E1 Guided Report_Edwin_s Report

8/13/2019 E1 Guided Report_Edwin_s Report

1/17

Page 1

PAP 118 Physics Lab 1aExperiment 1

Motion in a Viscous Medium

Name of Student : EDWIN ANG CHING JITTName of Partner : CHENG SHENG DA JOWELLDate of Experiment: 24-AUG-2011

Date of Report : 27-AUG-2011

Aim

The aim of this experiment is to measure the terminal velocity of spherical beads falling

under gravity in a liquid, and hence determine its viscosity using Stokes law.

Int roduct ion

When a stationary solid object is complete or partially immersed in a fluid, it experiences

an upthrust or buoyant force. According to Archimedes principle, this buoyant forceBis given by

,gVB s

where is the density of the fluid, Vsis the immersed volume of the solid object, andgis

the acceleration due to gravity. As its name implies, this force acting on the solid object

by the fluid is always directed upwards.If the solid object now moves through the fluid, it will have to push the fluid out

of the way. By Newtons third law, the fluid pushes back on the object with an equally

strong reaction in the opposite direction. This is experienced by the object as fluidresistancef to its motion. Depending on the speed vof the solid object, as well as the

nature of the fluid, this fluid resistance can be proportional to the speed, i.e. f~ v(skin

drag or viscous drag), or proportional to the square of the speed, i.e. f ~ v2(form drag or

inertial drag). Viscous drag is the dominant fluid resistance at low Reynolds numbers,

whereas inertial drag is the dominant fluid resistance at high Reynolds numbers. The

Reynolds number

vLRe

is a dimensionless ratio of inertial forces to viscous forces. Here, is the density of the

fluid, vis the typical speed of the fluid flow, Lis the typical distance the fluid has to flow

around, and is the viscosity of the fluid.


2/17

Page 2

For a sphere of radius rmoving with speed vin an infinite fluid with viscosity ,

the viscous drag has been worked out by Sir George Stokes in 1851 to be

,6 vrf (1)

if the fluid at the surface of the sphere is always at rest with respect to the sphere. This

has since come to be known as Stokes law. In this experiment, spherical beads aredropped into a highly viscous liquid detergent and allowed to reach terminal velocity vT.

At terminal velocity, the three forces acting on the bead (see Figure 1) balance each other,

and we have

,3

46 3 grvrBfmg T

which can be rewritten as

.6

3

4 3

Trv

grmg

(2)

Through careful measurements of m, the mass of the bead, r, the radius of the bead, and

vT, the terminal velocity of the bead, this equation can be used to determine , theviscosity of the liquid detergent.

Figure 1. Freebody diagram of spherical bead falling under gravity through a viscous fluid. Besides its

weight W= mg, the bead is also acted upon by the buoyant forceB, and the drag forcef.

Experimental

Preliminary Observations

In the first part of the experiment, we weighed beads of 3 different sizes (B1, B2 and B3)

immersed in a beaker of liquid detergent, when it is (a) suspended on a thread; (b) resting

on the bottom of the beaker; (c) suspended on the same thread as in (a); (d) pulledupwards slowly; and (e) allowed to fall slowly. These are recorded as (a) R1, (b) R2, (c)

R3, (d) R4 and (e) R5 respectively, where R1 to R5 represents the readings on the

electronic balance for each of the 5 cases from (a) to (e) respectively. Before each

reading is taken, the beaker of liquid detergent is allowed to settle on the electronicbalance without the bead until a steady reading is observed; then, the electronic balance is

tared for calibration. Also, the side window of the casing on top of the electronic balance

is closed before any reading is taken to minimize the effect of moving air on the accuracyof the readings by applying undue downward force on the balance, thus, introducing new

variables that might add on to the true readings.


