Angular impulse = change in angular momentum; T ∆t = ∆ Iω where T is…

The idea

Angular impulse is the rotational analogue of linear impulse. When a constant torque T acts for a time ∆t, the angular impulse is T ∆t and it equals the change in angular momentum, ∆(Iω). This follows from Newton's second law in rotational form, T = ∆(Iω)/∆t, rearranged to T ∆t = ∆(Iω). If I is constant, the change is I∆ω; if I changes, the change is ∆(Iω) as a whole. For example, a flywheel with I = 0.50 kg m² accelerated from rest to 12 rad s⁻¹ by a constant torque of 3.0 N m takes ∆t = I∆ω/T = (0.50 × 12)/3.0 = 2.0 s. To solve problems, identify the torque, the time interval, and the initial and final angular momenta, then apply T ∆t = ∆(Iω) with consistent signs.

What earns the marks

• State that angular impulse is the product of torque and the time for which it acts, with units N m s.
• Apply the relationship T ∆t = ∆(Iω) for a constant torque, recognising that ∆(Iω) may involve changes in I, ω, or both.
• Rearrange the equation to find torque, time interval, or change in angular momentum as required.
• Use consistent sign conventions for torque and angular momentum about the chosen axis.
• Link the equation to Newton's second law in rotational form, T = ∆(Iω)/∆t, to justify why angular impulse equals the change in angular momentum.
• Calculate ∆ω from initial and final angular velocities in rad s⁻¹, converting units where necessary.

Common mistakes and how to put them right

• Confusing angular impulse with linear impulse.

  Put it right by using torque T and angular momentum Iω rather than force F and linear momentum mv.

• Assuming T ∆t = I∆ω always, even when I changes.

  Put it right by writing the change as ∆(Iω) and evaluating initial and final values separately.

• Forgetting that torque must be constant for the simple form T ∆t = ∆(Iω).

  Put it right by checking the question states or implies a constant torque.

Exam tips

• Show the substitution into T ∆t = ∆(Iω) with units, so the examiner can follow the calculation and check the final unit.
• If the torque is not constant, do not use T ∆t directly; instead consider the area under a torque–time graph or use the average torque.
• Keep angular velocities in rad s⁻¹ and torques in N m throughout to avoid unit conversion errors.