Sir George Gabriel stokes found the formula for the viscous retarding force. It is given by,
F=6πηvr
Where η= the viscosity of the fluid.
From this formula, we can see that the viscous drag on the bigger ball is bigger than the viscous drag on the smaller ball. This is because the velocity increases as well as the radius. The weight downward also increases and the graph implies that this does so on a greater scale than the velocity and the radius. Hence, the time taken is less for a bigger ball bearing than for a smaller one.
The resultant force acting on the ball bearing is given by:
W-U-F
The ball bearing continues accelerating downwards until:
W-U-F=0.
At this point, the ball has attained a terminal velocity and is in a state of free fall.
Now, W=4/3πr3ρg
And U=4/3πr3σg
Sir George Gabriel Stokes also gave the formula for finding terminal velocity. This is given by:
4/3πr3ρg – 4/3πr3σg – 6πηrvt =0
∴ vt = (2r2(ρ-σ)g) / 9η
This can be compared to the formula:
Y = M x + C
Therefore, I will plot a graph of the terminal velocity against the radius squared (r2).
The gradient of this straight-line graph will be 2((ρ-σ) g) / 9η.
Where ρ is the density of the sphere, which is equal to 7930 kg/m3
and σ is the density of glycerine, which is equal to 1260 kg/m3
The table below shows the conversions of the diameter in mm to the radius squared in m2.
The distance between the two rubber bands has been given and is equal to 162mm. This is equal to 0.162m. I will assume that the ball has attained terminal velocity as it falls between the two bands. This is acceptable because the two bands were placed reasonably below the surface of the glycerine. This gives it time to accelerate and reach a terminal velocity.
The table below gives the quantities that I will plot on my graph.
My graph has a gradient of 7826m-1s-1. This is equal to 2((ρ-σ)g) / 9η.
Rearranging this to make η the subject of the formula gives,
η=2((ρ-σ)g) / 7826*9
- η= (7930-1260)/ 3593.6
- η= 6670/3593.6
- η=1.8560775Pa.
ERRORS
Errors are inevitable in all experiments. There are many possible errors that could have occurred in this experiment and these may be grouped into measurement and procedural errors.
Possible measurement errors could include
- The measurement of the distance between the two rubber bands. A metre rule was used and this has a reading accuracy of 0.05cm. Therefore there could be an error of + or – 0.05cm. Furthermore, the rubber bands are about 1mm thick each. This gives a further probability of an error of + or – 0.02cm.
- The measurement of the time taken for the ball bearings to travel between the two rubber bands is another possible source of error. All humans have a reaction time. This is the time taken between observation and reaction (in this case starting and stopping the stopwatch). The fact that there is variation between the repeated readings indicates that there is a measurement error of this nature. However taking an average of repeated readings helps to null the effect of this error.
- The stopwatch measures to an accuracy of 0.01s. This is a source of error. However it is so slight that its effect can be ignored. It probably affects the big ball bearings more because these take a shorter time to complete their travel.
- The ends of the metre rule used could be chipped or the stop clock could have a zero error. This gives rise to systematic error. However it does not affect results as long as the same instruments are used throughout the experiment.
Possible procedural errors include
- Not wetting the steel balls with glycerine properly could give rise to air bubbles sticking to their surface. This affects the measurements taken for the steel balls.
- Not dropping the steel balls through the centre of the tube could cause them to have contact with the walls and generate friction, which affects the results.
- Inconsistency which gives rise to differences in temperature could change the viscosity. This will affect the readings.
PRECAUTIONS
- The steel balls should be dropped carefully down the centre of the tube and vertically. This prevents additional friction from being set up between the ball and the walls of the tube.
- The balls should be dropped just above the surface of the glycerine to reduce the possibility of contamination by air.
- The balls should be wetted with glycerine before it is used also to avoid air bubbles clinging to its surface.
- They should be dropped carefully and the bottom of the tube should be fitted with a rubber bung so that it does not break on impact
- Care should be taken to avoid errors as much as possible. Common errors to watch out for are parallax errors and zero errors in the stopwatch.
SOME CALCULATIONS TO BACK UP MY WORK.
HOW DID I DETERMINE THAT THE VELOCITY BETWEEN THE TWO RUBBER BANDS WAS EQUAL TO THE TERMINAL VELOCITY?
This can easily be proved with Stokes law. It states that:
vt = (2r2(ρ-σ)g) / 9η
Therefore η=(2r2(ρ-σ)g) / 9 vt
I then chose two random corresponding vales of vt and r2.
I chose the values r2= 2.5*10-7 vt =2.04*10-3
And the values r2= 40*10-7 vt =31.25*10-3
Using the formula above, if indeed, the velocity I calculated is equal to the terminal velocity; I would obtain the same value for η in both cases.
Calculating gives me η=1.78012 pa for the first set and 1.8593 for the second set. Within the limits of experimental error, these are the same and so I can safely assume that the velocity between the two rubber bands is equal to the terminal velocity.
CONCLUSION
This experiment was a successful one. The results obtained were within the limits of experimental error. Though errors occurred, this is unavoidable and they were detected and reduced to as bare a minimum as possible. The expected relationship between the velocity and the radius squared was obtained from the graph.
I can safely conclude that the viscosity of the glycerine used in the experiment is 1.8560775Pa within the limits of experimental error.