(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I need to develop an equation for the damping factor with respect to time for a pendulum that consists of a spherical mass attached to a string that is damped by air resistance.

2. Relevant equations

We are given that the curve fit for the drag coefficient isc=0.45+30/_{D}Re^{0.886}, wherecis the drag coefficient and_{D}Reis the Reynold's number.

The equation for the Reynold's number isRe=VD/v, whereVis the velocity of the sphere,Dis the diameter of the sphere, andvis the kinematic viscosity of air.

We know that the drag force isF=(_{D}c)/2, where_{D}ρV^{2}Aρis the density of air, andA=piD^{2}/4.

3. The attempt at a solution

Since the damping force relative to velocity isF=-_{D}bV, we then have:

b=(c)/2._{D}ρVA

wherebis the damping factor. Substituting our curve fit for the drag coefficient we have:

b=[(0.45+30/Re^{0.886})ρVA]/2.

Substituting the Reynold's number formula we have:

b=[(0.45+30/(VD/v)^{0.886}))ρVA]/2.

I am stuck after this point. I need to find the velocity with respect to time and substitute it into the above equation. However, the Reynold's number changes with respect to velocity. I'm not sure how to proceed. I would appreciate any advice.

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# Homework Help: Pendulum Damped by Air Resistance

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