- #1
Frobar
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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!
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!