Knowlpedia

Definition

Centripetal acceleration for circular motion

The inward normal acceleration v²/r required by motion along a circle.

For motion along a circle of fixed radius r>0r>0, the centripetal acceleration is the inward radial component

ar=−vθ2r=−rΩ2.a_r=-\frac{v_\theta^2}{r}=-r\Omega^2.

Indeed, differentiating vθeθv_\theta e_\theta gives the radial term vθθ˙(−er)v_\theta\dot\theta(-e_r), and vθ=rθ˙v_\theta=r\dot\theta. A changing speed also gives a tangential acceleration.

Radial pressure balance

For a stationary inviscid purely circular flow u=V(r)eθu=V(r)e_\theta with no external force, the radial momentum equation is

∂rp=V(r)2r.\partial_rp=\frac{V(r)^2}{r}.

The pressure therefore increases radially outward, so the pressure acceleration −∇p-\nabla p points inward. This sign follows from the inertial-frame material acceleration.