Things You'll Need:
- Oscillating system you need the damping ratio for
- Scale (for mass measurement)
- Timer
- Ruler
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Step 1
Determine what you know. In an oscillating system, there are three primary quantities used to describe the differential equation: mass m, spring constant k and damping constant c. Each of these needs to be known to determine the damping ratio, which is given as: Z = c / (2*√(k*m)).
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Step 2
Measure the mass. Using your scale, weigh the object to get the mass in grams. If your scale measures in pounds, divide by 2.2 to convert pounds to kilograms (for increased accuracy, use 2.20462262).
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Step 3
Measure the frequency of oscillation. To determine k, we need the frequency of oscillation, or cycles per second. The best way to get that is to measure the period of oscillation, or how long it takes to return to the same position. Good practice will involve taking several measurements and averaging the results. Once you have a period (T) in seconds, invert it (1/T) to get the frequency w.
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Step 4
Convert frequency to wave number. In an oscillating system, the frequency is related to the mass and spring constant by the equation w²=k/m, or to rearrange terms, k=w²*m. Given the frequency and mass, you can compute the spring constant.
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Step 5
Calculate the logarithmic decrement d. This is the natural logarithm of the extent of two oscillations of the system. If your system is oscillating about a point x=0, using the ruler, measure the distance from 0 of two successive oscillations, x1 and x2. Take the natural logarithm of their ratio: d = ln(x1/x2). Do this a number of times to ensure you get an accurate result.
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Step 6
Calculate the damping ratio. If you know k, m and c, you can use the above equation Z = c / (2*√(k*m)). If you do not know c, however, you can use the logarithmic decrement d by the equation Z = d/√((2*pi)^2+d^2). Now you have the damping ratio.
