Find the unit normal vector of r(t)

In summary, the conversation was about a problem involving finding the values of various components in a given equation. The person had already attempted the solution multiple times with different answers and was looking for help to figure out what they had done wrong. They shared the components they found and evaluated them at a specific value of t, but forgot to check if the answer was a unit vector. With some guidance, they were able to correct their mistake and solve the problem.
  • #1
Unicow
14
0

Homework Statement


upload_2017-7-15_15-33-28.png


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))
 
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  • #2
Unicow said:

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.
 
  • #3
Sa mmyS said:
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...
Untitled.png

I got this as the answer, but it's wrong. I'm also not too sure how to simplify this... lol
 
Last edited:
  • #4
Unicow said:
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 ?
 
  • #5
SammyS said:
What do you get for v⋅r'' − v'⋅r', the numerator of N ?

I get
upload_2017-7-15_23-41-44.png
for the numerator
 
  • #6
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  • #7
SammyS said:
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...
 
  • #8
Unicow said:
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 !
 
  • Like
Likes Unicow

FAQ: Find the unit normal vector of r(t)

What is the definition of a unit normal vector?

A unit normal vector is a vector that is perpendicular to a curve or surface at a specific point, and has a magnitude of 1.

How do you find the unit normal vector of a parametric curve?

To find the unit normal vector of a parametric curve, first find the tangent vector by taking the derivative of the parametric equations. Then, divide the tangent vector by its magnitude to get the unit tangent vector. Finally, take the derivative of the unit tangent vector and divide it by its magnitude to get the unit normal vector.

Can the unit normal vector change along a curve?

Yes, the unit normal vector can change along a curve, as it is dependent on the tangent vector, which can change as the curve changes direction.

How is the unit normal vector used in physics?

The unit normal vector is used in physics to calculate the direction of acceleration and to determine the curvature of a path or surface. It is also used in the calculation of force vectors and centripetal force.

Can the unit normal vector be negative?

Yes, the unit normal vector can be negative, as its magnitude is always 1, but its direction can vary depending on the direction of the curve or surface. Negative values indicate that the vector is pointing in the opposite direction of the positive value.

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