Thomas Calculus 13th [Solutions]

Section 13.5 Tangential and Normal Components of Acceleration 953
4.
2
r t cost i t sin t j t k v cost t sin t i sin t t cos t j 2tk
v
(0) 0; a T
2 2 2 2 2
1 2
cost t sin t sin t t cost 2t 5t 1 a 1 5 1 (10 ) 5t
T t t
2
2
5t
1
2 2
a 2sin t t cos t i 2cost t sin t j 2 k a(0) 2j 2 k a(0) 2 2 2 2
aN
2 2
2
2
a aT
2 2 0 2 2 a(0) (0) T 2 2N 2 2N
5.
2 3 3 2 2 2 2
2
2
2
r t i t 1 t j t 1 t k v 2ti 1 t j 1 t k v 2t 1 t 1 t
3 3
4 2 2
2 t 2t 1 2 1 t aT
2t 2 aT
(0) 0; a 2i 2tj 2 tk a(0) 2 i a(0) 2
aN
2 2 2 2
a aT
2 0 2 a(0) (0) T 2N 2N
t t t t t t t t
6. r e cost i e sin t j 2e k v e cos t e sin t i e sin t e cos t j 2e
k
v
t t
2
t t
2
t
2
2t t t
e cos t e sin t e sin t e cos t 2e 4e 2e aT
2 e aT
(0) 2;
t t t t t t t t t
a e cos t e sin t e sin t e cost i e sin t e cos t e cost e sin t j 2e
k
2
2
2e t sin t i 2e t cos t j 2 e
t k a(0) 2j 2 k a(0) 2 2 6
aN
2 2
2
2
a aT
6 2 2 a(0) 2T 2N
7.
2 2
r cos t i sin t j k v sin t i cos t j v sin t cost
1
T v
2 2
sin t cos t
;
v
i j T 4 2 i 2
j d T
cos t i
dt
(sin t ) j
d T
dt
d T
2 2 dt
2 2
cost sin t 1 N cos t i sin t j N i j;
d T
4 2 2
i j k
B T N sin t cost
0 k B
4
k , the normal to the osculating plane;
cos t sin t 0
dt
2 2
r i j k
4 2 2
2 2
P , , 1 lies on the osculating plane
2 2
osculating plane; T is normal to the normal plane
2 2
0 x 0 y z ( 1) 0 z 1 is the
2 2
2 2 2 2
x y 0 z ( 1) 0
2 2 2 2
2 2
x y 0 x y 0 is the normal plane; N is normal to the rectifying plane
2 2
2 2 2 2 2 2
x y 0 z ( 1) 0 x y 1 x y 2 is the rectifying
2 2 2 2 2 2
plane.
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