Formulas

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 F_ik be the force that particle i exerts on particle k. Newton's third law states that F_ik= -F_ki.
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 F_ext= 0; F_ext= 0 <--> P = constant.
Angular momentum (L = r × p) is conserved if the torque τ_ext= 0; τ_ext= 0 <--> L = constant.
Energy (E = T + U) is conserved if all forces are conservative; ∫F∙dr = 0 <--> T + U = constant.

Formulas:

Laws:
Newton's 2nd law:	F = dp/dt
Newton's third law:	F_ik= -F_ki

Forces:
Static and kinetic friction:	f_s ≤ μ_sN, f_k = μ_kN
Gravity:	F₁₂ = -Gm₁m₂r₁₂/r₁₂³
Coulomb's law	F₁₂ = k_eq₁q₂r₁₂/r₁₂³, k_e = 1/(4πε₀)
Hooke's law:	F = -kr, F_x= -kx

Concepts:
Uniform circular motion:	F = mv²/r
Work:	W = F∙d
Kinetic energy:	K = ½mv²
Work-kinetic energy theorem:	W_net= ΔK = ½m(v_f²- v_i²)
Elastic potential energy:	U = ½kx²
Gravitational potential energy:	U_f - U_i = -∫_r12f^r12iF₁₂∙dr₁₂ = -Gm₁m₂(1/r_12f - 1/r_12i)
Conservative systems:	E = K + U, F= -∇U
Power:	P = F·v or P = dW/dt
Momentum:	p = mv
Impulse:	I = Δp = F_avgΔt
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 = F_avgΔt.
The integral of force over time as Δt approaches 0 is called the impulse of the force.

Angular impulse: dL = τdt, ΔL = ∫τdt, ΔL = τ_avgΔt.
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.

Elastic collision: momentum is conserved, mechanical energy is conserved.
Inelastic collisions: momentum is conserved, mechanical energy is not conserved.

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.