Yes, the doubling time differs for different media. Also, it differs between different growth stages of the same culture, so it is most commonly measured when it remains relatively constant (i.e. during exponential growth).
Assuming that OD is directly proportional to the number of cells (it not always is), the simplest procedure is to grow the culture and take OD measurements at fixed intervals, then plot the growth with lnOD as the y-axis and time in the x-axis to find what part of the curve is linear and then calculate growth rate from two points in the linear portion (use your favorite interpolation method there) by using r = (ln [OD2/OD1 ]) / (T2-T1). Doubling time corresponds to ln2/r.
Yes, the doubling time differs for different media. Also, it differs between different growth stages of the same culture, so it is most commonly measured when it remains relatively constant (i.e. during exponential growth).
Assuming that OD is directly proportional to the number of cells (it not always is), the simplest procedure is to grow the culture and take OD measurements at fixed intervals, then plot the growth with lnOD as the y-axis and time in the x-axis to find what part of the curve is linear and then calculate growth rate from two points in the linear portion (use your favorite interpolation method there) by using r = (ln [OD2/OD1 ]) / (T2-T1). Doubling time corresponds to ln2/r.
I hate an OD method, if I've got a microscope and a possibility to apply DAPI staining, much more accurate.
If you need to use OD be careful with the condition: Lag phase is impredecible.
Never put the bacterial colony from agar to liquid medium to measure a curve. You should "revive" a colony in small amount of medium (e.g., 10mL to be cultivated in 0.5 - 1L of culture). But still, growth rate will be affected by the pre-cultivation treatment.
Even using chemostat cultures, growth rate in such batch experiment could be different according to an original dilution rate in chemostat (articles from 70')