section_path: Rigid-body motion and rotation/Rotational analogues of Newton's laws
section_name: Rotational analogues of Newton's laws
paragraph_text: Rotational analogues of Newton's laws
  
  When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the moment of inertia, the counterpart of momentum is angular momentum, and the counterpart of force is torque.
  
  Angular momentum is calculated with respect to a reference point. If the displacement vector from a reference point to a body is r {\displaystyle \mathbf {r} } and the body has momentum p {\displaystyle \mathbf {p} }, then the body's angular momentum with respect to that point is, using the vector cross product, L = r × p. {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p}.} Taking the time derivative of the angular momentum gives d L d t = (d r d t) × p + r × d p d t = v × m v + r × F. {\displaystyle {\frac {d\mathbf {L} }{dt}}=\left({\frac {d\mathbf {r} }{dt}}\right)\times \mathbf {p} +\mathbf {r} \times {\frac {d\mathbf {p} }{dt}}=\mathbf {v} \times m\mathbf {v} +\mathbf {r} \times \mathbf {F}.} The first term vanishes because v {\displaystyle \mathbf {v} } and m v {\displaystyle m\mathbf {v} } point in the same direction. The remaining term is the torque, τ = r × F. {\displaystyle \mathbf {\tau } =\mathbf {r} \times \mathbf {F}.} When the torque is zero, the angular momentum is constant, just as when the force is zero, the momentum is constant. The torque can vanish even when the force is non-zero, if the body is located at the reference point (r = 0 {\displaystyle \mathbf {r} =0}) or if the force F {\displaystyle \mathbf {F} } and the displacement vector r {\displaystyle \mathbf {r} } are directed along the same line.
  
  The angular momentum of a collection of point masses, and thus of an extended body, is found by adding the contributions from each of the points. This provides a means to characterize a body's rotation about an axis, by adding up the angular momenta of its individual pieces. The result depends on the chosen axis, the shape of the body, and the rate of rotation.