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SI base units used in mechanics: metre (length), kilogram (mass), second (time). Derived: newton (N = kg m/s^2) for force, joule (J = N m) for work and energy, watt (W = J/s) for power, pascal (Pa = N/m^2) for pressure. Always check that both sides of an equation have the same dimension. Angles are measured in degrees or radians (pi radians = 180 degrees). For a right triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Useful identity: sin squared plus cos squared equals 1.

A vector can be written by components: v = (vx, vy, vz), with magnitude the square root of the sum of squares of its components. Unit vectors i, j, k point along the axes. Operations: sum and difference (component by component, or the parallelogram rule); multiplication by a scalar; scalar (dot) product a . b = |a||b| cos(theta) (a number, zero for perpendicular vectors, gives the projection of one vector on another); vector (cross) product a x b (a vector perpendicular to both, with magnitude |a||b| sin(theta); used for the moment of a force r x F).

1. Convert 90 degrees to radians. 2. Express the newton in base units. 3. What is the dot product of two perpendicular vectors? 4. How is the moment of a force about a point defined?

1. pi/2 radians. 2. kg m s^-2. 3. Zero. 4. The cross product r x F, where r is the vector from the point to the point of application of the force.

Position vector r(t). Velocity v = dr/dt (tangent to the trajectory) and acceleration a = dv/dt. Straight-line motion: uniform (v constant); uniformly accelerated (a constant): v = v0 + a t and s = s0 + v0 t + a t^2/2, with v^2 = v0^2 + 2 a (s - s0). Projectile (ballistic) motion: constant gravity, horizontal motion uniform and vertical motion uniformly accelerated. From height h with speed v0 at angle alpha, the range follows by solving for the time when the vertical position reaches zero. The maximum range on level ground is at 45 degrees (without air resistance). Motion along a curve with path coordinate s(t): tangential acceleration d^2s/dt^2 (changes the speed) and centripetal (normal) acceleration v^2/R directed toward the centre of curvature (changes direction).

Angle theta(t); angular velocity omega = d theta/dt; angular acceleration alpha = d omega/dt. Linear speed v = omega R, tangential acceleration alpha R, centripetal acceleration omega^2 R = v^2/R. Uniform circular motion has constant omega, no tangential acceleration, and period T = 2 pi/omega.

A rigid body keeps the distances between its points constant. In the plane, the general motion is a translation plus a rotation, with three degrees of freedom (two for position and one for orientation). The velocity of two points A and B of the body: v_B = v_A + omega x (B - A). Instantaneous centre of rotation: at each instant the planar motion is a pure rotation about a point with zero velocity.

Constraints (joints) remove degrees of freedom: a revolute joint (hinge) leaves one rotation, a prismatic joint one translation. Grubler equation for planar mechanisms: DOF = 3 (n - 1) - 2 j1 - j2, where n is the number of links including the frame, j1 the number of one-degree-of-freedom joints and j2 of two-degree-of-freedom joints. Kutzbach's equation is its 3D version: DOF = 6 (n - 1) - sum of constraints. A four-bar linkage (n = 4, j1 = 4) has DOF = 1. Biomechanical use: joints of the human body are modelled as ideal joints.

1. Write the equations of uniformly accelerated motion. 2. Which acceleration changes the speed and which the direction in curvilinear motion? 3. How many degrees of freedom has a rigid body in the plane? 4. Compute the mobility of a four-bar linkage. 5. What is the relation between v and omega in circular motion?

1. v = v0 + a t; s = s0 + v0 t + a t^2 / 2. 2. Tangential acceleration changes the speed; centripetal (normal) acceleration changes the direction. 3. Three. 4. DOF = 3 (4 - 1) - 2 x 4 = 1. 5. v = omega R.

