Curvature forumula of a planar trajectory

In summary, the conversation discusses a problem with a planar trajectory and the first two Frenet-Serret equations are included. The solution is presented in a separate document and involves finding the normalized tangent and the curvature of the curve parameter. The equations and solution can be found on a website provided by the speaker.
  • #1
Rhawk187
1
0

Homework Statement



http://steam.cs.ohio.edu/~cmourning/problem1.jpg

If the image doesn't load (and it might not, although I'm not sure why), it can be found at:

http://steam.cs.ohio.edu/~cmourning/problem1.jpg

Homework Equations



Part of the problem is I'm not entirely sure what all the relevant equations are. This is a planar trajectory so the torsion is 0, so I've just included the first two Frenet-Serret equations at:

http://steam.cs.ohio.edu/~cmourning/equations1.pdf

The Attempt at a Solution



It is rather complicated equationally, so I felt more comfortable typing it in something other than this, you can find the work at:

http://steam.cs.ohio.edu/~cmourning/physics1.pdf

If I should put this somewhere else on the forums let me know, this is my first time.
 
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  • #2
Just call the curve parameter s=x. Then the curve is (x,y(x)). The tangent vector is (1,y'(x)). That makes the normalized tangent T=(1,y'(x))/(1+y'(x)^2)^(1/2). Now kappa=|dT/dx|. You have that in your solution. You have to find the vector that is dT/dx and find it's magnitude. Start differentiating.
 

What is the curvature formula for a planar trajectory?

The curvature formula for a planar trajectory is given by k = |dT/ds|, where k is the curvature, T is the unit tangent vector, and s is the arc length parameter.

How is the curvature of a planar trajectory calculated?

The curvature of a planar trajectory is calculated by taking the derivative of the unit tangent vector with respect to the arc length parameter, and then taking the magnitude of this derivative.

What does the curvature of a planar trajectory represent?

The curvature of a planar trajectory represents the rate of change of the direction of the tangent vector along the curve at a given point. It measures how much the curve is bending at that point.

Can the curvature of a planar trajectory be negative?

Yes, the curvature of a planar trajectory can be negative. This indicates that the curve is bending in the opposite direction of the tangent vector at that point.

How is the curvature of a planar trajectory related to the radius of curvature?

The curvature and radius of curvature are inversely proportional. The radius of curvature is equal to 1/k, where k is the curvature at a given point. This means that a larger curvature corresponds to a smaller radius of curvature, and vice versa.

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