#3096 G496 - Pratical Investigation Sherman Ip
G496 Researching Physics: Practical Investigation
Investigating damping due to liquid resistance
Sherman Ip
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Investigation 1
This investigation investigates the exponential decay of the amplitude of an oscillating system due to damping from water resistance.
Analysis
An oscillating system under resistance will dampen due to energy being transferred to the surroundings, this leads to the amplitude of the oscillating system decreasing over time. By plotting a graph, displacement of the oscillating system against time, there should be a gradual decrease of the amplitude over time due to damping.
A pendulum can be used to show this because it undergoes simple harmonic motion as a result of a continuous transfer of energy from kinetic energy to gravitational potential energy and back again as it oscillates. However not all of the energy is transferred to one form to another as some of the energy is transferred to the surroundings; in the pendulum case, some of the energy is transferred to the air as a result from collisions with air particles and transferring the kinetic energy to the air particles, therefore the pendulum has less energy and decreases its amplitude.
From this, it is suggested that the amplitude of the pendulum will decrease exponentially because the number of collisions with air particles (ie amount of energy transferred from the oscillator to the air particles) is proportional to the amplitude of the oscillator (ie distance travelled in the air). So the bigger the amplitude of the pendulum is, the more energy of its energy is transferred to the air particles.
Measuring the amplitude of the pendulum is difficult because the length from the pendulum's bob to the equilibrium position is the displacement of the pendulum and cannot be measured accurately using an everyday straight ruler. Instead the angular amplitude of the pendulum during oscillation can be used to model the exponential decrease of the amplitude because the angular displacement is almost directly proportional to the linear displacement of the pendulum.
θ x∝sin
⇒ A∝sin
where is the angular amplitude
∴ A∝ ⇐sin ≈for small x
Because the angular displacement is almost directly proportional to the linear displacement of the pendulum, this means the angular amplitude during the oscillation is also almost directly proportional to the linear amplitude. Therefore the angular amplitude during oscillation will also decay exponentially. From this we can develop equations and theories to model this.
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∝ A
− − ∝e kt ⇐ A∝e kt
⇒= −kt ⇐ 0 e where 0 is the initial angular amplitude = −kt e 0 =− ln kt 0 − =− ln ln 0 kt
∴ =− ⇐ = ln kt ln 0 in the form y mx c
So to prove that the amplitude of the pendulum decreases exponentially over time, plotting a graph of =− ln kt ln 0 should make a straight line graph with a negative gradient, where the y-intercept is the natural logarithm of the initial angular amplitude and the value of the gradient is the constant damping coefficient which is dependent on the type of damping or resistance the pendulum undergoes, for example air resistance and water resistance. The value of the damping coefficient can be used to compare the magnitude of damping of different types of resistance.
Experiment plan
The angular displacement of the pendulum can be measured electronically using a data logger, the pendulum is attached to the data logger and it gives readings of the angular displacement of the pendulum at a very high frequency. The data from this can be used to plot a graph of the angular displacement over time, the graph should have a periodic curve because the angular displacement oscillates from one value to the other. From the graph, the value of α can be found by finding the value of the angular displacement at each turning point of the graph because this is were the angular velocity is zero and so where the angular amplitude occurs as it is a simple harmonic motion d = ⇒ = oscillator, ie when 0 at time t. By using the values of α and t, the graph of dt =− ln kt ln 0 can be plotted to find the damping coefficient k, ie the gradient of the line.
In this experiment, the value of k in water and air resistance is to be investigated and found.
Equipment
• 30cm string
• Data logger
• Water tank
• Mass (50g)
• Clamps
Method
1. Attach the string to the data logger so it is loosely hinged at the pivot
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2. Attach the data logger to the clamp
3. Attach mass to the end of the string so you have made a pendulum
4. Submerge the mass in the environment
◦ Inside a water tank for water resistance
◦ Inside a room for air resistance
5. Pull the pendulum back 30 degrees (simple harmonic motion only works for small amplitudes of oscillation)
6. Get the data logger to record and release the pendulum
Variables
Independent variables :
• Time, this is measured using a data logger
• Types of resistance; water and air resistance
Dependent variables :
• Angular displacement, this can be measured in degrees (calculus and trigonometric techniques won't be used in this investigation so it is safe to use degrees) by using a data logger. Because it is measured using a data logger, there is very little uncertainty unless the value is very small. The angular displacement is taken at a sample of 10Hz. For a more reliable result, the investigation should be carried out 5 times on the pendulum in a chosen environment and the most stable curve is then chosen to be analysed in the calculations section and published in the results section.
