What is the equation to calculate an angular momentum for a particle and for a rigid object?

Question

BlackTom AI · Accepted Answer

The question asks for the equations used to calculate angular momentum for a particle and for a rigid object. We will evaluate each option in turn, noting what is correct and what is incorrect.

Option a: 'p = mv for a particle, L = I ω for a rigid object' 
- For a particle, the linear momentum p = m v is not the angular momentum; angular momentum about a chosen origin is L = r × p, which equals m v r sinθ, not simply p. This makes the first part incorrect. 
- For a rigid body, L = I ω is correct provided the rotation is about a principal axis or the axis through the center of mass with the appropriate moment of inertia. However, since the particle part is wrong, the overall statement is incorrect.

Option b: 'L = mvr sin θ for a particle, L = I ω for a rigid object' 
- This matches the standard expressions: angular momentum of a point particle about a given origin is L = r × p = m r × v, which in magnitude is L = m v r sin θ, where θ is the angle between r and v. 
- For a rigid body (rotating rigid object), angular momentum about the axis of rotation is L = I ω, where I is the moment of inertia about that axis. This option correctly assigns the formulas to each case.

Option c: 'L = I ω for a particle, L = mvr sin θ for a rigid object' 
- This reverses the correct forms. A single particle does not have L = I ω unless you consider a continuous mass distribution and the appropriate parallel axis, but the standard point-particle formula is L = r × p = mvr sin θ, not I ω. The second part would be valid only if the rigid body’s rotation is described about a fixed axis, but the first part makes this option incorrect.

Option d: 'L = I ω for a particle, p = mv for a rigid object' 
- Again, the particle case is incorrect because a single particle’s angular momentum is not generally given by L = I ω; a point particle uses L = r × p = mvr sin θ. The second part, p = mv, is the linear momentum formula for a particle, but interpreting it as the rigid object’s momentum here is misleading since a rigid body can have both translational and rotational motion; this pairing is inconsistent with standard definitions of angular momentum for rigid bodies as L = I ω about the rotation axis. Therefore this option is incorrect.

To recap, option b correctly assigns L = mvr sin θ for a particle and L = I ω for a rigid object, while the other options mix incorrect pairings of angular momentum formulas with particle or rigid-body cases. The reasoning above highlights the standard definitions: point-particle angular momentum from r and p, and rigid-body angular momentum from I ω about the rotation axis.

What is the equation to calculate an angular momentum for a particle and for a rigid object?Single choice

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