Independent variables
I repeated the experiment of the metre ruler but this time I reduced the length (from the edge of the desk to itd end. I noticed it was faster than when I used the full length and the sound got larger. I reduced the length evn more and what I could conclude is that <the longer the length, the slower it will oscillate>.
I now used a steel blade (with a smaller thickness and width) and this was too fast and oscillated many times but and stopped in an unmeasurable time to me. This oscillated more than the ruler. Additonaly, I attached a mass at the one end of the both the steel blade and the ruler. For the same length (about), I noticed that the blade oscillated much longer than the ruler. So I could conclude that fot a same length, <the larger the thickness of a material, the quicker it will oscillate>, <the larger the mass attached to the material, the slower it will oscillate>, the smaller the width, the longer it will oscillate. All these in bold above will affect the period of oscillations. And so this will be considered here as independent variable since they won’t affect damping.
What might affect damping (dependant variable)
The decrease in displacement is caused to the lost in kinetic energy and potential enrgy. Energy is lost because of the resistance opposed by air molecules. And I know that the air resistance will increase or decrease depending of the size of the surface area in contact with air molecules. Damping depends them to the starting value of the amplitude. And I guess, if the initial maximum amplitude is small, then the oscillator is damped slowly. Oppositely, if it is large, it will experience a large damping. So the dependant variables are the maximum amplitude and the surface area.
Hypothesis (predictions)
The larger the surface area of the damping card, the more the collisions with air molecules, thus the more the damping that occurs.
Theory
Damped Harmonic Oscillator
Underdamped Oscillator
a is , it approaches zero faster than in the case of , but oscillates about that zero.
Critical Damping
Critical damping provides the quickest approach to zero amplitude for a . Critical damping occurs when the is equal to the of the oscillator.
Overdamped Oscillator
Overdamping of a will cause it to approach zero amplitude more slowly than for the case of . The is greater than the .
How to control
Our eyes cannot respond rapidly enough if the frequency of oscillation is more than about 5Hz (five oscillation per second). Anything faster than this appears as a blur. In order to be able to measure the damping, I needed to use suitable equipment to set a system that would oscillate slowly (Attaching a big mass at one end of the steel blade or ruler would store a lot of potential energy and so slows they oscillation down). This is where an adequate choice of the mass, thickness, length, width and even the material because the chosen material should not get deform when proceding the expeiment. The period of oscillations should be considerably larger so that the nuber of oscillations can be count.
The reaction time is very important. Reading the amplitude is a big deal here and the chosen method should allow the easy measurement of the amplitude.
Available apparatus
- A G-clamp,
- Wood anchor points,
- A metre ruler (accurate to 1mm),
- A large damping card board,
- A graph paper,
- Some blue tack,
- 2 stopwatchs (one accurate to 0.01s and the other to 0.2s)
- A rectangular wooden board,
- Slotted masses 50grams each,
- A steel blade,
- A pair of scissors,
- A big paper clips,
- A black (marker) pen,
- Rubber bands,
- A vernier caliper,
- A micrometre screw gauge.
Choice of equipment
Each apparatus should be tested and the way to use it should be defined and carefully process.
- I mesaured using a micrometer the thickness of the ruler and the one of the
steel blade and I obtained respectively……..and ……….Using the vernier caliper I measured their width and respectively obtained………and………. The steel blade has a smaller width and thickeness. Besides, it bends more easily than the ruler so I chose to use it.
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I chose the stopwatch accurate to 0.01s. However, reading the time on it wouldn’t be easy. When my partner gives the signal, I noticed the clock will be running and reading the decimal places would just be impossible. This would be terrible because the time would be measurable only to ±1s. By manipulating the clock, I noticed that while the clock was running, pressing the reset button stops the reading on screen and while the clock was still running. Pressing on it again just gets back to the running clock. Hence I would able to measure the time accurately and only my reaction time would matter.
- I used the ruler to measure the amplitude. I laid it down on the table under the system as shown below (system1). Reading the amplitude was not easy as it would be explained below in preliminary experiment. So I had to change the measuring apparatus; I then used the graph paper (system2).
- The wood anchor point has to be chosen causiously since they were some, which were not leveled up accurately. I chose one that was well leveled up.
Preliminary experiment (System1)
I did four sets of experiments.With no mass, 50g mass, 100g mass and 150g mass. We worked with and extra partner. I had to read the time.
- I set the apparatus as above.
- Chose as mid-point 25.0cm. So if the reading is r, the amplitude would be a = 25.0 - r.
- My partner displaced the end of the steel blade sideways just above the reading 14.0 for the set (without mass) and16.0cm (for the rest) then released it. So maximum amplitude was 9.0cm. Reading was taken on one end of the oscillations.
- My partner stood, looked at and read, in front (to avoid paralax error) of the device the amplitude on metre ruler scale. The extra partner had to write down the amplitude.
- My partner had to call the reading of amplitude as signal and I had to press the reset button on the stopwatch, record the reading and press back to the reset button again. At the same time the extra partner had to record reading of the amplitude.
- We changed the roles from a set of experiment to another.
- We ended up with these tables:
t1/2 t1/2
t1/2
t1/2
t1/2
Interpretation
We observed this and besides looking at the graph, we can clearly see that the gradient is large at the beginning and gets less steeper towards the end so the change in amplitude is fast at the beginning but slower towards the end. And this makes sense since the amplitude decreases with the amount of starting amplitude.
