Newton’s Second Law for Rotation

When you push a door open, you are applying a force — but what actually matters for rotating the door is the torque you exert. Torque is the rotational equivalent of force, and just as Newton's Second Law tells us that a net force causes a linear acceleration, there is a direct rotational counterpart: a net torque causes an angular acceleration. This relationship is the cornerstone of rotational dynamics and governs everything from spinning wheels to the motion of planets. The rotational form of Newton's Second Law states that the net torque acting on a rigid body equals the product of its moment of inertia and its angular acceleration. Moment of inertia plays the role that mass plays in linear motion — it measures how difficult it is to change the rotational state of an object. A large moment of inertia means the object resists angular acceleration strongly, just as a large mass resists linear acceleration. The compact equation τ_net = Iα encapsulates all of this elegantly. This law applies to any rigid body rotating about a fixed axis, from a spinning flywheel to a gymnast tucking into a somersault. Mastering it requires understanding three linked ideas: how torque is calculated, what moment of inertia depends on, and how to apply the law systematically to solve real problems. Each of these ideas builds directly on Newton's original insights about force and motion.

Key takeaways
  • From Linear to Rotational: Making the Analogy Precise
  • What is Torque?
  • What is Moment of Inertia?
  • Applying the Law: A Systematic Approach
  • Connection to Linear Dynamics

From Linear to Rotational: Making the Analogy Precise

Newton's Second Law for linear motion is F_net = ma. Every term has a rotational counterpart. Force F becomes torque τ (tau), measured in newton-metres (N·m). Mass m becomes moment of inertia I, measured in kg·m². Linear acceleration a becomes angular acceleration α (alpha), measured in rad·s⁻². The structure of the law is identical: the 'cause' (net torque) equals the 'resistance' (moment of inertia) multiplied by the 'effect' (angular acceleration).

What is Torque?

Torque is the turning effect of a force. If you apply a force F at a perpendicular distance r from a pivot (called the moment arm), the torque is τ = rF sinθ, where θ is the angle between the force vector and the position vector from the pivot. Pushing a door at the edge (large r) produces far more torque than pushing near the hinge (small r) with the same force. The direction of torque follows a right-hand rule: curl the fingers of your right hand from the position vector toward the force vector, and your thumb points in the direction of the torque vector.

What is Moment of Inertia?

Moment of inertia I is the rotational analogue of mass — it quantifies rotational inertia. For a single point mass m at distance r from the axis, I = mr². For a rigid body made of many pieces, you sum over all of them: I = Σmᵢrᵢ². For common uniform shapes, standard formulae apply: a solid disc of mass M and radius R has I = ½MR², while a hollow ring has I = MR². Crucially, the same object can have different moments of inertia depending on which axis it rotates about — mass far from the axis contributes much more than mass close to it.

Applying the Law: A Systematic Approach

To apply τ_net = Iα, follow these steps: (1) Identify the axis of rotation and all forces acting on the object. (2) Calculate the torque due to each force using τ = rF sinθ, assigning positive values to anticlockwise torques and negative to clockwise (or choose a consistent sign convention). (3) Sum all torques to find τ_net. (4) Determine or calculate the moment of inertia I about the chosen axis. (5) Solve for α = τ_net / I. If α is positive, the angular acceleration is in the positive (anticlockwise) direction.

Connection to Linear Dynamics

The linear and rotational equations of motion run in perfect parallel. Just as a net force in the same direction as velocity speeds up a body, a net torque in the same direction as the angular velocity speeds up rotation. For a point on the rotating body at radius r, the tangential linear acceleration is a_t = αr, providing a direct bridge between the two descriptions. This connection is essential when problems involve both rolling motion and rotation simultaneously.

The Key Equation

τnet=Iα\tau_{\text{net}} = I\alpha
  • \tau_{\text{net}} — Net torque (N·m (newton-metre)): The vector sum of all torques acting on the rigid body about the chosen axis of rotation. Positive torque is conventionally anticlockwise.
  • I — Moment of inertia (kg·m²): A measure of the body's resistance to angular acceleration about the specified axis. Depends on both the total mass and how that mass is distributed relative to the axis.
  • \alpha — Angular acceleration (rad·s⁻²): The rate of change of angular velocity. Positive α indicates increasing anticlockwise rotation; negative α indicates increasing clockwise rotation (or deceleration anticlockwise).

Worked Example

Problem. A uniform solid disc of mass 4.0 kg and radius 0.30 m is free to rotate about a fixed horizontal axis through its centre. A rope is wrapped around the rim and pulled with a constant tangential force of 12 N. Calculate (a) the moment of inertia of the disc, and (b) the angular acceleration produced.

I=0.18kg ⋅ m2,α=20rad ⋅ s−2I = 0.18 \text{kg}\,\cdot\,\text{m}^2, \quad \alpha = 20 \text{rad}\,\cdot\,\text{s}^{-2}

Answer. The moment of inertia is 0.18 kg·m² and the angular acceleration is 20 rad·s⁻².

Figure
Diagram
Axis of rotationr = RF = 12 Nατnet = Iα = 3.6 N·mI = ½MR² = 0.18 kg·m²α = 20 rad·s⁻²Newton's Second Law for Rotation
Origins · 1 minLeonhard Euler and how this was discovered
LE
Leonhard Euler
1707–1783 · Swiss

Euler was the mathematician who rigorously formulated the equations of rigid body rotation, extending Newton's work from particles to extended bodies. He derived the rotational equations of motion — including the form τ = Iα — and introduced the concept of the moment of inertia as a tensor quantity in his landmark treatise Theoria Motus Corporum Solidorum seu Rigidorum (1765). His work transformed rotational mechanics from a collection of special cases into a unified, systematic theory. Euler lost sight in his right eye around age 31 and became almost totally blind later in life, yet he produced nearly half of his total mathematical output after losing his vision — dictating papers to his sons and assistants.

  • Torque and Moment of a Force — Torque is the driving quantity in τ_net = Iα; understanding how torques are calculated from forces and moment arms is a prerequisite for applying Newton's Second Law for rotation.
  • Moment of Inertia — Moment of inertia I is the rotational inertia term in τ_net = Iα. Knowing how to calculate I for different shapes and axes is essential for solving rotational dynamics problems.
  • Angular Velocity and Angular Acceleration — Angular acceleration α is the output of Newton's Second Law for rotation. A firm grasp of angular kinematics is necessary to interpret and use the result of τ_net = Iα.
  • Conservation of Angular Momentum — When τ_net = 0, angular momentum is conserved — a direct consequence of the rotational Second Law that explains phenomena such as a spinning skater pulling in their arms.