Polar coordinates and radius of curvature

[Telemachus]

Homework Statement

I've got this problem on polar coordinates which says:

A particle moves along a plane trajectory on such a way that its polar coordinates are the next given functions of time:

$$r=0.833t^3+5t$$ $$\theta=0.3t^2$$

Determine the module of the speed and acceleration vectors for this particle and its radius of curvature at the instant $$t=2s$$.

And I don't know how to solve the radius of curvature part.
For intrinsic coordinates I know that: $$\rho=\displaystyle\frac{v^2}{a_n}$$

Where $$a_n$$ is the normal acceleration. Now how do I find the radius of curvature? do I have to take the trajectory to the intrinsic form?

For the first part I have that:
$$\dot r=v(t)=\dot r\vec{e_r}+r\dot \theta\vec{e_{\theta}}$$
$$v(2s)\approx{}10\hat{e_r}+20\hat{e_{\theta}}$$

$$\ddot r=a(t)=(\ddot r-r\dot \theta^2)\vec{e_r}+(r\ddot \theta+2 \dot r \dot \theta)\vec{e_{\theta}}$$
$$a(2s)\approx{}-10\hat{e_r}+46\hat{e_{\theta}}$$

From there I can get the modules, but I don't worry about that, I want to know how to find the radius of curvature at the point.

Bye there, and thanks for posting.

 

[tiny-tim]

Hi Telemachus!

See the polar-coordinates formula at http://mathworld.wolfram.com/RadiusofCurvature.html (and then see if you can prove it! )

 

[Telemachus]

Can it be done by using the velocity vector? considering its on the same direction than the tangent acceleration, then just looking a vector perpendicular to it, and projecting the acceleration on that vector I could have the normal acceleration, which I need to find the radius of curvature.

I've seen the wolfram page, but I don't want to learn a new formula, I think it can be done with some geometry.

Thank you tim.

 

[Telemachus]

Normal acceleration in polar coordinates

Homework Statement

Well, I've already created a topic with this inquietude here: << Mentor note -- threads merged >>

But as I couldn't find the answer I'm looking for, I thought that maybe on this section I could find some help.

The thing is I believe there is a way to fin the normal acceleration in a trajectory given in polar coordinates with using some algebra. But I've been trying. I think I must use the vector of velocity, which is tangent to the trajectory, but it confuses me, cause I'm not on the Cartesian plane. So I wanted some tips or advices from someone with more experience.

I'll quote here the original problem:

Homework Statement

I've got this problem on polar coordinates which says:

A particle moves along a plane trajectory on such a way that its polar coordinates are the next given functions of time:

$$r=0.833t^3+5t$$ $$\theta=0.3t^2$$

Determine the module of the speed and acceleration vectors for this particle and its radius of curvature at the instant $$t=2s$$.

And I don't know how to solve the radius of curvature part.
For intrinsic coordinates I know that: $$\rho=\displaystyle\frac{v^2}{a_n}$$

Where $$a_n$$ is the normal acceleration. Now how do I find the radius of curvature? do I have to take the trajectory to the intrinsic form?

For the first part I have that:
$$\dot r=v(t)=\dot r\vec{e_r}+r\dot \theta\vec{e_{\theta}}$$
$$v(2s)\approx{}10\hat{e_r}+20\hat{e_{\theta}}$$

$$\ddot r=a(t)=(\ddot r-r\dot \theta^2)\vec{e_r}+(r\ddot \theta+2 \dot r \dot \theta)\vec{e_{\theta}}$$
$$a(2s)\approx{}-10\hat{e_r}+46\hat{e_{\theta}}$$

From there I can get the modules, but I don't worry about that, I want to know how to find the radius of curvature at the point.

Bye there, and thanks for posting.

I've tried making the projections, but couldn't find the way, cause I get a vector, but its in the polar form, and that confuses me. I've got the speed in the polar form, and I've found the acceleration for the point in question too. So I need to project that acceleration on the line perpendicular to the speed vector. I need some help please.

Here I made a plot of the trajectory and the versors for the different coordinate systems:

So, I have the acceleration in terms of the red versors, and I want it with the green versors, particularly the acceleration projected over $$e_n$$

I hope the moderators don't get angry cause I've already posted this in the other section, I really need help with this.

Bye there!

 

[berkeman]

I hope the moderators don't get angry cause I've already posted this in the other section, I really need help with this.

What you should have done is click the "Report" button on your post in Intro Physics, and ask the Mentors to move your thread to Advanced Physics. Multiple posting is not allowed here.

I've merged the two threads for now.

 

[Telemachus]

Ok. Thanks, and sorry.