Let a particle of mass m travel along a differentiable path x in a Newtonian vector

Chapter 3, Problem 41

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Let a particle of mass m travel along a differentiable path x in a Newtonian vector field F (i.e., one that satisfies Newtons second law F = ma, where a is the acceleration of x). We define the angular momentum l(t) of the particle to be the cross product of the position vector and the linear momentum mv, that is, l(t) = x(t) mv(t). (Here v denotes the velocity of x.) The torque about the origin of the coordinate system due to the force F is the cross product of position and force: M(t) = x(t) F(t) = x(t) ma(t). (Also see 1.4 concerning the notion of torque.) Show that dl dt = M. Thus, we see that the rate of change of angular momentum is equal to the torque imparted to the particle by the vector field F.

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