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For a rigid body: sum of external forces = m a_cm; sum of moments about the centre of mass = I alpha (planar motion). The kinetic energy is m v_cm^2/2 + I omega^2/2. Work of a force is the integral of F . dr and the power is P = F . v (or M omega for a moment). The work-energy theorem equates total work to the change of kinetic energy; with conservative forces the mechanical energy is conserved. D'Alembert's principle: adding inertia forces converts a dynamics problem into a static one, used in the analysis of machine mechanisms and in biomechanics (inverse dynamics).

Model: mass m, spring stiffness k, viscous damper c, external force F(t): m x'' + c x' + k x = F(t). Natural frequency omega_n = sqrt(k/m); damping ratio zeta = c/(2 sqrt(k m)). Free vibration: undamped (zeta = 0) gives sustained sinusoidal oscillation; underdamped (zeta below 1) gives decaying oscillation at the damped frequency omega_d = omega_n sqrt(1 - zeta squared); critical damping (zeta = 1) returns to rest fastest without oscillating; overdamped (above 1) returns slowly. Forced vibration by a harmonic force: the steady-state amplitude depends on the frequency ratio r = omega/omega_n; near r = 1 resonance occurs, with a large amplitude limited only by damping (quality factor about 1/(2 zeta)). Vibration isolation requires r above sqrt(2). Biomedical relevance: whole-body vibration, mechanical models of tissue and shock absorbers in prostheses.

1. Write the equation of motion of a mass-spring-damper. 2. Give the natural frequency and the damping ratio. 3. What happens at resonance? 4. What frequency ratio is needed for vibration isolation?

1. m x'' + c x' + k x = F(t). 2. omega_n = sqrt(k/m); zeta = c/(2 sqrt(k m)). 3. The response amplitude is maximal, limited only by damping. 4. A ratio above sqrt(2).

Linear momentum p = m v. In an isolated collision the total momentum is conserved. The coefficient of restitution e = (relative speed after)/(relative speed before) along the line of impact: e = 1 for a perfectly elastic collision (kinetic energy conserved), e = 0 for a perfectly plastic collision (bodies stick together, maximum energy loss), and 0 < e < 1 for real collisions. For a head-on collision of masses m1 and m2 with initial velocities u1 and u2, the final velocities follow from momentum conservation and the restitution relation. Impulse J = integral of F dt = change of momentum, so a longer collision time reduces the peak force (the principle of crumple zones, helmets and padding). Application: injury biomechanics.

1. What is conserved in all collisions? 2. Define the coefficient of restitution. 3. Why does padding reduce injury?

1. Total linear momentum (of an isolated system). 2. The ratio of relative speed of separation to relative speed of approach along the line of impact. 3. It lengthens the collision time, so for the same impulse the peak force is lower.

Power P = torque x angular velocity (rotation) or force x velocity (translation). Efficiency eta = P_out/P_in = 1 minus losses/P_in; in a chain of machines the efficiencies multiply. Gears: a pair of gears with radii (or numbers of teeth) z1 and z2 has transmission ratio tau = omega2/omega1 = z1/z2 (speed inversely proportional to the size), and torque ratio T2/T1 = z2/z1 x eta. A reduction gear lowers speed and multiplies torque. Types: spur, helical, bevel and worm gears. Other transmissions: belts and chains. Brakes and clutches use friction (Coulomb law: friction force = mu N) to dissipate kinetic energy as heat, and their design must limit temperature rise.

Example: the total hip prosthesis with a femoral stem, a head (metal or ceramic) and an acetabular cup (metal back with a polyethylene or ceramic liner). Loads reach several times body weight during walking. Design issues: stress shielding (a stiff stem carries load away from bone, causing resorption), wear of the bearing surfaces and the resulting particles, fixation (cemented or press-fit with osseointegration), and fatigue of the components. Materials: titanium alloys, cobalt-chromium, stainless steel, ultra-high-molecular-weight polyethylene, ceramics.

1. Define the efficiency of a machine and of a chain of machines. 2. Give the transmission ratio of a gear pair. 3. What is stress shielding?

1. Ratio of output to input power; for a chain the efficiencies multiply. 2. omega2/omega1 = z1/z2. 3. Bone resorption caused by a stiff implant that carries load instead of the bone.