3/17

Page 3

Before recording R1, R2 and R3, we took care to allow the bead to come to rest, and the

reading on the electronic balance to stabilize. While recording R4 and R5, we noted the

extent of the fluctuations in the electronic balance readings, and also any increasing ordecreasing trends in the average reading. Consider 5 readings from R1 to R5 respectively

to be one complete set of readings, 3 sets of readings (i.e. 1st, 2

nd, 3

rdreading) are taken

for bead of each size. For reading R1, the average from the 3 sets of readings arecalculated; then, the averages of R2 to R5 are also calculated. The list of averages fromR1 to R5 are tabulated as the 4

thset of reading. The 4

threading for beads B1, B2, and B3

respectively are taken as the data to support the explanation of the physical forces present

for all cases (a) to (e).

Measurement of Terminal Velocity

To measure the terminal velocities of beads falling through the Mama Lemon liquid

detergent, we set up the experiment as shown in Figure 2. First we ensured that the

column is vertical, by viewing it from three different angles. Then we dropped five beads

of each size, each as close to the centre of the column as possible. If a bead drifted tooclose to the wall of the column, the trial was rejected, and another bead of the same size

was dropped again to replace the trial.

Figure 2. Experimental setup to measure the terminal velocity of a bead falling through a liquid detergent.

For each accepted trial, we recorded the stopwatch times t when the bottom of the bead

reached each 2-cm mark x on the metric taped stuck to the column. We then plotted agraph of x against t for each accepted trial. Fitting the graph for large times to a straight

line, we determined the terminal velocity vT for a trial as the slope of the straight line.

For each bead size, we averaged vTover the five trials, before we use Equation (2) todetermine the viscosity of the liquid detergent.


4/17

Page 4

Resul ts & Discussions

Preliminary Observations

Small Bead (B1)

Case (1stReading)/g0.001

(2n

Reading)/g0.001

(3r

Reading)/g0.001

(Mean Reading)/g0.001

Compare

R1 0.079 0.078 0.078 0.078 ~Same

R2 0.067 0.068 0.069 0.068

R3 0.080 0.081 0.079 0.080 ~Same

R4 0.022 0.019 0.027 0.023 Lowest

R5 0.086 0.084 0.083 0.084 Highest

Medium Bead (B2)

Case (1stReading)/g0.001

(2n Reading)/g0.001

(3r Reading)/g0.001

(Mean Reading)/g0.001

Compare

R1 0.138 0.123 0.130 0.130 Same1R2 0.191 0.189 0.188 0.189 ~Highest

R3 0.134 0.129 0.127 0.130 Same1R4 0.000 0.016 0.042 0.019 Lowest

R5 0.190 0.189 0.191 0.190 ~Highest

Large Bead (B3)

Case (1stReading)/g

0.001

(2n

Reading)/g

0.001

(3r

Reading)/g

0.001

(Mean Reading)/g

0.001

Compare

R1 0.702 0.703 0.704 0.703 ~Same

R2 0.920 0.924 0.923 0.922

R3 0.700 0.702 0.701 0.701 ~Same

R4 0.470 0.599 0.522 0.530 Lowest

R5 0.946 0.949 0.935 0.943 Highest

Firstly, a beaker of detergent is placed on an electronic balance. Then, the balance is tared

so that the effect of the weight of beaker is not reflected in the reading and the change isreading is solely due to the cases R1 to R5 respectively.

The small bead (B1) was first used to test for cases R1 to R5. However, the data in R2

showed a contradiction to the predicted result from the analysis of physical forces.


5/17

Page 5

Analysis of Physical Forces

R1/R3In figure 3, as the bead is in equilibrium, there

should be zero net force acting on the bead. Thus,

the forces follow the equation:

such that, W=Weight of bead

B =Buoyant Force on bead

T =Tension of thread on bead

According to Newtons 3rd

Law, the buoyant

force by detergent on bead will have an equal

and opposite force by bead on detergent and, thus,

on the electronic balance (B).

