Question

Why the inertia torque acts in the opposite direction to the accelerating couple?

• A. Bring the body in equilibrium
• B. To reduce the accelerating torque
• C. Acts as a constraint torque
• D. Increase the linear acceleration

Hint:

D'Alembert's Principle (Rotation). Introduces an inertia torque \(\tau_{inertia = -I\alpha\) acting opposite to the angular acceleration \(\alpha\). This allows the dynamic equation \(\sum \tau = I\alpha\) to be written as a dynamic equilibrium equation \(\sum \tau + \tau_{inertia = 0\).

Answer 1

The Correct Option is A

This question relates to D'Alembert's principle applied to rotation.
The equation of motion for rotation is \(\sum \tau_{ext} = I \alpha\), where \(\sum \tau_{ext}\) is the net external torque (the accelerating couple), \(I\) is the mass moment of inertia, and \(\alpha\) is the angular acceleration.
D'Alembert's principle reformulates this by introducing an "inertia torque" or "inertial couple" equal to \(-I\alpha\).
This inertia torque is fictitious and acts in the direction opposite to the angular acceleration.
The principle states that the sum of the external torques and the inertia torque is zero: $$ \sum \tau_{ext} + (-I\alpha) = 0 $$ This transforms the dynamic problem (\(\sum \tau = I\alpha\)) into a problem of dynamic equilibrium (\(\sum \tau_{effective} = 0\)).
The reason for defining the inertia torque as acting opposite to the accelerating couple (\(\sum \tau_{ext}\)) is precisely to achieve this state of dynamic equilibrium, allowing static analysis methods to be applied.
Therefore, the inertia torque is introduced to conceptually bring the body into equilibrium under the action of real external torques and the fictitious inertia torque.