Finding unit normal vectors and normal/tangent components of accelerat

  • Thread starter icesalmon
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  • #1
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Homework Statement


given r(t) = <t, 1/t,0> find T(t) N(t) aT and aN at t = 1


Homework Equations


T(t) = r'/||r'||
N(t) = T'/||T'||
aT = a . T = (v . a)/||T||
aN = a . N = ||v x a||/||v|| = sqrt(||a||2 - aT2)


The Attempt at a Solution


for my T(t) I get <t2, -1 , 0>/(sqrt(1+t4) (I like keeping things in 3 dimensions even if there is no contribution in the z direction)
and I am not calculating the normal vector if there isn't some algebra I can use to simplify this greatly , which I am not seeing, t4 + 1 I don't believe I can factor and I can't think of any other way to simplify this one so i'm just moving around through the problem set looking for some r(t) = <cost, sint, t> type of vector that I can simply differentiate and use identities with, desperately trying to avoid those other problems
 
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Answers and Replies

  • #2
jbunniii
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Homework Statement


given r(t) = <t, 1/t,0> find T(t) N(t) aT and aN at t = 1


Homework Equations


T(t) = r'/||r'||
N(t) = T'/||T'||
aT = a . T = (v . a)/||T||
aN = a . N = ||v x a||/||v|| = sqrt(||a||2 - aT2)


The Attempt at a Solution


for my T(t) I get <t2, -1 , 0>/(sqrt(1+t4)
How did you get that? Your first equation in the "relevant equations" section is ##T(t) = r' / \|r\|##. What is ##r'## in this case?
 
  • #3
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r' = <1,-1/t2>
|r'| = sqrt(1 + 1/t4)
did I mess something up when differentiating/doing algebra/both?
 
  • #4
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|r'| = sqrt((t4+1)/t4) = (1/t2)sqrt(t4+1)
which should be t2r'/(|r'|)
which I believe is actually <t2,-1>/(sqrt(t4+1)
which is the same thing...oops i'm not sure what i'm messing up here.
 
  • #5
jbunniii
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|r'| = sqrt((t4+1)/t4) = (1/t2)sqrt(t4+1)
which should be t2r'/(|r'|)
which I believe is actually <t2,-1>/(sqrt(t4+1)
which is the same thing...oops i'm not sure what i'm messing up here.
OK, that looks fine. So what is ##T'##? Just use the quotient rule, it shouldn't be too horrible.
 
  • #6
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well that's what i'm saying, are there any ways I can simplify this particular unit tangent vector before differentiating? I know how to differentiate it, but on an exam I will chew up a ton of time making sure I keep everything straight with the differentiation in terms of signs and cancellations.
 

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