The motion of single particles

Inertial frames:

Newton's first law defines a class of inertial frames.  Inertial frames are reference frames for which the trajectories for force-free motion are solutions to d2r/dt2 = 0.  With respect to inertial frames Newton's second law has the form
F = dp/dt.  (r = coordinate, F = force, p = mv momentum)

Let Fik be the force that particle i exerts on particle k.  Newton's third law states that Fik = -Fki.
Newton's laws are well suited for the study of unconstrained mechanical systems.  Constraints, such as requiring a particle to follow a given curve in space, tell us that there are external forces, but do not tell us what these forces are.  The forces are only known in terms of their effect on the motion.

Conservation laws are very important tools in solving mechanics problems.

• For a system of particles momentum is conserved if Fext = 0;  Fext = 0 <--> P = constant.
• Angular momentum (L = r × p) is conserved if the torque τext = 0;  τext = 0 <--> L = constant.
• Energy (E = T + U) is conserved if all forces are conservative;  ∫Fdr = 0 <--> T + U = constant.

Formulas:

 Laws: Newton's 2nd law: F = dp/dt Newton's third law: Fik = -Fki Forces: Static and kinetic friction: fs ≤ μsN,  fk = μkN Gravity: F12 = -Gm1m2r12/r123 Coulomb's law F12 = keq1q2r12/r123,  ke = 1/(4πε0) Hooke's law: F = -kr, Fx= -kx Concepts: Uniform circular motion: F = mv2/r Work: W = F∙d Kinetic energy: K = ½mv2 Work-kinetic energy theorem: Wnet = ΔK = ½m(vf2 - vi2) Elastic potential energy: U = ½kx2 Gravitational potential energy: Uf - Ui = -∫r12fr12iF12∙dr12 = -Gm1m2(1/r12f - 1/r12i) Conservative systems: E = K + U,  F= -∇U Power: P = F·v or P = dW/dt Momentum: p = mv Impulse: I = Δp = FavgΔt Angular momentum: L = r × p Torque τ = r × F Angular momentum and torque: dL/dt = d/dt(r×p) = r×dp/dt + dr/dt×p = r×dp/dt = r×F = τ   dL = τ dt

Non-inertial frames:

In non-inertial frames fictitious forces appear.
Consider a particle moving with velocity v in a reference frame K which moves with velocity V(t) relative to the inertial frame K0 and rotates with angular velocity Ω(t).
The equations of motion are

mdv/dt = -∂U/∂r - mdV/dt + mr × dΩ/dt - 2mΩ × v - mΩ × (Ω × r).

Here

• -∂U/∂r = force as observed in an inertial frame
(If the force in an inertial frame is is not derivable from a potential, replace -∂U/∂r by Finertial.)
• -mdV/dt = fictitious force due to acceleration of frame
• mr × dΩ/dt = fictitious force due to non-uniform rotation of frame
• -2mΩ × v = Coriolis force
• -mΩ × (Ω × r) = centrifugal force

For a uniformly rotating frame dΩ/dt = 0, dV/dt, and the equations of motion are
mdv/dt = Finertial - 2mΩ × v - mΩ × (Ω × r).