Control variables :
• Volume of water
• Temperature of the water
• The water is stationary and has no waves
• Depth the weights are submerged in
• Length of string, 30cm
• Mass, 50g
Safety and hazards
• The tank of water may spill and could cause slipping, ensure the tank of water is secured and not near the desk edge. Keep people away from the water tank.
• The clamps may tilt and fall over because weight of the mass will cause weight distribution to be uneven, ensure the clamps are secured using G-clamps to clamp the clamps down.
• The pendulum will travel and might hit an obstacle, ensure the path of the pendulum is clear from obstacle other than the desired resistance.
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Water resistance Air resistance
Data logger Data logger
30cm string 30cm string
Mass Mass Air Glass tank of water
G-clamp
Clamp
Clamp
G-clamp
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Results
Pendulum under air resistance
30
25
20
15 )
s 10 e e r g e
d 5 (
t n e
m 0 e c a l p
s -5 i d
r a l
u -10 g n A -15
-20
-25
-30 0 10 20 30 40 50 60
Time (seconds)
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Pendulum under water resistance
30
25
20
15 )
s 10 e e r g e
d 5 (
t n e
m 0 e c a l p
s -5 i d
r a l
u -10 g n A -15
-20
-25
-30 0 10 20 30 40 50 60
Time (seconds)
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Calculations
From the graphs, we can see the natural angular amplitude is about 20 degrees and not 30 degrees because the curve of the pendulum under air resistance becomes stable when the displacement is -20 degrees at t=1, therefore we assume the biggest angular amplitude at the start of the experiment is 20 degrees, this shall be the same for water = resistance since the same pendulum was used for both experiments, ie 0 20 and time t starts at that value.
One problem with the graphs above is that time t does not start when the pendulum started oscillating but instead started when the data logger started its stop-clock and so included the reaction time of the data logger starting its stop-clock till the time the oscillation started; this causes a systematic error because the actual value of t is smaller. The reaction time in both experiments in 0.6 seconds so subtracting 0.6 to the values of t will solve this problem.
Air resistance
There are too many turning points to calculate so a sample of 10 turning points will be used for the calculations which are selected using systematic sampling. t α ln α 5.5 19.5° ±0.4° 2.97 ±0.02 11.0 18.9° ±0.4° 2.94 ±0.02 (The uncertainty is defined by the range of the value of α plus the 15.8 16.5° ±0.4° 2.81 ±0.02 resolution of the data logger) 21.9 16.1° ±0.4° 2.78 ±0.03 26.8 15.7° ±0.4° 2.76 ±0.03 32.9 15.1° ±0.4° 2.71 ±0.03 (The uncertainty of the natural logarithm is found by subtracting 38.3 13.9° ±0.4° 2.63 ±0.03 the natural logarithm of the mean value by the natural logarithm 43.8 12.1° ±0.4° 2.49 ±0.03 of the lowest value) 49.2 12.2° ±0.4° 2.50 ±0.03 54.7 11.0° ±0.4° 2.40 ±0.04
Water resistance
It is very hard to judge what is the natural maximum angular displacement because the oscillation in never stable and = the damping is very heavy. As mention before, assume 0 20 because the same pendulum was used in both experiments.
t α ln(αmax) 0.6 2.8° ±0.4° 1.03 ±0.15 1.1 0.1° ±0.1° -2.30 ±∞ As before the uncertainty is worked out the same way as the previous experiment with air resistance.
The uncertainty for α at t=1.1 is 100% of the value due to the value of α being as big as the resolution of the data logger and so cannot to used to plot a graph. This leaves a problem of not having enough plots to plot a graph for = water resistance, however the y intercept is ln 0 ln 20 as we have assumed the natural angular amplitude is 20 degrees.
Unfortunately this assumption is not valid because we do not know the true value of α₀, we have only assumed the value of α₀. The value of α₀ should have been a smaller value, ie the initial angular amplitude, as simple harmonic motion works best with small amplitudes.
The assumption and invalidity is carried forward to allow the process of this investigation to go on.
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Angular amplitude over time (Air resistance)
3.5
3.0
2.5
) 2.0 s e e r g e d (
α 1.5 n l
1.0
0.5
0.0 0 5 10 15 20 25 30 35 40 45 50 55 60
Time (seconds)
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Angular amplitude over time (Water resistance)
3.5
3.0
2.5
) 2.0 s e e r g e d (
α 1.5 n l
1.0
0.5
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Time (seconds)
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Conclusion =− The graph of ln kt ln 0 was plotted successful and the value of k was found for each of the resistance. This shows that damping will have a exponential decay on the amplitude of the oscillator.