Modifications and improvement
- Reading of amplitude
The reading of the amplitude was generally bad since marks on the scale are too close together an the chances to read (human judgement) the wrong one were high. Using a graph paper stuck on the table (it would move if not), we can draw lines of fixed amplitudes. A maximum amplitude of 10.0cm to allow the steel blade to oscillate for a considerable time before we start recording measurements.
- Timing
I could not feel confident with the time since each signal was suddent, and so the reaction time of the timer high. We wanted reaction to be as small as possible so that our experiment would be reliable. To do that, when I had to record the time, my partner would say to me <get ready>, <get ready>, <it’s near>, <it’s near>, <almost there>, <NOW!>. I would have had enough time to concentrate and focus on starting and reseting the stopclock and so my reaction time would be very small. I had to the same for my partner when she had the turn to measure the time. To estimate the reaction time, we did a very simple test using a ruler. I vertically hold the lower end of the ruler (assumed to be 0cm). My parner hold a stopclock. She said: <get ready>, <steady>, <NOW!> and I released the ruler until she said:<STOP!>, and as soon as I heard that signal, I hold the the ruler back again while she stopped the clock. I did this test then we reversed the role and she did it too. For our experiment, we would used the average of reaction times.
- Cardboard
Since damping is affected to the surface area of the cardboard, the use of masses is not relevant for measuring damping but only to slow the oscillations down. So we used a 100g mass (2 sloted masses 50g each) and used cardboard of different surface areas.
- Only one half life (roughly) could be measured. The range of the measurements should be here a suitable number of measurable half lives so that an average value can be taken from them. A minimum of 3 measurable half lives should be obtained from first order exponential graphs so that an average can be calculated. To make this possible, we chose a maximum of 10.0cm and a minimum of 0.5cm in order to expect a least (10.0cm-5.0cm, 5.0cm-2.5cm, 2.5cm-1.25cm, 1.25cm-0.6cm) 3 half lives.
- We managed to work only 2.
Procedure (system2)
- We were provided with a large piece of damping card (cardboard) of uniform thickness, which we cut into different small rectangular pieces of length 10.0cm but of different widths. Same length for a fair test; we stuck that length along the steel blade using blue tack.
- We chose 6 different widths for the damping cards: 2.5cm, 4.5cm, 5.0cm, 6.5cm, 7.5cm, and 10.0cm. We cut the pieces of damping card using the pair of scissors and we calculated the correspondent surface areas and marked them on the pieces.
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From preliminary experiments, we noticed that the change in amplitude is fast at the beginning but slower towards the end. We then chose a larger range between points of higher amplitude and smaller range between points towards the rest position of the steel blade. The points we used were of amplitudes: 10.0cm, 7.5cm, 4.0cm, 3.0cm, 2.0cm, 1.5cm, 1.0cm, and 0.5cm.
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We marked these distances on the graph paper from a mid-point O representing each distance with a letter from A to I; A being the farthest and I the nearest from O.
- I unwound the big paper clip and bent it so it looked like this:
- I fixed the big deformed paper clip, together with a 2x 50g mass (100g is big so a lot of potential energy would be stored) onto the steel blade with aim of slowing down the oscillations.
- We set the rest of the apparatus as in the diagram above that is we fixed the one end of the steel blade tight into the wood anchor point and set it up across the table and not out of the table for health and safety raisons.
- We stuck the graph paper (under the steel blade) on the table using some blue tack and aligned it with the mid-point O (line) on the graph paper.
- By pulling the steel blade head from the position O and beyong the position A and releasing it, we were able to read measurements on each letter on the graph paper by lying our thin against the edge of the table and in front of the device. The eye aligned with the line of a out letter (depending on the position of the big pin of paper clip) to avoid paralax error. We proceed as mentioned in (modification and improvement) for the reading of the time.
- We stuck the cardboard each time on the steel blade, just behind the mass. We started with the smallest one and carried on in ascending amount of surface area. We recorded every reading on a table.
- At the end we cleared everything up. And we ended up with table:
I converted the time in second. 1minute is 60second so I multiplied the number of minutes by 60 and then added the number of seconds.
I plotted these graphs:
17.5 75.0 132.5 192.5 250.0
57.5s 57.5s 60.0s 57.5s
To find the average half life, I summed them all then divided by 4.
t1/2(average) = (57.5s + 57.5s+ 60.0s + 57.5s)/4 = 58.1s
The relation relating half life and decay constant is t1/2(average) = 0.693/λ so
58.1= 0.693/λ and then λ = 0.693/58.1s = 0.0119s-1
10 55 102.5 150 250
45 45 47.5 50
t1/2(average) = (45.0s + 47.5s + 47.5s + 50.0s)/4 = 47.5s
λ = 0.693/47.5s = 0.0146s-1
7.5 42.5 75 110 145
35s 32.5s 35s 35s
1/2(average) = (35.0s + 32.5s + 35.0s + 35.0s)/4 = 34.4s
λ = 0.693/34.4s = 0.0201s-1
12.5 50 87.5 124.5 162.5
37.5s 37.5s 37s 38s
t1/2(average) = (37.5s + 37.5s + 37.0s + 38.0s)/4 = 37.5s
λ = 0.693/37.5s = 0.0185s-1
5 29 53 76 100
24s 24s 23s 25s
t1/2(average) = (24.0s + 24.0s + 23.0s + 25.0s)/4 = 24.0s
λ = 0.693/24.0s = 0.0289s-1
0 24 46 68 94
24s 22s 22s 26s
t1/2(average) = (24.0s + 22.0s + 22.0s + 26.0s)/4 = 23.5s
λ = 0.693/23.5s = 0.0295s-1
Theory