A force is a vector applied at a point. The moment of a force about a point O is M = r x F (units N m). A couple is two equal and opposite parallel forces, whose moment is the same about any point. Equilibrium conditions: the sum of forces is zero and the sum of moments about any point is zero. In the plane this gives three scalar equations (sum of Fx, sum of Fy, sum of moments), which can determine three unknowns. Procedure: choose the body, draw the free-body diagram with all external forces and reactions, choose the axes and the moment point (a point where unknown forces pass simplifies the algebra), write the equations and solve.

Common supports in the plane: a roller or smooth surface (one reaction force normal to the surface), a hinge (two force components), a fixed support or clamp (two force components and a moment), a cable (tension along the cable). Statically determinate problems have as many unknowns as equations; more unknowns make the problem statically indeterminate. Biomechanical use: muscle forces and joint reactions can be estimated by treating the limb as a lever in equilibrium (for example the biceps holding a load).

Dissipation of energy occurs by friction between surfaces, internal friction of materials and fluid resistance (aerodynamic drag). Coulomb friction: when two solids in contact are at rest relative to each other the friction force adjusts up to a maximum F_max = mu_s N, where N is the normal force and mu_s the static coefficient. When sliding, the force is F = mu_d N in the direction opposing the motion, with mu_d usually less than mu_s. Friction does not depend on the contact area in this model. Aerodynamic drag increases with the square of speed and depends on the shape and frontal area.

Weight W = m g, with g about 9.81 m/s^2, acting at the centre of mass (centre of gravity), whose position is the mass-weighted average of positions: r_cm = (sum of m_i r_i)/(sum of m_i). For a body composed of parts (as the human body divided into segments), this formula gives the whole-body centre of mass. A body is stable if the vertical through its centre of mass falls within its base of support.

1. State the conditions of equilibrium of a rigid body. 2. How many reaction components has a hinge in the plane? 3. What is the maximum static friction force? 4. Compute the centre of mass of two masses of 2 kg at x = 0 and 6 kg at x = 4 m. 5. What is a couple?

1. The sum of forces and the sum of moments about any point are both zero. 2. Two force components. 3. F_max = mu_s N. 4. (2 x 0 + 6 x 4)/(8) = 3 m. 5. Two equal, opposite, parallel forces with different lines of action, giving a pure moment.

1. A body stays at rest or in uniform straight motion unless a net force acts (inertial frames). 2. F = m a (in vector form, with F the resultant of all forces). 3. Action and reaction: forces between two bodies are equal and opposite. Method: isolate the body, draw the free-body diagram, choose axes, write F = m a for each component, add constraint or friction relations, solve. Linear momentum p = m v, with F = dp/dt.

Work of a force: W = integral of F . dr. Kinetic energy T = m v^2 / 2. The work-energy theorem: total work = change in kinetic energy. Gravitational potential energy m g h and elastic energy k x^2 / 2 are potential energies of conservative forces: with only conservative forces the total mechanical energy is constant. Friction dissipates energy. Power: P = dW/dt = F . v, in watts. Efficiency is the ratio of useful output power to input power.

Translation of the centre of mass: F_total = m a_cm. Rotation: M_total = I alpha in planar motion about a fixed axis or about the centre of mass, where I is the moment of inertia (I = sum of m r^2, or the integral of r^2 dm), the rotational analogue of mass. Kinetic energy of a rotating body: I omega^2 / 2; in general motion T = m v_cm^2 / 2 + I_cm omega^2 / 2. Parallel axis (Steiner) theorem: I = I_cm + m d^2. Biomechanical use: the same equations applied to body segments give joint forces and moments (inverse dynamics).

1. State Newton's second law in vector form. 2. State the work-energy theorem. 3. Write the rotational equation of motion of a rigid body. 4. What does the Steiner theorem give? 5. When is mechanical energy conserved?

1. F = m a. 2. The work of all forces equals the change in kinetic energy. 3. M = I alpha. 4. The moment of inertia about an axis at distance d from the centre of mass: I = I_cm + m d^2. 5. When only conservative forces do work.