12_curv_acc - Curvature Curvature measures the bend in a curve T 0 Curvature is a scalar v v t We will now use a new formula at 3 vt rt ex t t t 2 3 2

# 12_curv_acc - Curvature Curvature measures the bend in a...

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Curvature Curvature measures the bend in a curve. κ 0 . Curvature is a scalar. κ = T v We will now use a new formula κ = v t (29 × a t (29 v t (29 3 ex. r t ( 29 = t 2 , t 3 , t 2
ex. r t ( 29 = a cos t , a s in t Find κ .
Do: r t ( 29 = s in t , co s t , s in t . Find κ .
Acceleration Velocity has speed and direction. v t (29 = v t (29 v t (29 v t (29 Acceleration is the rate of change of velocity so a t ( 29 has a component of rate of change of speed and also a component of rate of change of direction. T = N = a t ( 29 = a T T + a N N a T is the tangential component of acceleration. a N is the normal component of acceleration. a T = dv dt = v a v a N = v 2 κ = v × a v
ex. r t ( 29 = t 2 , t 3 . Find a T and a N .
ex. r t ( 29 = 2co s t , 2s in t , 2 . Find a T and a N . Do: r t ( 29 = s in t , co s t , s in t . Find a T and a N .
ex. x = e t , y = t 3 r t ( 29 = e t , t 3 ex. ex.
ex. If a particle is moving on a path where a N = is it moving in a straight line? ex. If a particle moves with constant speed can it be accelerating? ex. Let point P move along curve C with constant speed c. Let
point Q move along C with speed 2c. Show that the magnitude of acceleration of Q is 4 times greater than that of P.