Relation Between Torque And Moment Of Inertia

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. Mathematically given by-

Torque

\(\tau =I\alpha\)

Where,

  • ?? is Torque(Rotational ability of a body).
  • I is the moment of inertia(virtue of its mass)
  • ?? is angular acceleration(rate of change of angular velocity).

Relationship between Torque and 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

\(\Rightarrow a=\frac{F}{m}\) ———(1)

Substitute linear acceleration a with angular acceleration. That is-

We know that, Acceleration \(a=\frac{d}{dt}\left ( \frac{ds}{dt} \right )\)

For rotatory motion s = rd??. Thus, Substituting we get-

\(=\frac{d}{dt}\left ( \frac{rd\theta }{dt} \right )\) \(=r\frac{d}{dt}\left ( \frac{d\theta }{dt} \right )\)

Thus, \(a=r\alpha\) is the angular acceleration ———-(2)

Simarelly replace force F by Torque ?? we get-

\(\tau =Fr\) \(\Rightarrow F=\frac{\tau }{r}\) ——–(3)

Substituting equation (2) and (3) in (1) we get-

\((1)\Rightarrow r\alpha =\frac{\left ( \frac{\tau}{r} \right )}{m}\) \(\Rightarrow r\alpha =\frac{\tau }{rm}\) \(\Rightarrow \tau =mr^{2}\alpha\)

We know that moment of inertia \(I =mr^{2}\)

Thus, substituting it in the above equation we get-

\(\Rightarrow \tau =I\alpha\)

Hope you understood the relation and conversion between the Torque and the Moment of Inertia of rotational motion.

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