfold change in intensity. 2. d = v t/2 from the echo delay t. 3. Shorter wavelength resolves finer detail but is attenuated more in tissue. 4. n1 sin(theta1) = n2 sin(theta2). 5. A diverging (concave) lens.

Production: electrons accelerated across a tube voltage (tens to hundreds of kV) strike a tungsten anode. Bremsstrahlung gives a continuous spectrum with maximum photon energy equal to the tube voltage in keV; characteristic X-rays add lines. Most energy becomes heat. Interaction with matter: photoelectric effect (dominant at low energy, strongly dependent on atomic number, gives bone contrast), Compton scattering (dominant at diagnostic to therapeutic energies, degrades images) and pair production at very high energy. Attenuation: I = I0 exp(-mu x), where mu is the linear attenuation coefficient. Bone attenuates more than soft tissue; contrast media (iodine, barium) increase contrast. Computed tomography reconstructs slices from many projections.

Absorbed dose: energy per unit mass (gray, Gy = J/kg). Equivalent dose weights for the radiation type and effective dose (sievert, Sv) weights for tissue radiosensitivity. Typical effective doses: chest X-ray about 0.1 mSv, CT abdomen about 10 mSv; natural background about 2-3 mSv per year. Radiation protection: justification, optimisation (ALARA: as low as reasonably achievable) and dose limits. Measures: time, distance (inverse square law) and shielding (lead).

Nuclei with spin behave like small magnets. In a static field B0 they align and precess at the Larmor frequency f = gamma B0/(2 pi) (42.6 MHz per tesla for hydrogen). A radiofrequency pulse at this frequency tips the net magnetisation, which then induces a signal in a receiver coil and decays (free induction decay). Relaxation: T1 (recovery along B0) and T2 (loss of transverse coherence) differ between fat, water and pathological tissue, giving contrast. Position is encoded by magnetic field gradients. No ionising radiation. Contraindications: pacemakers not rated for MRI and ferromagnetic implants.

1. What is the maximum photon energy of an X-ray tube operated at 100 kV? 2. Which interaction produces bone contrast? 3. Write the attenuation law. 4. What is ALARA? 5. What determines the Larmor frequency?

1. 100 keV. 2. The photoelectric effect (strongly dependent on atomic number). 3. I = I0 exp(-mu x). 4. As low as reasonably achievable dose consistent with the diagnostic aim. 5. The gyromagnetic ratio of the nucleus and the magnetic field strength.