Find the unit normal vector of r(t)

[Unicow]

Homework Statement

Homework Equations

The equation has already been given in the question.

The Attempt at a Solution

So what I did was find r'(t), r"(t), v(t), v'(t) and plug it into the equation. I've done 3 different full pages of this and have gotten 3 different answers. I'm guessing due to simple arithmetic errors, can someone help me figure this out and what I may have done wrong? I will post the full work I did on here but I don't think it will be very legible or even worth trying to go through because of how annoying this problem gets. I'm going to be working through it once more but I'd appreciate it if someone else could share what answer they got... I normally don't like taking straight up answers and I like solving it myself but I think I'm going to lose my damn mind.

I forgot to share atleast the components I found.
Since r(t) = <t - sin(t), 1 - cos(t)>
r'(t) = <1 - cos(t), sin(t)>
r"(t) = <sin(t), cos(t)
v(t) = sqrt(2 - 2cos(t))
v'(t) = sin(t) / sqrt(2 - 2cos(t))

 

[SammyS]

Homework Statement

View attachment 207235

Homework Equations

The equation has already been given in the question.

The Attempt at a Solution

So what I did was find r'(t), r"(t), v(t), v'(t) and plug it into the equation. I've done 3 different full pages of this and have gotten 3 different answers. I'm guessing due to simple arithmetic errors, can someone help me figure this out and what I may have done wrong? I will post the full work I did on here but I don't think it will be very legible or even worth trying to go through because of how annoying this problem gets. I'm going to be working through it once more but I'd appreciate it if someone else could share what answer they got... I normally don't like taking straight up answers and I like solving it myself but I think I'm going to lose my damn mind.

I forgot to share atleast the components I found.
Since r(t) = <t - sin(t), 1 - cos(t)>
r'(t) = <1 - cos(t), sin(t)>
r"(t) = <sin(t), cos(t)
v(t) = sqrt(2 - 2cos(t))
v'(t) = sin(t) / sqrt(2 - 2cos(t))

Evaluate those quantities at t = π/3 .

What do you get?

Notice that your answer for N is not a unit vector.

 

[Unicow]

Evaluate those quantities at t = π/3 .

What do you get?

Notice that your answer for N is not a unit vector.

Wow, I can't believe I forgot to check that. The current answer on the picture shown is the final answer I had gotten, before that I think I made some mistakes.

Edit: I just realized what you asked me to do.
r(t) = <pi/3 - sqrt(3)/2, 1/2>
r'(t) = <1/2, sqrt(3)/2>
r"(t) = <sqrt(3)/2 , 1/2>
v(t) = 1
v"(t) = sqrt(3)/2
Can I simply use these and plug it into the equation...? I will test this now...

I got this as the answer, but it's wrong. I'm also not too sure how to simplify this... lol

 

[SammyS]

Wow, I can't believe I forgot to check that. The current answer on the picture shown is the final answer I had gotten, before that I think I made some mistakes.

Edit: I just realized what you asked me to do.
r(t) = <pi/3 - sqrt(3)/2, 1/2>
r'(t) = <1/2, sqrt(3)/2>
r"(t) = <sqrt(3)/2 , 1/2>
v(t) = 1
v"(t) = sqrt(3)/2
Can I simply use these and plug it into the equation...? I will test this now...
View attachment 207246
I got this as the answer, but it's wrong. I'm also not too sure how to simplify this... lol

What do you get for v⋅r'' − v'⋅r', the numerator of N ?

 

[Unicow]

What do you get for v⋅r'' − v'⋅r', the numerator of N ?

I get

for the numerator

 

[SammyS]

I getView attachment 207247 for the numerator

That's not what I get.

 

[Unicow]

That's not what I get.

I made a small mistake with the equation. I got the answer now, thank you so much. I didn't know I could do it that simply...

 

[SammyS]

I made a small mistake with the equation. I got the answer now, thank you so much. I didn't know I could do it that simply...

Good !