Thomas Calculus 13th [Solutions]

952 Chapter 13 Vector-Valued Functions and Motion in Space
Clear[r, t, x, y]
_ r[t ] {3 Cos[t], 5 Sin[t] }
t0 /4; tmin 0; tmax 2 ;
_ r2[t ] {r[t][[1]], r[t][[2]]}
pp ParametricPlot[r2[t], {t, tmin, tmax}];
_ mag[v ] Sqrt[v.v]
_ vel[t ]
r'[t]
_ speed[t ] mag[vel[t]]
_ acc[t ]
_ curv[t ]
_ unittan[t ]
vel'[t]
_ unitnorm[t ]
3
mag[Cross[vel[t],acc[t]]]/speed[t] //Simplify
vel[t]/speed[t]//Simplify
unittan'[t] / mag[unittan'[t]]
ctr r[t0] (1/ curv[t0]) unitnorm[t0] //Simplify
{a,b} {ctr[[1]], ctr[[2]]}
To plot the osculating circle, load a graphics package and then plot it, and show it together with the original
curve.
Graphics`ImplicitPlot`
2
pc ImplicitPlot[(x a)^2 (y b)^2 1/curv[t0] , {x, 8, 8}, {y, 8, 8}]
radius Graphics[Line[{{a, b}, r2[t0]}]]
Show[pp, pc, radius, AspectRatio 1]
13.5 TANGENTIAL AND NORMAL COMPONENTS OF ACCELERATION
1.
2 2 2
r a cos t i a sin t j btk v a sin t i a cos t j bk v asin t a cost b
2 2
a b a d
T v 0;
dt
2 2 2
a a cost i asin t j a a cost asin
t a a
2 2 2 2
aN
a aT
a 0 a a a (0) T a N a N
2.
2 2 2
r 1 3t i t 2 j 3tk v 3i j 3k v 3 1 ( 3) 19 a d T v 0;
dt
2 2
a 0 aN
a aT
0 a (0) T (0) N 0
3.
2 2 2 2 2 2
1 2
r ( t 1) i 2tj t k v i 2j 2tk v 1 2 (2 t) 5 4t a 1
T 5 4 t (8 t)
2
4 2
1 2
t 5 4 t a (1) 4 4
T
9 3
; a 2 k a(1) 2 k a(1) 2
2 2 2
2
4 20 2 5 4 2 5
aN
a aT
2 a(1)
T N
3 9 3 3 3
Copyright
2014 Pearson Education, Inc.