The value of k is as followed:
• Air :
• Water :
The value of the damping coefficient for water resistance is about times bigger than air.
Evaluation
The investigation went as plan however there were two main problems:
• The initial angular amplitude was too big to create a valid simple harmonic motion oscillator. It was fine for air resistance as the oscillator managed to become stable after a while but for water resistance, the amount of damping was too great to allow the oscillator to become stable. This lead to an invalidity of the graph for water resistance because the value α₀ was only assumed. To improve this experiment, the oscillator should of have an initial amplitude of less than 20 degrees.
• The uncertainty is underestimated because the line of best fit, in the graph for air resistance, did not go through all the uncertainty bars. It was still valid because the plots do have a trend to make a straight line of best fit however it does suggest that the uncertainty is too small.
Investigation 2
This investigation investigates how the exponential decay of the amplitude of an oscillating system due to the damping is affected by the viscosity of a liquid.
Analysis
The term viscosity is how much friction there is between the liquid particles and hence the general term 'thickness', its units are N.s.m¯² or Pa.s and depends on the temperature; the hotter the liquid the less viscosity it has.
It is suggested that the viscosity of a liquid is directly proportional to the decay constant of the amplitude of an oscillating system. This is because the friction between the liquid particles is less in a liquid with little viscosity and so less energy is transferred from the simple harmonic motion oscillator to the liquid particles. Another way to explain it is that the liquid particles can move around easier in a liquid with less viscosity liquid and so require less energy to move them out of the way to allow the simple harmonic motion oscillator to continue its path.
Also if a liquid has completely no viscosity, this means very little energy is required to move the liquid particles out of the way and so very little energy will be transferred from the oscillator to the liquid particles which leads to a very small value of the damping coefficient k. This supports the suggestion of the direct proportionality between the decay constant k and the viscosity of the liquid because it means a graph of k over viscosity should go through the origin and have a uniform gradient.
Experiment plan
By submerging a pendulum under different temperatures of water, there should be a change of the damping coefficient k because the viscosity of the water depends on the temperature of the water. If the damping coefficient k depends on the viscosity of the water, plotting the damping coefficient k against the viscosity of the liquid should have a correlation. If the graph goes through the origin as well, the damping coefficient k and the viscosity is directly proportional.
From the previous investigation, it was found that the biggest natural angular amplitude of a pendulum with 30cm
11 #3096 G496 - Pratical Investigation Sherman Ip length was about 20°, therefore in this investigation the initial angular amplitude will be 15° to ensure a small amplitude for a simple harmonic motion oscillator.
To find the decay constant k, the mean value of α at time t is to be found for each turning point of the graph of the angular displacement of the pendulum over time using a data logger. This would produce a range of the value of k; the mean value of k with its uncertainty is then plotted on the graph with its corresponding viscosity.
Equipment
• 30cm string
• Data logger
• Water tank
• Mass (50g)
• Clamps
• Kettle
• Thermometer
• Rubber gloves
Method
1. Attach the string to the data logger so it is loosely hinged at the pivot
2. Attach the data logger to the clamp
3. Attach mass to the end of the string so you have made a pendulum
4. Heat up the water in the kettle to 100°C
5. Put on the rubber gloves
6. Pour the boiling water in the water tank, ensure the tank is full enough so that the pendulum's bob is completely submerged when oscillating
7. Submerge the mass in the water
8. Pull the pendulum back 15°
9. Get the data logger to record and release the pendulum
10. Repeat from step 8, 3 times
11. Select the most stable graph or data set
12. Wait till the water has cooled down 10°C and go to step 7 till the water has reached a stable temperature and cannot cool down further.
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Data logger
30cm string Themometer
Kettle Mass
Glass tank of hot water
Clamp
G-clamp
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Variables
Independent variables :
• Time, this is measured automatically by the data logger
• Temperature of water, this is recorded using a thermometer with range up to 100°C. It has resolution of 1°C but however the uncertainty shall not be recorded because it is an independent variable.
• Viscosity of the water, the temperature of the water is used to work out the viscosity of the water using an engineering reference book. Because other scientist have developed the engineering reference book, we assume it has no uncertainty.
