Mechanics Kinematics & Centres of Mass Definitions and Formulas

Kinematics:

Motion constant acceleration

$v=u+ft$

$s=ut+\frac{1}{2}ft^2=\frac{1}{2}(u+v)t$

$v^2=u^2+2f.s$

General solution of $\frac{d^2x}{dt^2}=-w^2x$ is

x=a.coswt + b.sinwt = R.sin(wt+φ)

where $R=\sqrt{a^2+b^2}$ and cosφ = a / R, sinφ = b / R.

In polar coordinates the celocity is  and the acceleration is

Centres of mass:

The following results are for uniform bodies:

• hemispherical shell, radius r    ⇾    $\frac{1}{2}r$                    ⇾     from centre
• hemisphere, radius r                 ⇾    $\frac{3}{8}r$                  ⇾     from centre
• right circular cone, height h      ⇾    $\frac{3}{4}h$                  ⇾     from vertex
• arc, radius r and angle 2θ         ⇾    (r sinθ) / θ       ⇾     from centre
• sector, radius r and angle 2θ    ⇾    ($\frac{2}{3}r$ sinθ) / θ    ⇾     from centre