In the $$List-I$$ below, four different paths of a particle are given as functions of time. In these functions, α and β are positive constants of appropriate dimensions and α ≠β. In each case, the force acting on the particle is either zero or conservative. In $$List-II$$, five physical quantities of the particle are mentioned: p→ is the linear momentum, L→ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in $$List-I$$ with those quantities in $$List-II$$, which are conserved for that path.

Question

In the $$List-I$$ below, four different paths of a particle are given as functions of time. In these functions, α and β are positive constants of appropriate dimensions and  α ≠β. In each case, the force acting  on the particle  is either zero or conservative. In $$List-II$$, five physical quantities  of the particle  are mentioned:   p→ is the linear momentum,  L→ is the  angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in $$List-I$$ with those quantities in $$List-II$$, which are conserved for that path.

Infinity Learn · Accepted Answer

(P) r→(t)=$$α$$ti^+$$β$$tj^$$                           ⇒V→$$=dr$$→$$(t)dt=$$α$$i^+$$β$$j^{$$ Constant $$}$$⇒a→$$=dv$$→$$dt=0$$                                   P→$$=mv$$→ $$(Remain$$ $$constant) $$K=12mv2{$$ Remain constant $$}$$       F→$$=ma$$→$$=-$$∂U∂$$xi^+$$∂U∂$$yj^=0$$⇒U→ Constant ∴$$E=K+U=$$ Constant dL→$$dt=$$τ→$$=r$$→×F→$$=0$$⇒L→$$=constant$$  (Q) r→$$=$$α$$cos$$⁡$$($$ω$$t)i^+$$β$$sin$$⁡$$($$ω$$t)j^$$⇒v→$$=dr$$→$$dt=-$$αω$$sin$$ω$$ti^+$$βω$$cos$$ω$$tj^$$⇒a→$$=dv$$→$$dt=$$−αω$$2cos$$⁡$$($$ω$$t)i^$$−βω$$2sin$$⁡$$($$ω$$t)j^$$⇒a→$$=$$−ω$$2r$$→ Path of the particle is elliptical                 ∴F∝r⇒U∝$$r2$$∵Δ$$U=-$$∫F→.dr→$$=$$∫mω$$2rdr=m$$ω$$2r22U$$ depends on r hence it will change with timeTotal energy remain constant because force is central  (R) r→(t)=$$α$$($$$$cos$$⁡ω$$i^+$$$$sin$$⁡$$($$ω$$t)j^)⇒v→(t)=dr$$→$$(t)dt=$$α$$($$−ω$$sin$$⁡$$($$ω$$t)i^+$$ω$$cos$$⁡$$($$ω$$t)j^)⇒a→(t)=$$−αω$$2$$[$$cos$$⁡$$($$ω$$t)i^+$$$$sin$$⁡$$($$ω$$t)j^$$] $$(S) r→$$=$$α$$ti^+$$β$$2t2j^$$⇒v→$$=$$α$$i^+$$β$$tj^$$⇒a→$$=dv$$→$$dt=$$β$$j^$$ Constant F→$$=ma$$→ Constant τ→$$=r$$→×F→$$=m$$αβ$$tk^L$$→$$=$$∫τ→$$dt=12m$$αβ$$t2k^$$    Depends on timeΔ$$U=$$−∫F→⋅dr→$$=$$−m∫$$0t$$ β$$j^$$⋅$$($$α$$i^+$$β$$tj^)dt=-12m$$β$$2t2$$ Δ$$K=12mv2=12m$$α$$2+$$β$$2t2$$∴$$E=$$Δ$$K+$$Δ$$U=12m$$α$$2$$[ Remain constant]

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