Notes on drag and viscosity for the experimental study



Notes for the experimental study of the air-damped motion of a mass on a spring using a ULI system

Basic Theory

The problem of the motion of a mass on a spring is well worked in the elementary math and physics literature1. For a mass m on a perfect massless spring with spring constant k, equilibrium position x=0, and in the absence of drag and friction, the differential equation is

[pic] and the solution is [pic]. (1.1)

Damping corrections and estimations

When velocity-dependent damping is added, the problem can become much more complex. One simple behavior that is also treated in elementary textbooks1 is when the damping force is linear in velocity [pic]. (We will review below under what circumstances this is a good assumption.) The differential equation becomes:

[pic] (1.2)

whose solution has the form, [pic](1.3)

where [pic]. So for simple linear damping we expect the amplitude to decay exponentially in time. The rate of decay depends on 1/m. There is also a (usually small) shift in the natural frequency.

There is a large literature on drag and frictional forces acting on a body moving through a viscous liquid. There are two important limits of behavior, low velocity and high velocity, corresponding to whether turbulent flow is present. The complexities of the system are hidden in a fudge factor known as the coefficient of drag [pic]. The drag force is thus given by

[pic] (1.4)

where A is the cross-sectional area of the object,[pic] is the mass density of the fluid, and [pic] is the velocity of the object in the fluid.

An empirical measure of low and high flow is the Reynolds number, defined as

[pic], (1.5)

where [pic]represents a characteristic length of the object and [pic] is the viscosity of the fluid.

For very low Reynolds numbers (Re ................
................

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