Damped Harmonic Motion of a Spring Patrick McAtee | Thermodynamics Thermodynamics | 11 November 2010
Introduction
A spring-mass system is one of the simplest systems s ystems in physics to both set-up and to describe mathematically, assuming assuming no friction f riction or outside forces. However, damping the spring-mass system shows an oscillation that is not as simple as an undamped case. Below, the damped situation of the spring-mass system is explored. Apparatus y
Spring with mass hanging from is vertically from a vertical bar raised in the air
y
Beaker with water
y
Beaker with shampoo water mixture
y
Stopwatch
Methods
The spring-mass system was set-up in a manner similar to that p icture below:
Source: http://www.radian.com.hk/alphyprac/expt%20graphics/a0301a.jpg
In one beaker, water was added to fill it up to about seventy-five percent capacity. Another beaker was filled up to the same capacity but with a water-shampoo water-shampoo mixture, where about half the mixture was shampoo.
For the first trial, the mass was pulled down a reasonable displacement and set to oscillate. The
time it took for the mass-spring system to t o complete twenty oscillations was measured on a stopwatch. The mass-spring mass-spring system was then oscillated in the water, with the time to complete twenty oscillations measured again. The same procedure was completed again w ith the system oscillating in the water-shampoo mixture. Results
Below are presented the periods for the mass-spring system. In trial 1, the spring is pulled down and let to oscillate in air. In I n trial 2, the mass-spring system s ystem oscillates in water. In trial 3, the mass-spring system oscillates in a shampoo-water s hampoo-water mixture. Trial 1
(no damping)
Trials
1
Periods (seconds) 13.71
2
14.5
3
13.3 13.84
Average
Trial 2 (damped in water)
Trials
1 3
14.3 15.4
Average
14.7
2
Trial 3
Period (seconds) 14.4
(damped in water-shampoo mixture)
Trials
1
Periods (seconds) 15.9
2
16.2
3
15.5
Average
15.87
Now using the basic equations of simple harmonic motion, the frequency of the oscillations will be determined using the period of the oscillations. The frequency of an oscillating system can be found by
Where T is the period and f is the frequency. Since 20 oscillations were allowed to occur, we can find the frequency from (for trial 1)
Then the frequency for each trial is Trial
Frequency
1 2 3
1.44 1.36 1.26
To find the b value (damping ratio) the equation is
[ [ [
where [ is the frequency of the damped oscillator and [ 0 is the frequency of the oscillator in liquid. M is simply mass. Calculating the b values y ields Trial
b-value (damping ratio) 1 0* 2 0.141986 3 0.209141 *although air acts like a fluid, the experiment was not run in a vacuum to get a better estimate
of an omega-nought value. Thus, since air can be presumed to have damped the spring little in either case, its frequency is used as omega-nought. Discussion
In this experiment, it was found that the liquid which one would presume to be the thickest, the shampoo-water mixture, was in fact the most viscous fluid because it had the highest calculated damping ratio. Shampoo, by itself, is quite a thick fluid, so it is obvious that when something tries to move through it, or oscillate through it, the shampoo will slow the motion of that object. Although water seems like it is thinner, or has less body than shampoo, it had a
comparable damping damping ratio to that of the shampoo. The densities of the fluids were not measured. Doing so would have allowed the finding of a correlation between density of the fluids and their damping ratio. From a theoretical standpoint, it would not matter how much the spring is compressed when
calculating the frequency. There is no (x term in any of the equations used. However, from a practical standpoint, standpoint, the less the spring the less time there will wil l be for the spring to ccomplete omplete twenty full oscillations. Since the spring is in fa ct damped, it must be pulled down far enough e nough that it completes twenty full, noticeable oscillations before it is damped. When the oscillations of a damped harmonic oscillator are graphed as the position vs. time, it will look something similar to this:
In this diagram, the damping ratio b will determine the function that represents the blue dotted line. From theory, it is known that the damped harmonic oscillator has an amplitude that will diminish exponentially with time. Thus, the amplitude is most likely related to the b factor in some manner such that
where A is the amplitude, c is some constant, b is the damping coefficient, and t is time. Then b could possibly be found as
The damping ratios found for all trials are most likely somewhat accurate, maybe within the range of twenty percent error. Errors will arise mostly around the timing the oscillations, the fact that the spring does not uniformly compress and decompress, and the movement of the spring in any way that is not just up and down. Most error is probably from the spring swaying back and forth. The math of the system is for two-dimensions when the spring obviously moves in three. Conclusion
Above it is shown how the damping ration can be calculated by simply measuring the pe riod of a mass-spring system oscillating twenty time. Although the values are not the most accurate that could have been found, they agree with common sense, but also the theory behind how a damped spring would work.
