Tangential velocity of an object rotating around a rotating axis

[physical angler]

TL;DR Summary

Fishing rod casting dynamics

Hi all,

It has been some time since I've done physics. I wish to model some projectile motion of a lure being cast from a fishing rod. The setup is very similar to that of a trebuchet.

The fishing rod - we'll assume a perfectly rigid beam - is rotating about a fixed axis. I can calculate the angular and tangential velocity of the tip of the rod.

From the rod tip hangs the line and lure - a sling arm and object to be projected.

As the rod rotates about it's fixed axis, the sling arm rotates about the moving rod tip. How do I model and evaluate the influence that the rod's tip has on the angular and tangential velocity of the object at the end of the sling arm?

 

[ergospherical]

I'm not sure exactly what model you have in mind, but some intuition might be gained by considering a system of two rods: the first with one end hinged about a fixed axis, and the second hinged to the other end of the first [i.e. essentially a double pendulum]. If the angles of either rod to the horizontal are denoted by $\varphi_1$ and $\varphi_2$ respectively, then the end of the second rod is at the position \begin{align*}
\mathbf{r} = \begin{pmatrix} l_1 \cos{\varphi_1} + l_2 \cos{\varphi_2} \\ l_1 \sin{\varphi_1} + l_2 \sin{\varphi_2} \end{pmatrix}
\end{align*}and has velocity\begin{align*}
\dot{\mathbf{r}} =\begin{pmatrix} -l_1 \dot{\varphi}_1 \sin{\varphi}_1 - l_2 \dot{\varphi}_2 \sin{\varphi_2} \\ l_1 \dot{\varphi}_1 \cos{\varphi_1} + l_2 \dot{\varphi}_2 \cos{\varphi_2} \end{pmatrix}
\end{align*}

 

[physical angler]

Thank you very much for this, double pendulum model would fit nicely.

This setup looks complex to me, like nothing I've seen before. Are these matrices or a different way to describe coordinates in the cartesian plane? Is l the length of the pendulum? If so , what is r if it is not radius?

My physics is limited to high school level in Canada, where we have what is called 11th grade university prep and 12th grade university prep physics. We covered kinematics, some dynamics, and some rotational mechanics, but nothing this complex. This is the extent of my instruction in the methods. My mathematics is also limited to 12th grade advanced functions (essentially pre-calc) and some single variable calculus in the 12th grade. Both of these were about 20 years ago. At best, I know that velocity is the derivative of position the equations you've provided.

Can you explain the symbol that you use to denote the angles of either rod to the horizontal? How is it different from what I'm used to θ (theta)? I'm sure I can understand with just some more labelling of the variables and layout. In the mean time, I will look into further instruction into double pendulum methods and brush up on my skills.

Again, thank you for your input.

 

[jrmichler]

The setup is very similar to that of a trebuchet.

Except that a trebuchet has a rigid rod. If you ever tried to cast a panfish lure using a musky rod, you found that it did not work at all. Rod flexibility must be matched to the mass of the lure for best casting action. So you are absolutely correct in your understanding of the problem.

I suggest starting with modeling a rigid rod, then adding flexibility. There is an excellent reference to the design of trebuchets, almost all of which is understandable by a person of your background. The web site: http://www.algobeautytreb.com/, and that site links to this document: http://www.algobeautytreb.com/trebmath356.pdf. This document has a November 2013 date on it. The document looks very similar to the original document dated March 2001. A simple trebuchet made with a one pound weight was able to throw a nickel to the far corner ceiling of my office.

If I was modeling a fishing rod, I would start with a trebuchet model, then add a massless spring to the tip of the rod. You could get a good estimate of the spring constant by pulling on the tip of a rod, and measuring the deflection.

It will be a fun project. I am almost tempted to try it myself. Almost.

 

[physical angler]

If I was modeling a fishing rod, I would start with a trebuchet model, then add a massless spring to the tip of the rod. You could get a good estimate of the spring constant by pulling on the tip of a rod, and measuring the deflection.

It will be a fun project. I am almost tempted to try it myself. Almost.

Awesome, thank you for your links and input!

 

[physical angler]

To those who’ve answered previously, and those who may be reading anew, I would like to simplify and clarify my goal here:

I would to develop a table or simple method of determining the initial velocity of the projectile motion of the object (the lure) at the point it is released from the rod depending on rod length and sling length.

I would also like to determine the optimal launch angle (ignoring the mathematics of wind resistance, but knowing that it has effect) and considering the launch is occurring from some height above ground level (rod plus human body).

I would also like to determine the optimal sling length for a given rod length.

My process will include obtaining a tangential velocity of the rod tip at the end of the cast experimentally and hopefully calculate the subsequent tangential velocity of the object at the end of the sling arm. The more I describe it, the more complex I understand it to be. Perhaps some can recommend ways to relax the conditions and simplify enough to make it more easily workable, at the very least simply to understand the effects that different relationships between rod length, sling arm, casting velocity, and release angle have? Am I chasing a unicorn?

I am at work right now, but I will have a play around with the trebuchet links provided above.