In figure 4, as the bead is in equilibrium, there

should be zero net force acting on the bead. Thus,the forces follow the equation:

() ()

such that: R =reaction force by table on

electronic balanceB =reaction force of B

W(eb) =weight of electronicbalanceW(bk)=weight of beaker

From figure 4, the reading on the electronic

balance is proportional to the sum of W(bk) andB, as both forces act downwards on the sensor of

the balance. However, as the balance is tared, W(bk) equals constant zero. As such,

reading is dependent only on B. From this, I can also deduce that, since B=B and B isconstant anywhere in the detergent, the reading should be independent of the relative

position of bead in the detergent, provided the bead is fully submerged and the density of

detergent is the same throughout.

With this deduction, cases R1 and R3 are the same and can be represented in the same

way. However, there is a possibility that R3 values may be lower than R1. As R2

involved submerged part of the thread in the detergent, when the thread is then raised totake reading for R3, some detergent may stick to the thread and be lifted above the

surface of the detergent. Thus, there may be a net decrease in the mass of detergent in the

beaker. This can be proven if by lifting the bead out of the detergent, and allowing alldetergent to drip back into the beaker, the balance shows a negative reading.

Figure 3: Free-body diagram of Bead(R1/R3)

Figure 4: Free-body diagram of

Electronic Balance (R1/R3)


6/17

Page 6

R2In figure 5, as the bead is in equilibrium, there



B =Buoyant Force on beadN =Reaction Force by Electronic

Balance on Bead


Law, N will have anequal and opposite force by bead on electronic

balance (N).

In figure 6, as the bead is in equilibrium, thereshould be zero net force acting on the bead. Thus,


() ()

such that: B =Reaction force of B

N =Reaction force of N

From figure 6, the reading on the electronicbalance is proportional to the sum of Band N,

as all 3 forces act downwards on the sensor of thebalance. Comparing this to the reading in R1

which is proportional to only B, the reading in

R2 should be larger than R1. However, the R2

reading for B1 in the table (mean = 0.068g) is

lower than R1 (mean = 0.078g), thus, reflecting acontradiction with the predicted result.

To address this contradiction, we regarded the reading for R2 in B1 as an anomaly, andrepeated the experiment with B2 and B3. The results in B2 and B3 show conclusively

that the reading for R2 should be larger than R1.

Figure 6: Free-Body Diagram of

Electronic Balance (R2)

Figure 5: Free-body diagram of Bead (R2)


7/17

Page 7

R4

In figure 7, as the bead is in equilibrium when itis travelling at constant velocity, there should be

zero net force acting on the bead. Thus, the

forces follow the equation:


B =Buoyant Force on bead

T =Tension of thread on bead

D = Drag on bead


Law, D will have an

equal and opposite force by bead on electronic

balance (D).

In figure 8, as the bead is in equilibrium, there


()()

such that: B =Reaction force of B

D =Reaction force of D

From figure 6, the reading on the electronicbalance is proportional to (B-D), as D acts in

the direction opposite to B. As it is impossible

for a human hand to lift bead up continuously at

exactly the same velocity, there is a large margin

of human error, resulting in a large range offluctuating readings. However, the readings show a general decreasing trend as the

upward velocity increases from rest. This is because Drag (D) is proportional to speed at

relatively slow speeds; thus, as the magnitude of velocity (i.e. the speed) increases, D alsoincreases, resulting in a decrease in (B-D) for D=D.

In addition, it is possible for the reading to turn negative because B


8/17

Page 8

R5In figure 9, as the bead is initially in

disequilibrium, there should be non-zero netforce acting on the bead. Thus, the forces follow

the equation:

such that, W =Weight of beadB =Buoyant Force on bead

D =Drag of detergent on bead

Drag Force(D) changes as the bead acceleratedownwards in the detergent due to gravity. As

speed of bead increases, D increases and, thus,

(B+D) increases. Equilibrium is attained when:

Such that, = Drag on bead at its

terminal velocity

Then, bead will travel downwards at constant

speed towards the base of the beaker.

