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Theme A — Space, Time and Motion

A.4.6 Newton’s Second Law for Rotation — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 48 of 63 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Curriculum Equation Key قوانين نيوتن للحركة - Newton Laws of Motion FormulasFast Exam Solution
F = m × a | W = m × g | F₁₂ = -F₂₁
Word Equation & Conditions: القوة المحصلة = الكتلة × التسارع | Force = Mass × Acceleration
Location & Specifications: الميكانيكا الكلاسيكية والتحصيلي وفيزياء 3 ثانوي و IGCSE Physics.
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Lesson summary

A.4.6 Newton’s Second Law for Rotation — IB PHYSICS HL notes: Apply τ = Iα to rotating bodies and compare it with F = ma.Linear: F = ma — the force needed depends on mass.What the symbols mean: F = resultant force (N) · m = mass (kg) · a = acceleration (m s−2)Rotational: τ = Iα —…

  • Torque on a point mass: τ
  • Angular acceleration α
  • Substitute: τ
  • The moment of inertia I
  • A block of mass m on a string wrapped round a pulley of mass M and radius R (I
  • Newton's second law for rotation

Part 1 of 6Linear and rotational forms

Linear: F
Linear: F ma — the force needed depends on mass.
What the symbols mean: F = resultant force (N) · m = mass (kg) · a = acceleration (m s−2)
Rotational: τ
Iα — the torque needed depends on moment of inertia.
  • τ = torque (N m), I = moment of inertia (kg m2), α = angular acceleration (rad s-2).
Torque is the rotational equivalent of force.

Part 2 of 6Derivation

Derivation
Torque on a point mass: τ
Fr, with F = ma.
Angular acceleration α
Angular acceleration α a/r, so a = αr.
Substitute: τ
Substitute: τ r(mαr) = (mr2)α = Iα.
The moment of inertia I
The moment of inertia I mr2 plays the role of mass.
Resistance to rotation is the ability to resist changes in angular acceleration.

Part 3 of 6Analogy table

LinearRotational
Force FTorque τ
Mass mMoment of inertia I
Acceleration aAngular acceleration α
Newton's second law F ∝ aτ ∝ α
F = maτ = Iα

Part 4 of 6Applying the law

1
Draw the forces and identify the torque about the axis.
2
Write τ = Iα with the correct moment of inertia.
3
Relate α to linear acceleration using a = αr where a string unwinds.
4
Solve with the linear equation for any attached mass.
Applying the law
A larger torque gives a larger angular acceleration.
A larger I gives a smaller α for the same torque.

Part 5 of 6Linked linear and rotational systems

Linked linear and rotational systems
A block of mass m on a string wrapped round a pulley of mass M and radius R (I
½MR2).
  • Blockmg − T = ma.
  • PulleyTR = Iα, with α = a/R.
  • Resulttension T = ½Ma and acceleration a = mg / (m + M/2).
Apply the second law separately to the block and to the pulley.
The tension on the string provides the torque on the pulley.

Part 6 of 6More from the specification

More from the specification
Rotational dynamic analogue of Newton's law
  • Rotational dynamic analogue of Newton's law net torque equals rotational inertia multiplied by angular acceleration in an accelerating mechanical system.
Rotational Analogue of Newton's Second Law
  • The net external torque acting on a rigid object is directly proportional to its angular acceleration: Sigma tau = I * alpha,
  • where I is the moment of inertia about the axis of rotation and alpha is angular acceleration.
Torque, Couple, and Rotational Equilibrium
  • A couple consists of two parallel forces equal in magnitude and opposite in direction separated by a perpendicular distance d;
    • rotational equilibrium requires the sum of clockwise torques to equal the sum of anticlockwise torques (Sigma tau = 0).
Kinematics of Rotational Motion
  • Rotational motion variables mirror translational variables, relating angular displacement Delta theta, initial angular velocity omega_i, final angular velocity omega_f,
  • and angular acceleration alpha via equations such as omega_f2 = omega_i2 + 2 * alpha * Delta theta.
Rotational Kinetic Energy
  • A rotating rigid body possesses rotational kinetic energy defined by Ek,rot = 1/2 * I * omega2, contributing to total mechanical energy alongside translational kinetic energy (1/2 * m * v2.
Constant
  • Not changing.
Kinetic energy
  • K = ½mv2, energy of motion.
Velocity
  • Rate of change of displacement.
Distance
  • Total length of path travelled, a scalar.
Displacement
  • Change in position.
Couple
  • A pair of equal opposite forces with different lines of action.
Clockwise
  • Turning in the direction of a clock's hands.
Equilibrium
  • State with zero net force.
Perpendicular
  • At right angles.
Impulse
  • J = FΔt = Δp.
    What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s−1); FΔt is the impulse (N s)

Key terms

TermMeaning
TorqueTurning effect of a force, τ = Fr.
Moment of inertiaResistance of a body to angular acceleration.
Angular accelerationRate of change of angular velocity.
Newton's second law for rotationτ = Iα.
PulleyWheel over which a string runs.
TensionForce in a string pulling on bodies at its ends.

Avoid these mistakes

1
Using τ = Fr without checking the force is perpendicular.
2
Using the mass in place of the moment of inertia.
3
Forgetting α = a/R for a string on a pulley.
4
Mixing units — α is in rad s-2, not m s-2.
5
Applying one equation to the combined system instead of separately to block and pulley.

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