Cross product

calculator

Law of Cosines

c=a2+b22abcosγ

Small angle approximations:

sinθθ,tanθθ,cosθ1θ22 sin(u)sin(v)=12[cos(uv)cos(u+v)] cos(u)cos(v)=12[cos(uv)+cos(u+v)]

Physics prequisite

force to torque:

τ=r×F=rFsinθ

friction torque(viscous) >Wiki:

τ=kθ˙

弹性力(k 为弹簧的劲度系数,x 为振子偏离平衡位置的位移): F=kx 阻尼力(c 为阻尼系数,v 为振子速度): F=cv

Potential energy

Springs: 12k(xx0)2

Gravity: mgh

Kinetic energy

Linear motion: 12mvTv

Angular motion: 12ωTIω

Inertia

I is always symmetric. Diagonal if CoM aligns with primary axes.

For cylinder: Iz=12MR2, Ix=Iy=14MR2+112ML2