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 d^{2}r/dt^{2} = 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.
Laws: | |
Newton's 2nd law: | F = dp/dt |
Newton's third law: | F_{ik }= -F_{ki} |
Forces: | |
Static and kinetic friction: | f_{s} ≤ μ_{s}N, f_{k} = μ_{k}N |
Gravity: | F_{12} = -Gm_{1}m_{2}r_{12}/r_{12}^{3} |
Uniform circular motion: | F = mv^{2}/r |
Hooke's law: | F = -kr, F_{x}= -kx |
Concepts: | |
Work: | W = F∙d |
Kinetic energy: | K = ½mv^{2} |
Work-kinetic energy theorem: | W_{net }= ΔK = ½m(v_{f}^{2 }- v_{i}^{2}) |
Elastic potential energy: | U = ½kx^{2} |
Gravitational potential energy: | U_{f} - U_{i} = -∫_{r12f}^{r12i}F_{12}∙dr_{12} = -Gm_{1}m_{2}(1/r_{12f} - 1/r_{12i}) |
Conservative systems: | E = K + U, F_{x }= -∂U/∂x |
Power: | P = F·v or P = dW/dt |
Momentum: | p = mv |
Impulse: | I = Δp = F_{avg}Δt |
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 |
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 K_{0} 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
For a uniformly rotating frame dΩ/dt = 0, dV/dt, and the
equations of motion are
mdv/dt = F_{inertial} - 2mΩ × v - mΩ × (Ω
× r).