- #1

cboystills

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## Homework Statement

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.

## Homework Equations

We are given that the curve fit for the drag coefficient is

*c*=0.45+30/

_{D}*Re*

^{0.886}, where

*c*is the drag coefficient and

_{D}*Re*is the Reynold's number.

The equation for the Reynold's number is

*Re*=

*VD/v*, where

*V*is the velocity of the sphere,

*D*is the diameter of the sphere, and

*v*is the kinematic viscosity of air.

We know that the drag force is

*F*=(

_{D}*c*)/2, where

_{D}ρV^{2}A*ρ*is the density of air, and

*A*=pi

*D*

^{2}/4.

## The Attempt at a Solution

Since the damping force relative to velocity is

*F*=-

_{D}*bV*, we then have:

*b*=(

*c*)/2.

_{D}ρVAwhere

*b*is 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.