General Planar Motion

General Motion

Translation + rotation about an accelerating axis restricted to a plane

Rigid bodies move in a plane passing through its centre of mass and symmetrical with respect to the plane.

Motion: Translation and rotation about an axis perpendicular to the plane.

Basically for a rigid body, acceleration and motion occurs about the CoM but forces can act anywhere on the body.

Equations of Force

$\sum F_{x} = m a_{G x}$
$\sum F_{y} = m a_{G y}$ $M_{G} = I_{G} α$ Rotation about a fixed axis that pass through point O
$M_{o} = I_{o} α$

$f_{G} = \sum f_{x G} + \sum f_{y G}$ $M_{p} = M_{G} + f_{g} d$
$M_{p} = M_{G} + m a_{G} d$ $M_{p} = I_{G} α + m_{a G} d$

For a wheel that rolls down a hill, we can analyze translation…

Sum forces in the X and Y directions to get the tangential force at the point of contact
Take sum of moments about the CoM now that you know the tangential force
No you can find acceleration of the CoM

If F < F_friction (no slipping) If F > F_friction (slipping)

See OneNote for a rolling cylinder equation of motion derivation