Thomas Calculus 13th [Solutions]

Section 13.6 Velocity and Acceleration in Polar Coordinates 959
N[{curv[t0], torsion[t0]}]
{at[t0], an[t0]}
N[{at[t0], an[t0]}]
To verify that the tangential and normal components of the acceleration agree with the formulas in the book:
at[t] speed'[t]//Simplify
an[t]
2
curv [t] speed[t] //Simplify
13.6 VELOCITY AND ACCELERATION IN POLAR COORDINATES
1. d 3 0, 1 cos sin d 3 sin 3 cos d 9 cos
dt
r a r a dt
a r a dt
a
v 3a sin ur
a 1 cos (3) u 3a sin ur
3a
1 cos u
2
a 9a cos a 1 cos (3) ur
a 1 cos 0 2 3a
sin (3) u
9a cos 9a 9a cos ur
18a sin u 9a 2cos 1 ur
18asin
u
2. d 2t 2, sin 2 cos 2 2 d 4 cos 2 4 sin 2 2 d 4 cos 2
dt
r a r a dt
ta r ta dt
a
2
16t asin 2 4a cos 2 ; v 4ta cos 2 ur
asin 2 (2 t) u 4ta cos 2 ur
2ta
sin 2 u
2 2
a 16t asin 2 4a cos 2 asin 2 (2 t) ur
asin 2 (2) 2 4ta cos 2 (2 t)
u
2 2 2
16t asin 2 4a cos 2 4t asin 2 ur
2asin 2 16t a cos 2
2 2
20t asin 2 4a cos 2 ur
2asin 2 16t a cos 2 u
2 2
4a cos 2 5t sin 2 ur
2a sin 2 8t
cos 2 u
u
3. d
a a 2
2 0,
d a a
2 2 d a
4
dt
r e r e a dt
a e r a e a dt
a e
a a a a
v 2 a e ur
e (2) u 2a e ur
2e
u
2 a a 2 a a 2 a a a
a 4 a e e (2) ur
e (0) 2 2 a e (2) u 4a e 4e ur
0 8a e u
a 2
a
4e a 1 ur
8a e
u
t t t
4. 1 e e e , r a(1 sin t) r a cos t r asin
t
t
t
v a cos t ur
a 1 sin t e u a cost ur
a e 1 sin t u
t
2
t t
a a sin t a 1 sin t e ur
a 1 sin t e 2 a cost e u
2t t t
a sin t a e 1 sin t ur
a e 1 sin t 2a e cost
2t
t
a sin t e 1 sin t ur
a e 1 sin t 2cos t u
2t
t
a sin t e 1 sin t ur
a e 2cos t 1 sin t u
u
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