Dependent variables :
• Angular displacement
• k, the damping coefficient
Control variables :
• Volume of water
• The water is stationary and has no waves
• Depth the weights are submerged in
• Length of string, 30cm
• Mass, 50g
By having these control variables controlled, it allows the pendulum to naturally dampen exponentially and reduce the uncertainty of the damping constant k.
Safety and hazards
• The tank of water may be spilled and could cause slipping, ensure the tank of water is secured and not near the desk edge. Keep people away from the water tank.
• The clamps may tilt and fall over because weight of the mass will cause weight distribution to be uneven, ensure the clamps are secured using G-clamps to clamp the clamps down.
• The pendulum will travel and might hit an obstacle, ensure the path of the pendulum is clear from obstacle other than the desired resistance.
• The water is VERY hot, ensure rubbers gloves are ALWAYS on.
• Glass can crack when under a a big change of temperature, DO NOT pour boiling water in the tank at a fast rate and allow the glass water tank to heat up before pouring more boiling water into it.
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Results Pendulum in 70°C water
20
15 ) s
e 10 e r g e d (
t n e m
e 5 70 C c a l p s i d
r a l u
g 0 n A 0 1 2 3 4 5
-5
-10
Time (seconds)
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Pendulum in 60°C water
20
15 ) s
e 10 e r g e d (
t n e m
e 5 60 C c a l p s i d
r a l u
g 0 n
A 0 1 2 3 4 5
-5
-10
Time (seconds)
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Pendulum in 50°C water
20
15 ) s
e 10 e r g e d (
t n e m
e 5 50 C c a l p s i d
r a l u
g 0 n
A 0 1 2 3 4 5
-5
-10
Time (seconds)
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Pendulum in 40°C water
20
15 ) s
e 10 e r g e d (
t n e m
e 5 40 C c a l p s i d
r a l u
g 0 n
A 0 1 2 3 4 5
-5
-10
Time (seconds)
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Pendulum in 30°C water
20
15 ) s
e 10 e r g e d (
t n e m
e 5 30 C c a l p s i d
r a l u
g 0 n
A 0 1 2 3 4 5
-5
-10
Time (seconds)
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Pendulum in 20°C water
20
15 ) s e
e 10 r g e d (
t n e m
e 5 20 C c a l p s i d
r a l u
g 0 n
A 0 1 2 3 4 5
-5
-10
Time (seconds)
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From the graphs of angular displacement against time, there were a lot of turning points in a big turning point like these:
These small turning points was probably caused by the pendulum making a small wave in the water when oscillating and the waves reflect off the walls of the water tank and interfere with the pendulum.
The graphs are rectified using pencil by drawing a smooth turning point over each main turning point. A cross has been drawn to show the coordinates of α. In the calculations, the two points of α will be used from the raw graph and the rectified graph separately to see if the rectified graph has rectified any calculations.
Calculations =− From the previous investigation, the equation ln kt ln 0 was derived, where α is the angular amplitude, α₀ is the initial angular amplitude which is 15 ° in this investigation, k is a constant and t is time. k can be found by rearranging the equation. =− ln kt ln 0 − =− ln ln 0 kt − = ln 0 ln kt 0 = ln kt
0 ln ∴ k= t
The average value of α and time is measured at each turning point of the graph, giving us a range of k and range of it.
TABLE OF CALCULATIONS
The uncertainty is the standard deviation of the data values because only a sample of data is being used as a result of the outliers being removed.
In these data values, assume all values of k using the first turning points are outliers, highlighted red, because there is a trend of all values of k using the first turning points being the biggest value and different magnitude, suggesting a systematic error. The outliers are ignored and are not used in further calculations.
For 30°C and 20°C, the 4th turning point cannot be recorded or used because the angular amplitude is very small and would cause too much uncertainty.
Values are rounded up to 2 decimal places, using a fixed amount of significant figures isn't suitable here as it loses precision of the values of k and they are around the same magnitude.
