Angular momentum = Iω Conservation of angular momentum.

The idea

Angular momentum is the rotational analogue of linear momentum. For a rigid body rotating about a fixed axis, it equals the product of moment of inertia I and angular velocity ω, so L = Iω. I depends on mass distribution about the axis; ω is the angular velocity in rad s⁻¹. Conservation of angular momentum states that if the resultant external torque on a system is zero, its total angular momentum remains constant. A skater pulling in their arms reduces I, so ω increases to keep Iω unchanged. Similarly, a planet in an elliptical orbit moves faster when closer to the Sun because its moment of inertia about the Sun is smaller. To solve problems, identify the axis, calculate I for the given shape, convert ω to rad s⁻¹, then apply L = Iω before and after any change.

What earns the marks

• State that angular momentum L is the product of moment of inertia I and angular velocity ω, with L = Iω and units kg m² rad s⁻¹ or kg m² s⁻¹.
• Explain that moment of inertia I depends on mass and how that mass is distributed about the chosen axis, so the same body can have different I values for different axes.
• Apply conservation of angular momentum when the resultant external torque is zero, setting I₁ω₁ = I₂ω₂ for a system whose mass distribution changes.
• Convert angular velocity between rev s⁻¹ and rad s⁻¹ using ω = 2πf before substituting into L = Iω.
• Interpret a decrease in I as causing an increase in ω when L is conserved, and link this to the skater or collapsing-star example.
• Use consistent sign conventions for clockwise and anticlockwise rotation when combining angular momenta of interacting bodies.

Common mistakes and how to put them right

• Treating angular momentum as a scalar and ignoring direction.

  Put it right by assigning positive and negative signs to opposite senses of rotation about the chosen axis.

• Using linear momentum p = mv instead of L = Iω for a rotating rigid body.

  Put it right by identifying the rotational variables and the axis of rotation.

• Assuming I is always mr² for any body.

  Put it right by selecting the appropriate expression for the shape, such as ½mr² for a solid disc about its central axis.

Exam tips

• Write the conservation equation explicitly as I₁ω₁ = I₂ω₂ before substituting values, so the examiner can see the physical principle being applied.
• Check that ω is in rad s⁻¹; if a frequency in Hz or rev s⁻¹ is given, convert using ω = 2πf.
• State the condition for conservation clearly: resultant external torque is zero, not merely that no forces act.