The Torque in rotational motion is equivalent to force in linear motion. It is the prime parameter which keeps an object under rotatory motion. The torque applied to an object begins to rotate it with an acceleration inversely proportional to its moment of inertia.
For simple understanding, we can imagine it as Newton’s Second Law for rotation. Where, torque is the force equivalent, a moment of inertia is mass equivalent and angular acceleration is linear acceleration equivalent. The rotational motion does obey Newton’s First law of motion.
Consider an object under rotatory motion with mass m, moving along an arc of a circle with radius r. From Newton’s Second Law of motion we know that,
F= ma
⇒ a = F/m ....(1)
Substitute linear acceleration a with angular acceleration. That is
We know that, Acceleration a = \(\frac{d}{dt}(\frac{ds}{dt})\)
For rotatory motion s = rd. Thus, Substituting we get
Thus, a = rα is the angular acceleration ...... (2)
Similarly replace force F by Torque we get.,
Substituting equation (2) and (3) in (1) we get
We know that moment of inertia I = mr2
Thus, substituting it in the above equation we get-
⇒ τ = Iα
