Lab_9
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1) You measure in your experiment on viscous fluid flow the resistance R of a capillary ("your tube"), which is equal to the ratio P/Q, where P is the pressure difference across the capillary (see Poiseuille's Law in Ch11 sheet 36) and Q is the flow rate (Ch11 sheet 25) through the capillary. Use R=P/Q and the expression in Ch11 sheet 36 for the flow rate Q to get the expression for the resistance R of the capillary. (The pressure difference, which is labeled P here, is labeled DP in Ch11 sheet 36.) What is the resistance R when the viscosity of the fluid is , the length of the capillary is cm, and the diameter of the capillary is mm ? Indicate with a positive (negative) sign whether the expression you use for R is valid only if the flow is frictionless (friction is present).
2) You calculate the resistance of a mm diameter capillary and the resistance of a mm diameter capillary, using your expression for R which you extracted from Poiseuille's Law (Ch11 sheet 36) in Quiz 1) above. Both capillaries have the same length L and the resistances are calculated for the same fluid, i.e. the same viscosity . What is the ratio of resistances (see Ch11 sheet 38,38') ? Indicate with a negative (positive) sign whether your calculated ratio depends (does not depend) on the value of the fluid viscosity and the length of the capillaries.
3) You calculate the resistance of a mm diameter capillary and the resistance of a mm diameter capillary, using your expression for R which you extracted from Poiseuille's Law (Ch11 sheet 36) in Quiz 1) above. Both capillaries have the same length L and the resistances are calculated for the same fluid, i.e. the same viscosity . You want to compare the ratio with the corresponding ratio you measured experimentally and need the error of the theoretical ratio for the comparison. You assume a % error for each of the radii r of the 2 capillaries. You neglect all other errors keeping only the error of r, which occurs in the 4-th power in the expression for the resistance R, and hence dominates the error of the resistance R. What is the absolute error of the ratio ? (Hint: write R in the form , as is done in Ch11 sheet 38,38', and use equation (8) in "Error and Uncertainty"("EU") to calculate the errors for and . Then use equation (7) in “EU” to calculate the error of the ratio ) Indicate with a negative (positive) sign whether for a given fixed relative error of the radius r the relative error of the resistance R increases (decreases) with increasing power n of the radius r.
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