Mathematical Methods for Physics and Engineering - Matematica.NET

10.3 SPACE CURVESso we finally obtaind ˆb = −τ ˆn. (10.14)dsTaking the dot product of each side with ˆn, we see that the torsion of a curve isgiven byτ = − ˆn · d ˆbds .We may also define the quantity σ =1/τ, which is called the radius of torsion.Finally, we consider the derivative d ˆn/ds. Since ˆn = ˆb × ˆt we haved ˆnds = d ˆbds × ˆt + ˆb × d ˆtds= −τ ˆn × ˆt + ˆb × κ ˆn= τ ˆb − κ ˆt. (10.15)In summary, ˆt, ˆn and ˆb and their derivatives with respect to s are related to oneanother by the relations (10.13), (10.14) and (10.15), the Frenet–Serret formulae,d ˆtds = κ ˆn,d ˆnds = τ ˆb − κ ˆt,d ˆbds= −τ ˆn. (10.16)◮Show that the acceleration of a particle travelling along a trajectory r(t) is given bya(t) = dvdt ˆt + v2ρ ˆn,where v is the speed of the particle, ˆt is the unit tangent to the trajectory, ˆn is its principalnormal and ρ is its radius of curvature.The velocity of the particle is given byv(t) = drdt = dr dsds dt = dsdt ˆt,where ds/dt is the speed of the particle, which we denote by v, and ˆt is the unit vectortangent to the trajectory. Writing the velocity as v = v ˆt, and differentiating once morewith respect to time t, weobtaina(t) = dvdt = dvdt ˆt + v d ˆtdt ;but we note thatd ˆtdt = ds d ˆtdt ds = vκ ˆn = v ρ ˆn.Therefore, we havea(t) = dvdt ˆt + v2ρ ˆn.This shows that in addition to an acceleration dv/dt along the tangent to the particle’strajectory, there is also an acceleration v 2 /ρ in the direction of the principal normal. Thelatter is often called the centripetal acceleration. ◭343