Answer the following questions related to pure (dimensionless) physical quantities:
a. (i) State what is meant by a dimensionless (pure) physical quantity and give two examples from mechanics.
b. (ii) Show algebraically that the argument of the natural logarithm in the expression S=kln(V0V) must be dimensionless, where S is entropy, V the volume of a gas and k a constant.
c. (iii) A student writes ln(3m) while solving a thermodynamics problem. Explain clearly why this expression is unacceptable and describe the correct way of writing a logarithmic term that involves a length.