A particle of constant mass moves in space with position vector for

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems

Problem

A particle of constant mass mm moves in space with position vector r(t)=(cos⁡t, sin⁡t, t)\mathbf r(t)=\bigl(\cos t,\,\sin t,\,t\bigr) for t≥0t\ge 0. Its linear momentum is p(t)=m v(t)\mathbf p(t)=m\,\mathbf v(t), where v(t)=r′(t)\mathbf v(t)=\mathbf r'(t), and its angular momentum about the origin is defined by L(t)=r(t)×p(t)\mathbf L(t)=\mathbf r(t)\times\mathbf p(t). (a) Determine the velocity vector v(t)\mathbf v(t) and express p(t)\mathbf p(t) in terms of mm and tt. (b) Compute the angular momentum vector L(t)\mathbf L(t). Present each component in fully simplified form. (c) Differentiate L(t)\mathbf L(t) to obtain L′(t)\mathbf L'(t) and interpret your result in the language of torque. Specifically, indicate whether the torque about the origin is zero, constant, or time–dependent, and explain why. (d) Verify that the magnitude ∥L(t)∥\lVert\mathbf L(t)\rVert is constant or determine its rate of change, as appropriate. Discuss the physical significance of your finding in terms of conservation laws.

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