Torque and rotational inertia are directly related: torque is equal to the rotational inertia of an object multiplied by its angular acceleration. This relationship mirrors Newton's Second Law of motion (F=ma) for linear motion.
Understanding the Relationship: τ = Iα
The fundamental relationship between torque (τ) and rotational inertia (I) is expressed by the equation:
τ = Iα
Where:
- τ (Tau) represents the torque applied to an object, which is the force that causes rotation. It's essentially the "rotational force".
- I represents the rotational inertia (also called the moment of inertia) of the object. It's a measure of an object's resistance to changes in its rotational motion. It depends on the object's mass distribution relative to the axis of rotation.
- α (Alpha) represents the angular acceleration of the object, which is the rate of change of its angular velocity.
Rotational Inertia Explained
Rotational inertia (I) is analogous to mass in linear motion. Just as mass resists changes in linear velocity, rotational inertia resists changes in angular velocity. The distribution of mass is critical; the further the mass is from the axis of rotation, the greater the rotational inertia.
For a single point mass m rotating at a radius r, the rotational inertia is:
I = mr2
For more complex objects, the rotational inertia is calculated by summing (or integrating) the contributions of all the individual mass elements. Rotational inertia is not an intrinsic property like mass; it depends on the axis of rotation.
Analogy to Newton's Second Law
The equation τ = Iα is the rotational analog of Newton's Second Law of Motion, F = ma.
| Linear Motion | Rotational Motion |
|---|---|
| Force (F) | Torque (τ) |
| Mass (m) | Rotational Inertia (I) |
| Acceleration (a) | Angular Acceleration (α) |
Examples
-
Spinning a Figure Skater: A figure skater increases their rotational speed by pulling their arms and legs closer to their body. This decreases their rotational inertia (I) because mass is brought closer to the axis of rotation. Since torque remains relatively constant (or at least doesn't increase as much as the reduction in I would require), angular acceleration and therefore angular velocity (spin speed) increases (since τ = Iα).
-
Opening a Door: Applying a force further from the hinges (axis of rotation) creates more torque, making it easier to open the door. While the door's rotational inertia remains constant, the increased torque results in greater angular acceleration.
In Summary
Torque is the rotational force causing angular acceleration. Rotational inertia is the resistance to that acceleration. The greater the rotational inertia, the more torque is required to achieve a given angular acceleration. They are directly proportional through the equation τ = Iα.