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70°C Raw graph Rectified graph Turning point Time (seconds) α (degrees) k α (degrees) k 1 0.62 6.5 1.35 7.5 1.12 2 1.40 4.0 0.94 5.0 0.78 3 2.15 1.8 0.99 3.0 0.75 4 2.85 1.1 0.92 2.0 0.71
60°C Raw graph Rectified graph Turning point Time (seconds) α (degrees) k α (degrees) k 1 0.60 6.3 1.45 7.8 1.09 2 1.40 3.8 0.98 5.2 0.76 3 2.15 2.0 0.94 3.0 0.75 4 2.90 1.0 0.93 2.0 0.69
50°C Raw graph Rectified graph Turning point Time (seconds) α (degrees) k α (degrees) k 1 0.60 7.0 1.27 7.8 1.09 2 1.40 3.6 1.02 4.9 0.80 3 2.10 2.6 0.83 3.3 0.72 4 2.95 0.7 1.04 1.5 0.78
40°C Raw graph Rectified graph Turning point Time (seconds) α (degrees) k α (degrees) k 1 0.60 7.8 1.09 8.0 1.05 2 1.35 4.1 0.96 5.0 0.81 3 2.20 2.0 0.92 2.9 0.75 4 2.90 0.7 1.06 1.5 0.79
30°C Raw graph Rectified graph Turning point Time (seconds) α (degrees) k α (degrees) k 1 0.65 6.9 1.19 7.5 1.07 2 1.40 3.0 1.15 4.0 0.94 3 2.15 2.0 0.94 2.0 0.94
20°C Raw graph Rectified graph Turning point Time (seconds) α (degrees) k α (degrees) k 1 0.65 5.9 1.44 6.0 1.41 2 1.50 2.2 1.28 3.0 1.07 3 2.15 1.7 1.01 2.2 0.89
Raw graph Rectified graph Temp (°C) Viscosity (N.s.m¯²) Mean k Uncertainty Mean k Uncertainty 70 0.40 0.95 ±0.04 0.75 ±0.04 60 0.47 0.95 ±0.03 0.73 ±0.04 50 0.55 0.96 ±0.12 0.77 ±0.04 40 0.65 0.98 ±0.07 0.78 ±0.03 30 0.80 1.05 ±0.15 0.83 ±0.16 20 1.00 1.15 ±0.19 0.98 ±0.13
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k aganist viscosity (Using the raw graph)
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0 k 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Viscosity (N.s.m¯²)
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k aganist viscosity (Using the rectified graph)
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0 k 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Viscosity (N.s.m¯²)
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Conclusion
The graphs above show there is a correlation between the rate of damping and the viscosity of the liquid. A straight line with an uniform positive gradient shows that the damping coefficient of an oscillator is dependent on the viscosity of the liquid if the oscillator is submerged in a liquid.
A straight line with no gradient cannot be drawn which shows there must be a correlation between k and the viscosity of the water.
However the best line of fit does not go through the origin which means the decay constant k is not directly proportional to the viscosity of the liquid. This suggests that a lot of energy is still being transferred to the liquid particles, even if they have zero viscosity and move freely around, because the momentum of the pendulum is transferred to the liquid particles as it collides with each other and so transfer its energy to the liquid particles. Compare the value of k in air (0.01), a liquid with no viscosity still has a decay constant 7 times more than the decay constant in air.
Evaluation
The experiment was carried out very well using a smaller initial angular amplitude as it gave stable curves which values of α was easy to read off the graph. The values of k in water were much more consistent compared to the value of k in Investigation 1; the values of k in water in this investigation were in the magnitude of 1 but the value of k in Investigation 1 was in the magnitude of 3, 3 times the value!
As mentioned before, in the table of calculations there seems to be a trend that the value of k using the 1st turning point to be the maximum value in the data set which suggests maybe a systematic error. One reason for this is that the pendulum is unstable at that point but becomes stable afterwards, this suggest that the initial angular displacement is still too big.
From the graphs of angular displacement against time, there were a lot of turning points in a big turning point and this was rectified by manually drawing a smoother turning point, comparing the graphs using the raw graph and the rectified graph, the uncertainty of k has decreased using the rectified graph. This suggest that the interference the pendulum is receiving is the main cause of uncertainty of the values of k, this could have been avoided if a bigger water tank was used to avoid interference of waves being created by the pendulum and could make reading values of turning points much easier, causing the readings of α to be more accurate.
The uncertainty of k increases as the viscosity of water increases for both graphs; this is because as the viscosity increases, the damping coefficient increases which causes a bigger decrease of the angular amplitude therefore increases the uncertainty of k as the angular amplitude approaches the resolution of the data logger.
The values of k using the rectified graph is lower than the values of k using the raw graph. This is because the values of α in the rectified graph is bigger than the values of α in the raw graph as a result of interference in the water tank which suggest the values of α is an underestimate because of the interference.
The experiment could have been improved if another liquid was used which has a bigger change of viscosity over temperature like olive oil, which has a bigger range of viscosity and therefore having a bigger spread of values of k. However using olive oil is very dangerous as it is extremely hazardous when it interacts with water.
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