Finding unit normal vectors and normal/tangent components of accelerat

[icesalmon]

Homework Statement

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

Homework Equations

T(t) = r'/||r'||
N(t) = T'/||T'||
a_T = a . T = (v . a)/||T||
a_N = a . N = ||v x a||/||v|| = sqrt(||a||² - a_T²)

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

 

[jbunniii]

Homework Statement

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

Homework Equations

T(t) = r'/||r'||
N(t) = T'/||T'||
a_T = a . T = (v . a)/||T||
a_N = a . N = ||v x a||/||v|| = sqrt(||a||² - a_T²)

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?

 

[icesalmon]

r' = <1,-1/t²>
|r'| = sqrt(1 + 1/t⁴)
did I mess something up when differentiating/doing algebra/both?

 

[icesalmon]

|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.

 

[jbunniii]

|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.

 

[icesalmon]

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.