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I was studying Principles of Flight, I came across a sentence which is my question.

_ the aileron is known as a rate control since a given aileron angle of deflection determines a rate of roll, not a roll displacement.

My 1st question is what is the difference between rate of roll and roll displacement? What is roll displacement?

My 2nd question is, I know rate of roll is the rate at which an aircraft executes a turn expressed in degrees per second, can we say that rate of roll is the rate of change of heading as the aircraft rolls about its longitudinal axis?

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    $\begingroup$ Your "2nd question" conflates roll rate with turn rate; heading change is turn rate; bank angle change is roll rate. In the case of an aileron roll, you could have a high roll rate with essentially zero heading change or turn rate; in a turn with a constant bank angle, you'd have zero roll rate with a constant turn rate. $\endgroup$
    – Ralph J ♦
    Apr 9 at 13:11

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Roll displacement is basically the bank angle of the aircraft. Rate of roll is the rate of change of bank angle per unit time: $\large \frac {∆bank \ angle}{∆time}$

A given aileron deflection does not result in a particular bank angle - it results in a particular rate of change of bank angle. In other words, holding the yoke at a particular deflection will not* result in the aircraft attaining a particular bank angle and staying there; instead, the aircraft will continuously roll at a constant roll rate for as long as the yoke is held at that angle.

*Ignoring any effects of lateral stability

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    $\begingroup$ (Also ignoring rotational inertia in the roll axis. Good answer, but you might consider qualifying with "to a first approximation") $\endgroup$
    – quiet flyer
    Apr 9 at 13:18
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    $\begingroup$ Good answer. Per Ralph's comment on the question, I would recommend you take the opportunity to clarify the misunderstanding in the second question by briefly defining "turn rate"... $\endgroup$
    – Michael Hall
    Apr 9 at 15:52

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