Съдържание

IPhO 1995, Експеримент — Задача 1. Terminal velocity in a viscous liquid

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Experimental Question 1 · 20 т.

Условие

An object falling in a liquid will eventually reach a constant velocity, called the terminal velocity. The aim of this experiment is to measure the terminal velocities of objects falling through glycerine.

For a sphere of radius $r$ falling at velocity $v$ through a viscous liquid, the viscous force $F$ is given by $F = 6\pi\eta r v$ . Here $\eta$ is a property of the liquid called the viscosity. In this experiment you will measure the terminal velocity of metal cylinders (because cylinders are easier to make than spheres). The diameter of each cylinder is equal to its length, and we will assume the viscous force on such a cylinder is similar to the viscous force on a sphere of the same diameter, $2r$ :

F_{cyl} = 6\pi\kappa\eta r^{m} v \qquad (1)

where $\kappa = 1$ , $m = 1$ for a sphere.

Preliminary Calculation of terminal velocity (2 marks)

If $\rho$ is the density of the cylinder and $\rho'$ is the density of the liquid, show that the terminal velocity $v_T$ of the cylinder is given by

v_T = C r^{3-m} (\rho - \rho') \qquad (2)

where $C$ is a constant and derive a expression for $C$ . [2 т.]

Experiment Use the equipment available to determine the numerical value of the exponent $m$ (10 marks) and the density of glycerine (8 marks). [18 т.]

Notes

For consistency, try to ensure that the cylinders fall in the same orientation, with the axis of the cylinder horizontal.
The tolerances on the diameter and the length of the cylinders are 0.05 mm (you need not measure them yourself).
There is a brass sieve inside the container that you should use to retrieve the metal cylinders. Important: make sure the sieve is in place before dropping objects into the glycerine, otherwise you will not be able to retrieve them for repeat measurements.
When glycerine absorbs water from the atmosphere, it becomes less viscous. Ensure that the cylinder of glycerine is covered with the plastic film provided when not in use.
Do not mix cylinders of different size and different material after the experiment.

Material	Density (kgm $^{-3}$ )
Aluminium	$2.70 \times 10^{3}$
Titanium	$4.54 \times 10^{3}$
Stainless steel	$7.87 \times 10^{3}$
Copper	$8.96 \times 10^{3}$

Отговори

Решение

Preliminary: Calculation of Terminal Velocity

When the cylinder is moving at its terminal velocity, the resultant of the three forces acting on the cylinder, gravity, viscous drag and buoyant force, is zero.

V\rho g - 6\pi\kappa\eta r^{m} v_T - V\rho' g = 0

where $V = 2\pi r^{3}$ is the volume of a cylinder (whose height is $2r$ ).

This gives

v_r = C r^{3-m} (\rho - \rho')

where

C = \frac{g}{3\kappa\eta}

Experiment

Determination of the exponent $m$

Aluminium cylinders of different diameters are dropped into the glycerine. Fall times between specified marks on the measuring cylinder containing the glycerine are recorded for each cylinder. A preliminary experiment should establish that the cylinders have reached their terminal velocity before detailed results are obtained. The measurements are repeated several times for each cylinder and an average fall time is calculated. Table 1 shows a typical set of data. To find the value of $m$ a graph of log(fall time) as a function of log(diameter) is plotted as in figure 1. The slope of the resulting straight line graph is $3 - m$ from which a value of $m$ can be determined. A reasonable value for $m$ is 1.33 with an uncertainty of order $\pm 0.1$ . The uncertainty is estimated by the deviation from the line of best fit through the data points obtained by drawing other possible lines.

Determination of the density of glycerine

Cylinders with the same geometry but different densities are dropped into the glycerine and timed as in the first part of the experiment. Table 2 shows a typical set of results. From equation (2) a linear plot of $1/t$ as a function of density should yield a straight line with an intercept on the density axis corresponding to the density of glycerine. Figure 2 shows a typical plot. Alternatively the terminal velocities could be calculated and plotted against density which would again lead to the same intercept on the density axis. The uncertainty in the measurement can be estimated by drawing other possible straight lines through the data points and noting the change in the value of the intercept.

Diameter (mm)	Individual readings (s)						Mean (s)
10	1.44	1.56	1.44	1.37	1.44	1.41	1.44
4	6.22	6.06	6.16	6.13	6.13	6.22	6.15
8	1.80	1.82	1.78	1.84	1.82	1.81	1.82
5	4.06	4.34	4.09	4.12	4.25	4.13	4.13

Table 1: Sample data set

Log-log plot: vertical axis 'Diameter of cylinders (mm)' from 1 to 10, horizontal axis 'Fall Time (s)' from 1 to 10; four open-circle data points lie on a descending straight line. — Figure 1: Sample plot

\text{Slope } = -\frac{58.2}{66.2} \div \frac{48.5}{93} = -1.67 \qquad \therefore m = 3 - 1.67 = 1.33

Material	Individual readings (s)						Mean (s)
Ti	3.00	2.91	2.97	2.91	2.84	2.75	2.91
Cu	1.25	1.25	1.28	1.25	1.22	1.22	1.25
S.Steel	1.31	1.32	1.38	1.44	1.31	1.34	1.33
Al	6.03	6.09	6.09	6.16	6.06	6.06	6.08

Table 2: Sample data set

Linear plot with grid: vertical axis '1/(fall time) (s-1)' from 0.0 to 1.0, horizontal axis 'Density (kg.m-3)' from 0 to 10000; four data points with error bars and three possible straight lines crossing the density axis near 1000. — Figure 2: Sample plot

\rho' = (1.1 \pm 0.2) \times 10^{3}\ \mathrm{kg.m^{-3}}

Detailed mark allocation

Section I
Reasonable range of data points with a scatter of $\sim 0.1$ s	[2]
Check that the cylinders have reached their terminal velocity
Visual check, or check referred to	[1]
Specific data presented	[1]
Labelled log-log graph	[2]
Data points for all samples, with a reasonable scatter about a straight line on the log-log graph	[1]
Calculation of $(3 - m)$ from graph	[1]
including estimate of error in determining $m$	[1]
Reasonable value of $m$ , $\sim 1.33$	[1]
Subtotal	[10]
Section 2
Reasonable range of data points	[1]
Check that the cylinders have reached their terminal velocity	[1]
Labelled graph of (falltime) $^{-1}$ vs. density of cylinder	[1]
Data points for all samples, with a reasonable scatter about a straight line on the (falltime) $^{-1}$ vs. density of cylinder graph	[1]
Calculation of the density of glycerine ( $\rho'$ ) from this graph	[1]
Estimate of uncertainty in $\rho'$	[1]
Reasonable value of $\rho'$ . “Correct” value is 1.260 kg.m $^{-3}$	[1]
Subtotal	[8]
TOTAL	20

Оригинал в Архива: exp1.pdf · официални решения: exp1_sol.pdf