Calculating Whip Speed: The Science Behind It [SOLVED]

[xJuggleboy]

[SOLVED] Speed of a whip

Does anyone know how to calculate the speed at the end of a whip? I know it goes faster than sound and the general ideia of how it does it. But I would like a formal explanation of how to caluclate it.

 

[rayjohn01]

Joking ?

The problem with this calculation is that the whole whip is moving , so whatever energy you imparted is distributed along it's whole length.
There is NO formula for this --- you would have to define all aspects of the whip at each point along it's length from mass to elasticity .
The only possible way is to create a simulation of described points using classical mechanics (ignoring air resistance ) and assume some propagation factor for the forces . GOOD LUCK .
Tis quite possible to ask the impossible question -- the art is to ask questions for which there is a possible answer.
Ray

 

[xJuggleboy]

Here is my bassic understanding of how the whip works. You have a mass at the beginning moving at a given speed. When the first end of the whip stops moving it causes the energy to ba transferd to the rest of the whip. So... While keeping the same amount of energy and the mass continualy going down (as the whip decreeses in diamiter and there is less of it moving). You have a increse in speed. THere must be a way to show this and even calculate it.

George

 

[pervect]

You can approximate the whip as a series of masses joined by strings, or even better by a series of jointed rods.

If you take the limit, you'll have a distributed system. The undistributed system could be represented by a Lagrangian with n variables - the distributed system will be represented by a Lagrangian density, in the limit as n-> infinity.

 

[xJuggleboy]

So how do you do that? Keep in mind I have not taken any higher level math cources.

 

[pervect]

So how do you do that? Keep in mind I have not taken any higher level math cources.

I'm afraid this is probably one of those "higher math" things - you need at least calculus.

There's a formulation of physics where all you need to solve a problem is to write down the Lagrangian. The Lagrangian is usually equal to the kinetic energy T, minus the potential energy, V - i.e. L = T-V. The Lagrangian is writtten down as a function of generalized coordinates, q, and genearilzed velocities, q' = dq/dt.

You can then write down the differential equations of motion as

$$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}}) = \frac{\partial L}{\partial q}$$

This is a much more mechanical procedure than writing down all the forces - all you need is the Lagrangian, and the equations of motion just pop out.

The Lagrangian density approach is similar, except that instead of ordinary differential equations, you get partial differential equations.

There's some more detial in "Classical Mechanics" by Goldstein on pg 548, including writing down the equivalent of Lagrange's equation for a Lagrangian density.

Google also finds

http://math.arizona.edu/~ura/031/Taft.Jefferson/Report.pdf

 

[xJuggleboy]

That web sit was EXTREMLY helpfull. Thank you very much

 

[Dr.D]

There is a problem with the work of Jefferson Taft referenced in a previous comment and particularly the application of the Lagrange equations of motion to this problem.

1. The Lagrange equations were incorrectly applied in that the energy expressions were not correctly written, thereby leading to an incorrect formulation of the Lagrangian function.

2. The Lagrange formulation is not applicable to this process. When we consider what happens in the bend where the direction of the whip changes, there are processes involved that are not modeled at all but that are essential to the operation taking place. If the whip is a continuum, such as a thread, a string, a rope, etc., then there is some amount of bending stiffness and some internal friction at work in this bend. If the whip is discrete, such as a chain, a ball chain, etc. then there is friction and impact happening in this bend that is not modeled but that are essential to a full understanding of the process. The Lagrange equations are not the way to approach this problem.