Formula & Calculator
Rotational Torque
Torque required to produce a given angular acceleration for a rotating body.
Interpretation
τ = Iα. Newton's second law for rotation: torque equals moment of inertia times angular acceleration. Used in dynamics of rotating systems, motors, and robotics.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| τ | Torque | N·m |
| I | Moment of inertia | kg·m² |
| α | Angular acceleration | rad/s² |
What it means
The rotational analogue of F = ma, this equation relates the net torque (τ) applied to a rigid body to its angular acceleration (α), with I being the moment of inertia about the axis of rotation. This is the fundamental equation of rotational dynamics. It is used to size motors, analyse flywheels, design robotic joints, and study the motion of wheels, gears, and pendulums. In robotics, it appears in inverse dynamics for calculating joint torques required for desired motion. Understanding this equation is essential for mechanical and robotics engineers to design actuation systems and to simulate dynamic behaviour.
Worked example
Rotational Torque – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| I (kg·m²) | 2.0 |
| α (rad/s²) | 3.0 |
| Parameter | Value |
|---|---|
| I | 5.0 |
| α | 4.0 |
Common mistakes
- Rotational torque: τ = I·α – where α is angular acceleration (rad/s²).
- Moment of inertia I: Must be about the axis of rotation – use the parallel axis theorem if needed.
- Units: I in kg·m², α in rad/s² → τ in N·m.
- Sign: Torque direction follows the right‑hand rule – consistent with α.
- External loads: This is the net torque – include friction and load torques as additional terms.
Applications
Rotational torque, τ = I·α, is the rotational analogue of Newton's second law, relating the net torque (τ) to the moment of inertia (I) and angular acceleration (α). This is fundamental for designing motors, gears, and robotic joints. Mechanical engineers use it to size actuators, to analyse the dynamics of rotating machinery, and to predict the angular response of systems under torque. In robotics, it helps determine the motor torque required to achieve desired joint accelerations. By understanding this relation, engineers can design efficient and responsive mechatronic systems. This formula is also used in automotive drivetrain design, turbine analysis, and flywheel energy storage. Mastering the torque‑acceleration relationship is essential for any engineer working with rotating bodies.
- Motor sizing and actuator selection for robotic joints
- Analysis of rotating machinery (gears, shafts, turbines)
- Design of flywheels and energy storage systems
- Vehicle drivetrain and powertrain dynamics
- Control system modelling for rotational dynamics
Frequently Asked Questions
The rotational analog of Newton's second law is τ = I·α, where τ is the net torque applied to a body, I is its moment of inertia about the axis of rotation, and α is the angular acceleration. It relates the cause (torque) to the effect (angular acceleration).
Torque τ is in N·m, moment of inertia I is in kg·m², and angular acceleration α is in rad/s². The equation is dimensionally consistent: N·m = kg·m² · rad/s².
For a solid sphere: I = (2/5)MR²; solid cylinder: I = (1/2)MR²; thin rod about centre: I = (1/12)ML²; hoop: I = MR². For composite bodies, use the parallel axis theorem to add contributions.
It states that the moment of inertia about any axis parallel to an axis through the centre of mass is I = I_cm + M·d², where d is the perpendicular distance between the axes. This is useful for calculating I about an offset axis.
For a given torque, a larger moment of inertia results in a smaller angular acceleration (slower change in rotational speed). This is why heavy flywheels resist changes in speed.
Torque is a rotational force (vector), while work is energy (scalar). The work done by a torque over an angular displacement is W = τ·θ (when torque and displacement are in the same direction).
It is used to size motors: the required torque is the sum of the torque needed for acceleration (I·α) and the torque to overcome friction and loads. It is also used to design shafts and gears to withstand the applied torque.
The moment of inertia I is the rotational analog of mass. It measures the resistance to angular acceleration, just as mass measures resistance to linear acceleration.
Torque is a vector; by convention, positive torque (counter‑clockwise) produces positive angular acceleration. The sign is determined by the right‑hand rule.
Using the wrong moment of inertia (e.g., about a different axis), forgetting to sum all torques, or using linear acceleration instead of angular acceleration. Also, assuming constant I when the mass distribution changes (e.g., a robotic arm).