Question

Find the curvature κ(t) of the curve X (t) = ( −4 sint )I + ( −4 sint ) j + ( 3 cost ) K

Answer 1

do you know how to derive the formula for k(t)

Answer 2

no i don't

Answer 3

ok derivation of k(t)
if you have r(t)
then r'(t)/||r'(t)||=T
and since ||dT/ds||=k
then |dT/dt(ds/dt)|=|dT/ds|
so 1/|ds/dt|(dT/dt|=1/|v|(|dT/dt)|=k
there you go

Answer 4

|dT/dt(dt/ds)|=|dT/ds|

Answer 5

!

Answer 6

so ill let you do this one,
take your velcocity or r'(t) then take the magnitude=1/|r'(t)|
then take your r'(t)/|r'(t)|=T
then take d/dt(T)=dT/dt and take that magnitude=|dT/dt|
1/|r'(t)|(dT/dt)|=k(t)

Answer 7

i dont know about this one BUT THESE TURN OUT UGLYYYYYYYY

Answer 8

it is ugly ! haha

Answer 9

do u mind giving me a numerical value too !?

Answer 10

ill do a very easy example
r(t)=(cos(t)i+(sin(t))j
r'(t)=(-sin(t))i+(cos(t))j
|r'(t)|=1
r'(t)/|r'(t)|=(-sin(t))i+(cos(t))j=T
d/dt(T)=(-cos(t))i+(-sin(t))j
|dT/dt|=1
k(t)=1/1(1)=1

Answer 11

N is pointing towards the center of curvature