SOLUTION: A piece of capillary tubing was calibrated in the following manner. A clean sample of the tubing weighed 3.247g. A thread of mercury, drawn into the tube, occupied a length of 23.7

This problem assumes to use grams as the measure unit of force. So, it is intended for the very old system of units 
which was in use before 1960.


It also assumes that the student knows the basics about "capillary action" and "surface tension" from Physics.


I am not going to teach you these basics here. Instead, I refer you to these Wikipedia articles


    https://en.wikipedia.org/wiki/Capillary_action

    https://en.wikipedia.org/wiki/Surface_tension    (*)

    https://en.wikipedia.org/wiki/Meniscus_(liquid)


Hope you will have interesting reading.



The idea of solution is that capillary force, that acts at the contact of the mercury column in the tubing 
with the tubing walls KEEPS in EQUILIBRIUM the weight of this vertical mercury column.


The weight of the mercury column is  3.489 grams - 3.247 grams = 0.242 grams.


The capillary force is  , where pi = 3.14,  d is the diameter under the question and  is the surface tension of Mercury.


So, your basic equation to find  the diameter "d" from this experiment is


     = 0.242 grams,  or  d = ,


The value of the surface tension coefficient for Mercury you can find in the second Wikipedia article in the list above,
marked by (*).