Summary
Highlights
In uniform rectilinear motion, the body moves along a straight line with constant velocity. The equation of motion is x = x0 + v(t - t0). The graph of position (x) versus time (t) is a straight line, and its slope (tangent of the angle) represents the velocity.
For uniformly varied rectilinear motion, acceleration is constant. The velocity equation is v = v0 + a(t - t0). The graph of velocity (v) versus time (t) is a straight line, and its slope represents the acceleration. The area under the velocity-time graph gives the displacement. The position equation is x = x0 + v0(t - t0) + (a/2)(t - t0)^2. The Galilei equation relates final velocity, initial velocity, acceleration, and displacement: v^2 = v0^2 + 2a(x - x0).
Dynamics studies the causes of motion. Newton's laws are fundamental: (1) Principle of Inertia (a body at rest or in uniform motion stays that way unless acted upon by a force); (2) Fundamental Principle (F = ma, force causes acceleration); (3) Principle of Action and Reaction (for every action, there is an equal and opposite reaction, F_action = -F_reaction); and the Principle of Superposition of forces.
Weight (G = mg) is the force of gravity, directed vertically downwards. The normal force (N) is the reaction force from a surface, perpendicular to the surface. On a horizontal surface, N = mg. On an inclined plane, weight is decomposed into tangential (Gt = mg sinα) and normal (Gn = mg cosα) components, and N = Gn.
Friction force (Ff = μN) acts parallel to the surface, opposing motion. The coefficient of friction (μ) is dimensionless and between 0 and 1. Tension force (T) arises in connecting wires or ropes and acts along the wire, applying equal and opposite forces to connected bodies.
Elastic force (Fe = -kΔl) arises in springs due to deformation (elongation or compression Δl), directed opposite to the deformation. The constant k is the spring constant. Hooke's Law states that the relative elongation (Δl/l0) is proportional to the normal stress (F/S), with the proportionality constant being the inverse of the Young's modulus (E): Δl/l0 = (1/E)(F/S).
Mechanical work (L = Fd cosα) is the scalar product of force and displacement. Work is positive if the angle α between force and displacement is acute (active force), negative if obtuse (resistive force), and zero if perpendicular. Mechanical power (P = L/Δt = Fv) is the rate at which work is done. Mechanical energy is the sum of kinetic energy (Ec = mv^2/2) and potential energy (Ep), which can be gravitational or elastic.
Momentum (p = mv) is a vector quantity with the same direction as velocity. Three important theorems are: (1) Work-Energy Theorem (ΔEc = L_total); (2) Theorem of Mechanical Energy Variation (ΔEm = L_resistive), which leads to the law of conservation of mechanical energy if L_resistive = 0; and (3) Impulse-Momentum Theorem (Δp = FΔt), leading to the law of conservation of momentum if F = 0.
Kinematics studies the motion of bodies without considering the causes. It starts by defining the position of a body relative to an inertial reference system. The position vector can be described by its magnitude and angles, or by its projections on the coordinate axes (Rx, Ry, Rz), using unit vectors (i, j, k).
Displacement (Δr) is the change in position vector from an initial to a final point. The magnitude of displacement can be calculated using a generalized Pythagorean theorem or the law of cosines. The law of motion describes how the position vector changes over time (R = f(t)).
Average velocity is the ratio of displacement to time interval (Δr/Δt). Instantaneous velocity is the derivative of the position vector with respect to time (dR/dt). Similarly, average acceleration is the ratio of change in velocity to time interval (Δv/Δt), and instantaneous acceleration is the derivative of velocity with respect to time (dv/dt).