Countering this upward force is the downward force of gravitation which is

Countering this upward force is the downward force of

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Countering this upward force is the downward force of gravitation, which is equal to the weight (mg dynes) of the raised water. The mass of the column of water raised by adhesion is equal to its volume (v) times its density (d). Substituting vd for m, we see that the weight of the water is vdg dynes. Since the raised column of water in the tube is in the form of a cylinder, we can make use of the geometrical formula for the volume of a cylinder and say chat the volume of the raised water is equal to the height of the column (h) multiplied by the cross-sectional area (pi r x r), where r is the radius of the column. Substituting [(pi) r x r x h] for v, we see that the weight of the water is (pi)r x r x h x d x g dynes. When the water in the narrow tube has been raised as high as it will go, the upward adhesive force is balanced by the downward gravitational force, so we have: 2 (pi) r (sigma) = (pi) r x r x h x d x g (Equation 9-4) Solving for h: H = 2 (sigma) /(r d g) (Equation 9-5)
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The acceleration due to gravity (g) is fixed for any given point on the earth; and for any particular liquid, the surface tension (sigma) and the density (d) are fixed for the particular conditions of the experiment. The important variable is the radius of the tube (r). As you see, the height to which a column of water is drawn upward in a narrow tube is inversely proportional to the radius of the tube. The narrower the tube, the greater the height to which the liquid is lifted. Consequently, the effect is most noticeable in tubes (natural or artificial) of microscopic width. These are capillary tubes (from a Latin expression meaning "hair-like"), and the rise of columns of water in such tubes is called capillary action. It is through capillary action that water rises through the narrow interstices of a lump of sugar or a piece of blotting paper, and it is at least partly through capillary action that water rises upward through the narrow tubes within the stems of plants. Again, if we know the value of the density of a liquid and the extent of its rise in a tube of known radius (both rise and radius being easily measured), it follows that since the value of g is also known, the value of the surface tension – sigma) can be calculated from Equation 9-5. In the case of mercury, where the adhesive forces with glass an exerted downwards, the level is pulled below the "natural level." The degree to which the level is lowered is increased as the radius of the tube is decreased. Viscosity We are accustomed to the notion of friction as a force that is exerted opposite to that which brings about motion when one solid moves in contact with another. Such friction tends to slow and eventually stop, motion unless the propulsive force is vigorously maintained. There is also friction where a solid moves through a fluid, as when a ship plows through water. For all that water seems so smooth and lacking in projections to catch at the ship, the ship once set in motion will speedily come to a halt, its
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