Задание №82170 ЕГЭ Физике

$1) \mathrm{m}\overrightarrow{\mathrm{g}}+{\mathrm{F}}_{эл}+\overrightarrow{\mathrm{T}}+\overrightarrow{{\mathrm{F}}_{\mathrm{л}}}=\mathrm{m}·\overrightarrow{\mathrm{a}}$
$\mathrm{Оу}: -\mathrm{mg}·\mathrm{cos\alpha }-\mathrm{Fm}·\mathrm{sin\alpha }+\mathrm{T}+\mathrm{Fл}=\mathrm{m}·{\mathrm{a}}_{\mathrm{y}}$
$$\begin{gathered} {\mathrm{a}}_{\mathrm{ц}}=\frac{v^2}{\mathrm{R}}=\frac{{\mathrm{v}}^2}{\mathrm{L}}, \mathrm{R}=\mathrm{L} \\ {\mathrm{F}}_{л}=qvB·\sin 90°=qvB \\ qvB-\mathrm{mg}·\mathrm{cos\alpha }-\mathrm{qE}·\mathrm{sin\alpha }=\frac{m{v}^2}{\mathrm{L}} \end{gathered}$$

$$\begin{gathered} \mathrm{Ох}: т.к. \mathrm{v}=\max , \mathrm{то} {\mathrm{a}}_{\mathrm{T}}=0 \\ -\mathrm{mg}·\mathrm{sin\alpha }+{\mathrm{F}}_{\mathrm{эл}}·\mathrm{сos\alpha }=0 \\ qE·\mathrm{сos\alpha }=\mathrm{mg}·\mathrm{sin\alpha } \\ \mathrm{cos\alpha }=\frac{\mathrm{mg}·\mathrm{sin\alpha }}{qE}, {\sin }^2\mathrm{\alpha }+{\left(\frac{\mathrm{mg}}{qE}\right)}^2·{\sin }^2\mathrm{\alpha }=1 \\ {\sin }^2\mathrm{\alpha }(1+\frac{{\left(\mathrm{mg}\right)}^2}{{\left(qE\right)}^2})=1, {\sin }^2\mathrm{\alpha }\frac{{\left(qE\right)}^2+{\left(\mathrm{mg}\right)}^2}{{\left(qE\right)}^2}=1 \\ \mathrm{sin\alpha }=\frac{qE}{\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}}\to \mathrm{сos\alpha }=\frac{\mathrm{mg}}{\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}} \end{gathered}$$

$$\begin{gathered} 2) \mathrm{A}=∆{\mathrm{E}}_{\mathrm{к}}: \mathrm{A}=\frac{m{v}^2}{2}-\frac{{m{v}_0}^2}{2}=\frac{m{v}^2}{2} \\ m{v}^2=2\mathrm{A} \\ \mathrm{A}={\mathrm{A}}_{\mathrm{mg}}+{\mathrm{F}}_{\mathrm{эл}}+{\mathrm{A}}_{\mathrm{T}}+{\mathrm{A}}_{\mathrm{FA} } \\ {\mathrm{A}}_{\mathrm{mg}}={-}_{∆}{\mathrm{E}}_{\mathrm{n}},{ }_{∆}{\mathrm{E}}_{\mathrm{n}}=\mathrm{mg}{·}_{∆}\mathrm{h},{ }_{∆}\mathrm{h}=\mathrm{L}-\mathrm{L}·\mathrm{cos\alpha }=\mathrm{L}(1-\mathrm{cos\alpha }) \\ {\mathrm{A}}_{Fэл}=\mathrm{Fэл}·\mathrm{S}·\mathrm{cos\beta }, \mathrm{S}·\mathrm{cos\beta }=\mathrm{L}·\mathrm{sin\alpha } \\ {\mathrm{A}}_{Fэл}=\mathrm{qEL}·\mathrm{sin\alpha } \\ \mathrm{A}=\mathrm{qEL}·\mathrm{sin\alpha }-\mathrm{mgL}(1-\mathrm{cos\alpha }) \\ m{v}^2=2\mathrm{qEL}·\mathrm{sin\alpha }-2\mathrm{mgL}(1-\mathrm{cos\alpha }) \\ qvB-\mathrm{mg}·\mathrm{cos\alpha }-\mathrm{qEsin}\alpha =\frac{2\mathrm{qEL}·\mathrm{sin\alpha }-2\mathrm{mgL}(1-\mathrm{cos\alpha })}{\mathrm{L}} \\ qvB-\mathrm{mg}·\mathrm{cos\alpha }-\mathrm{qEsin}\alpha =2\mathrm{qE}·\mathrm{sin\alpha }-2\mathrm{mg}(1-\mathrm{cos\alpha }) \end{gathered}$$
$$\begin{gathered} qvB=3\mathrm{qE}·\frac{\mathrm{qE}}{\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(\mathrm{gE}\right)}^2}}+3mg·\frac{\mathrm{mg}}{\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}}-2\mathrm{mg} \\ qvB=3·\frac{{\left(qE\right)}^2+{\left(\mathrm{mg}\right)}^2}{\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}}-2\mathrm{mg} \\ qvB=3\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}-2\mathrm{mg} \\ {\mathrm{v}}^2=\frac{2\mathrm{L}}{\mathrm{m}}·\left(qE·\mathrm{sin\alpha }-\mathrm{mg}(1-\mathrm{cos\alpha }\right)) \\ {\mathrm{v}}^2=\frac{2\mathrm{L}}{\mathrm{m}}·\left(qE·\frac{\mathrm{qE}}{\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}}-\mathrm{mg}+\mathrm{mg}·\frac{\mathrm{mg}}{\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}}\right) \\ {\mathrm{v}}^2=\frac{2\mathrm{L}}{\mathrm{m}}\left(\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}-\mathrm{mg}\right) , \sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}=\mathrm{A} \\ {\mathrm{v}}^2=\frac{2\mathrm{L}}{\mathrm{m}}(\mathrm{A}-\mathrm{mg}) \\ \mathrm{v}=\sqrt{\frac{2\mathrm{L}}{\mathrm{m}}(\mathrm{A}-\mathrm{mg})} \\ qvB=3\mathrm{A}-2\mathrm{mg} \to \mathrm{B}=\frac{3\mathrm{A}-2\mathrm{mg}}{qv} \\ \mathrm{B}=\frac{3\mathrm{A}-2\mathrm{mg}}{q}\sqrt{\frac{\mathrm{m}}{2\mathrm{L}\left(\mathrm{A}-\mathrm{mg}\right)}} \\ \mathrm{B}=\frac{3\sqrt{{\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2}-2\mathrm{mg}}{\mathrm{q}}·\sqrt{\frac{\mathrm{m}}{2\mathrm{L}({\left(\mathrm{mg}\right)}^2+{\left(qE\right)}^2) ·mg}}. \end{gathered}$$