Concepts and formulas

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.

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

Impulsive forces

Forces and torques that act so powerfully but so briefly that they produce finite changes in linear and angular momentum while the system undergoes negligible displacement are said to be impulsive.

Linear impulse: dp = Fdt, Δp = Fdt, Δp = FavgΔt.
The integral of force over time as Δt approaches 0 is called the impulse of the force.

Angular impulse: dL = τdt, ΔL = τdt, ΔL = τavgΔt.
The integral of torque over time as Δt approaches 0 is called the angular impulse of the torque.

Collisions

In collisions, it is assumed that the colliding particles interact for such a short time, that the impulse due to external forces is negligible.  Thus the total momentum of the system just before the collision is the same as the total momentum just after the collision.

Interacting objects

If no external forces act on a system of interacting objects, the center of mass of the system does not accelerate.  Any two interacting objects obey Newton's third law.