# Damped pendulum and sliding rod

1. Feb 26, 2006

### Frobar

Problem 1:

I have a mathematical pendulum with a mass m connected to a string of length l. The pendulum is damped by air resistance that is proportional to the velocity, Ffric = -k*v. I need to derive the damping effect the air resistance has on the pendulum - that is, the decrease of the total energy of the energy it causes per unit time. The damping effect should be expressed as a function of k, l and a, the angular velocity ("omega-dot" in common notation).

I know that the speed of the point mass must be a*l, because the angular velocity is expressed in radians. The force on it from air resistance must then be -k*a*l. I thought that, because work is force*displacement, and effect is work/time, maybe I will get the effect if I take the force times the speed (displacement/time). In that case, the effect from air friction would be -k*(a*l)^2. Is this correct? What worries me is that I may not have understood the ramifications of potential and kinetic energy being constantly swapped in this system.

Is there some better form I could get the solution in given that the next problem is to derive the movement equation for the pendulum (a differential equation in theta) by comparing dE/dt to the result? E is the total energy.

Problem 2:

A rod is lying still on a frictionless surface when a force is applied to one end of it. As a result of this the rod will start to slide and rotate. I need to calculate what point of the rod will remain static for small angles of rotation. If I knew how to calculate the slide speed, I could probably derive the point with a little trig, but I'm uncertain how to get at it. The rotation can probably be had using T=I*alpha.

Any help appreciated!

2. Feb 26, 2006

### Hootenanny

Staff Emeritus
Problem 2: Just to clarify you want to calculate the point at which if a force is applied and the rod rotates slightly it will not be displaced?

3. Feb 26, 2006

### Hootenanny

Staff Emeritus
For problem 1 you dont need to consider the effect of the force on the individual types of of energy (potential or kinetic)only its effect on the total energy of the system.

4. Feb 26, 2006

### Frobar

> Problem 2: Just to clarify you want to calculate the point at which if a
> force is applied and the rod rotates slightly it will not be displaced?

Yes, that's the point I'm after.

> For problem 1 you dont need to consider the effect of the force on the
> individual types of of energy (potential or kinetic)only its effect on the
> total energy of the system.

But is my derivation of the effect of the force on the total energy correct?