1. A circle of radius R and centered at O(0; 0) moves by curvature if at the time t, its radius R(t) changes with a rate -1/R(t):
R ' (t) = -1/R(t)
Recall that -1/R(t) is the curvature of the circle of radius R(t). The minus sign in the rate of change of R(t) indicates that radius of the circle is decreasing. In other words, the circle moves towards its center with the speed being equal to its curvature.
At the time t = 0, the radius of the circle is R. Thus R(0) = R:
a. Compute explicitly the radius R(t) at the time t in terms of R and t.
b. When does the circle extinct? (i.e, Find the time T such that R(T) = 0)
2. Suppose you start at the point (0,0,3) and move 5 units along the curve x=3 sin t , y= 4t , z=3 cos t in the postive direction. where are you now?
please explain in details. thankyou very much!!!
vector function (curvature)
2011-02-28 2:51 pm
回答 (2)
2011-02-28 5:24 pm
✔ 最佳答案
1a) R ' (t) = -1/R(t)∫R'(t) R(t) dt = - ∫dt
∫R(t) d[R(t)] = - ∫dt
[R(t)]2/2 = -t + C
[R(t)]2 = -2t + C
With R(0) = R, we have C = R2
So [R(t)]2 = -2t + R2
b) R(t) = 0
R2 = 2t
t = R2/2
2) The magnitude of the vector along the motion path is given by:
√[(3 sin t)2 + (4t)2 + (3 cos t)2]
= √(9 sin2 t + 16t2 + 9 cos2 t)
= √(9 + 16t2)
So for a magnitude = 5, t = 1
So the position now is:
(0, 0, 3) + (3 sin 1, 4, 3 cos 1) = (3 sin 1, 4, 3 + 3 cos 1)
參考: 原創答案
2011-02-28 7:04 pm
1a) R'(t) = -1/R(t)∫ R'(t) R(t) dt = - ∫dt∫ R(t) d[R(t)] = - ∫dt[R(t)]^2/2 = -t + CWhen t = 0, R(0) = R, we have C = R^2/2So [R(t)]^2 = -2t + R^2b) R(T) = 0R^2 = 2TT = R^2/2
2 arc length
= ∫ √{ [x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2 } dt
= ∫ √(9 + 16) dt
= 5t
At t = 0, I am at the point (0,0,3). After t = 1, I will at the point (3sin1, 4, 3cos1) by moving 5 units along the curve x=3 sin t , y= 4t , z=3 cos t from the calculation above. So,
2 arc length
= ∫ √{ [x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2 } dt
= ∫ √(9 + 16) dt
= 5t
At t = 0, I am at the point (0,0,3). After t = 1, I will at the point (3sin1, 4, 3cos1) by moving 5 units along the curve x=3 sin t , y= 4t , z=3 cos t from the calculation above. So,
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