In Figure 10, as the bead is in equilibrium, thereshould be zero net force acting on the bead. Thus,


() ()

Such that, B =Reaction force of BD =Reaction force of D

From Figure 10, the reading on the electronic balance is proportional to (B+D). As such,the reading R5 should stabilise when the terminal velocity is reached, when drag(D) takes

a fixed value .

From R2, From R5, As such, N=;

N= D at terminal velocity(vT);

(B+N)=(B+Dat vT).

Thus, reading R2=reading R5 at vT. This is supported by reading R2 (mean = 0.189g) and

R5 (mean = 0.190g) in B2.

Figure 10: Free-Body Diagram of

Electronic Balance (R5)

Figure 9: Free-Body Diagram of Bead (R5)


9/17

Page 9

Measurement of Terminal Velocity

The 3 tables below show the time taken for beads (B1, B2, B3) to travel a distance of5cm downwards for 9 different intervals along the plastic column. There are 5 repeated

measurements for bead of each size.

Reading x/cmT1/s for B1

1a 1b 1c 1d 1e

1 5--10 30.90 31.81 31.05 30.75 30.98

2 11--16 29.97 31.10 30.80 30.30 30.75

3 17--22 29.69 29.96 29.90 30.10 30.68

4 23--28 29.41 30.19 30.18 30.15 30.70

5 29--34 29.93 29.60 30.17 29.58 29.80

6 35--40 29.69 29.50 29.61 28.67 28.96

7 41--46 28.40 28.81 29.54 28.70 29.05

8 47--52 28.53 28.57 28.89 29.03 28.939 53--58 28.40 28.53 28.87 28.90 28.80


3a 3b 3c 3d 3e

1 5--10 8.87 9.08 9.04 9.07 8.97

2 11--16 8.74 8.34 8.56 8.89 8.89

3 17--22 8.37 8.53 8.71 8.80 8.764 23--28 8.49 8.31 8.30 8.53 8.81

5 29--34 8.29 8.37 8.19 8.19 8.56

6 35--40 8.41 8.50 8.56 8.20 8.30

7 41--46 8.30 8.75 8.74 8.75 8.42

8 47--52 8.47 8.76 8.40 8.64 8.68

9 53--58 8.23 8.65 8.35 8.56 8.54


2a 2b 2c 2d 2e

1 5--10 17.87 17.95 17.94 17.97 17.79

2 11--16 17.33 17.90 17.79 17.97 17.60

3 17--22 17.02 16.90 16.91 17.13 17.35

4 23--28 16.60 16.98 17.03 16.88 17.00

5 29--34 16.80 17.03 16.97 16.85 17.106 35--40 16.50 16.98 17.04 16.92 17.16

7 41--46 16.88 17.22 17.12 16.68 16.95

8 47--52 16.79 16.97 17.09 16.85 16.62

9 53--58 16.84 17.03 16.99 16.89 16.73


10/17

Page

10

Graph of time taken for a bead to travel 5cm (s) against reading number, which represents

the distance (x) along the plastic column, is plotted for B1, B2, and B3 respectively.

Small Bead B1

Medium Bead B2

T1

T2


11/17

Page

11

The 3 graphs above show the rough fit of a downward trend for the initial readings. From

the graphs, time (T) appears to approach a particular constant as the gradient of graph

tend towards zero. As velocity is displacement per unit time, by fixing the value of

displacement for each reading at 5cm, when the change in time taken(T) for multiple

subsequent readings (near the final reading9

th

reading) becomes increasingly smaller,this means that the velocity of bead also approaches a constant value i.e. terminal

velocity (vt).

Referring to all 3 graphs above, the time taken (T) generally appears to approach a

constant after the 5th

reading (29-34cm). Therefore, the average time taken can becalculated by taking only the values of (T) for the last 5 readings5

th, 6

th, 7

th, 8

th, and 9

th

reading.

Large Bead (B3)T3


12/17

Page

12

The 3 tables below show the mean time taken for a bead to travel 5cm for measurements

of time when value of x exceeds 29cmaverage of readings 5 to 9for all 5 sets of

measurements respectively.

Small Bead (B1)

Reading 1a 1b 1c 1d 1eAverage reading

for x>29/s28.99 29.00 29.42 28.98 29.11

Final Average/s 29.10

Medium Bead (B2)

Reading 2a 2b 2c 2d 2e

Average reading

for x>29/s16.76 17.05 17.04 16.84 16.91


Large Bead (B3)

Reading 3a 3b 3c 3d 3e

Average reading

for x>29/s8.34 8.61 8.45 8.47 8.50



13/17

Page

13

The table below shows the measurement of mass of bead B1, B2, and B3 respectively

with 3 repeated readings, in order to calculate the average mass of bead. Calibrated

electronic balance was used.

M1/g 0.001 M2/g 0.001 M3/g 0.001 /g 0.001

B1 0.069 0.070 0.070 0.070B2 0.182 0.182 0.182 0.182

B3 0.937 0.936 0.937 0.937

The table below shows the measurement of diameter of bead B1, B2 and B3 respectively

with 3 repeated readings, in order to calculate the average diameter of bead. Micrometer

Screw Gauge was used. Then, the average radius of bead for all 3 sizes are determined.

D1/mm 0.01 D2/mm 0.01 D3/mm 0.01 /mm 0.01 /mm 0.005B1 4.58 4.58 4.59 4.58 2.290

B2 6.31 6.31 6.32 6.31 3.155

B3 10.95 10.95 10.95 10.95 5.575

Computation of uncertainty values

Calculate of absolute uncertainty of VT

VT

x

x

2

t

t

2

V

T

For small bead (B1),

)17182.0(10.29

01.0

0.5

1.0 22

TV

= 3.436933 x 10-3

cm s-1

= 3.4 x 10-5

m s-1

For medium bead (B2),

)29551.0(92.16

01.0

0.5

1.0 22

TV

= 5.91278 x 10-3

cm s-1

= 5.9 x 10-5m s-1

For large bead (B3),

)590319.0(47.8

01.0

0.5

1.0 22

TV

= 1.18269 x 10-2

cm s-1

= 1.2 x 10-4

m s-1


14/17

Page

14

Calculation of absolute uncertainty of radius

rd

2, Where dis diameter of bead, r is radius of bead

For beads of all sizes

2

01.0r

= 0.005 mm

= 5 x 10-3

m

Calculation of absolute uncertainty of volume of sphere

Vb 3

r

r

2

Vb, Where Vbis volume of bead

For small bead (B1),

)3031.50()290.2(

)005.0(3

2

bV

= 0.329497 mm3

= 3.3 x 10-10

m3

For medium bead (B2),

)549.131()155.3(

)005.0(3

2

bV

= 0.625430 mm3

= 6.3 x 10-10m3

For large bead (B3),

)810.725()575.5(

)005.0(3

2

bV

= 1.95285 mm3

= 2.0 x 10-9

m3

Calculation of ViscosityUsing the average values of VT, the radius of the beads, mass of the bead, density of the

detergent, we can calculate the viscosity using equation (2);

For smallest bead

.)17182.0)(229.0(6

)981)(207.1()229.0(3

4)981(070.0

3

= 12.2800 g cm s-2

/ cm2s

-1

= 1.22800 Pa s


15/17

Page

15

= 1228.0 m Pa s

= 1228 cps

Difference compared to accepted reading = (14101228)/(1410) x 100%

= 12.9%

For medium bead

.)29551.0)(3155.0(6

)981)(207.1()3155.0(3

4)981(182.0

3

= 12.9620 g cm s-2

/ cm2s

-1

= 1.29620 Pa s

= 1296.20 m Pa s

= 1296 cps


= 8.08511%For large bead

.)590319.0)(5575.0(6

)981)(207.1()5575.0(3

4)981(937.0 3

= 9.63800 g cm s-2

/ cm2s

-1

= 0.963800 Pa s= 963.800 m Pa s

= 964 cps

Difference compared to accepted reading = (1410964)/(1410) x 100%= 31.6%

Average = (1228.0 + 1296.2 + 963.8) / 3

= 1162.7 m Pa s

= 1163 cps


= 17.5%


16/17

Page

16

Calculation of absolute uncertainty of viscosity

V

b

Vb

2

r

r

2

V

T

VT

2

For small bead,

1228172.0

10*4.3

290.2

005.0

10*0.5

10*3.32

522

8

10

= 8.5 cps

For medium bead,

1296296.0

10*9.5

3155.0

005.0

10*3.1

10*3.62

522

7

10

= 21.5 cps

For large bead,

964590.0

10*2.1

5575.0

005.0

10*3.7

10*0.22

422

7

9

= 9.04 cps

As such, average of = (8.5 + 21.5 + 9.04) / 3= 13.0 cps

From the calculated results above, we can see that the viscosity values for the small andmedium beads are relatively nearer to the actual value of 1410cps, whereas the viscosityvalue for the large bead drifted quite far away from the actual value, with a deviation of

31.6% (highest among the 3).

In overall, this deviation can also be seen as possible evidence to prove the direct causal

relationship between size of bead and the viscosity of the liquid detergent. This means

that change in bead size may directly cause a change in the viscosity of detergent aroundits motion pathway.

On the other hand, the constant negative deviation from the actual viscosity for all 3

values of viscosity being lower than the actual valuecan also suggest that detergent is anon-Newtonian fluid, making the procedure of determination of viscosity in this

experiment less ideal as the final value would be less predictable.


17/17

Page

Systematic errorThe true viscosity value of 1410cps is most determined in a highly controlled

environment where factors like surrounding temperature, and pressure are carefullymaintained to be constant. Besides, the equipment that we use may be inferior to the

equipment used to determine the true value in terms of accuracy and precision.

Human errorIn the measurement of time with a stopwatch, human reaction error can be a significant

factor in determining the accuracy of the measurement. For we have to take readings

while the bead is in motionfalling through the column of detergent, the accuracy of thereadings really depend on the reaction time of the person measuring the time and his

ability to multi-task as he needs to measure time with a stopwatch, while observing the

scale on column. This effect is amplified with the large bead, as it has a higher terminal

velocity and will travel faster through the column. As the ratio of human reaction time tothe final measurement of time increases, there is a higher percentage error which might

explain the large deviation for the large bead (B3) of 31.6% for calculated viscosity value

through this experiment.

Conclus ions

We measured the terminal velocities of beads of three different sizes falling through a

column of Mama Lemon liquid detergent, and found these to be vT= (171.83.4)*10-5

m/s, (295.55.9)*10

-5 m/s, and (59.01.2)*10

-4 m/s for the small, medium, and large

beads respectively. Assuming that the viscous drag experienced by the bead is given by

Stokes law in Equation (1), we inferred the viscosity of the liquid detergent to be1162.713.0 cps. This differs by 17.5% from the accepted value of 1410 cps. We

believe this deviation is largely due to human error as the time needed for humans toreact consistently produce a random error to our measurement of time. As such, we

cannot accurately determine the viscosity value as time is one of the few most importantfactors that is used to calculate the final value of viscosity.

References

Physics for Scientists and Engineers with Modern Physics 8th

Edition, by John W. Jewett,Jr. Raymond A. Serway.

Wikipedia.org

http://www.physics.unc.edu

E1 Guided Report_Edwin_s Report

Documents

Transcript of E1 Guided Report_Edwin_s Report