Question

un mobil efectueaza o miscare oscilatorie armonica. atunci cand miscarea elongatiei x1 este 2cm valoarea vitezei este 5m/s iar can elobgatia este 3 cm atunci vteza este 4 m/s. calculati amplitudinea perioada si frecventa

Answer 1

[tex]\displaystyle \text{Se da:}\\ \\ x_1=2cm=0,02m\\ \\ v_1=5\frac ms\\ \\ x_2=3cm=0,03m\\ \\ v_2=4\frac ms\\ \\ A=?m\\ \\ T=?s\\ \\ V=?Hz\\ \\ \\[/tex]


$\text{Formule:}\\ \\ \text{Ecuatia uzuala ce descrie miscarea unui oscilator armonic:}\\ \\ x(t)=A\times\sin{(w\times t+\varphi_0)}\\ \\ \text{Nu e nici o vorba despre faza initiala, deci reducem la:}\\ \\ x(t)=A\times\sin{(w\times t)},\text{ unde trebuie sa aflam A si w}\\ \\ \\ \text{Sa formam un sistem de ecuatii din cele 2 cazuri propuse in problema:}\\ \\ x_1=A\times\sin{(w\times t_1)}\\ \\ v_1=w\times A\times\cos{(w\times t_2)}(\text{formula vitezei})\\ \\ \text{Sistem:}$


[tex]\displaystyle \begin{cases}x_1=A\times\sin{(w\times t_1)}\\v_1=w\times A\times\cos{(w\times t_1)}\end{cases}\Leftrightarrow \begin{cases}\frac {x_1}{A}=\sin(w\times t_1)\\ \frac{v_1}{w\times A}=\cos(w\times t_1)\end{cases}\\ \\ \begin{cases}\frac {x_1^2}{A^2}=\sin^2(w\times t_1)\\ \frac{v_1^2}{w^2\times A^2}=\cos^2(w\times t_1)\end{cases}\\ \\ \text{Adunam ecuatiile:}\\ \\ \frac {x_1^2}{A^2}+\frac{v_1^2}{w^2\times A^2}=\sin^2(w\times t_1)+\cos^2(w\times t_1)\\ \\[/tex]


[tex]\displaystyle \text{Iar din trigonometrie stim ca: } \sin^2(x)+\cos^2(x)=1\\ \\ \frac {x_1^2}{A^2}+\frac{v_1^2}{w^2\times A^2}=1\text{ (1)}\\ \\ \\ \text{Analogic cu } x_2 \text{ si } v_2!\\ \\ \frac {x_2^2}{A^2}+\frac{v_2^2}{w^2\times A^2}=1\\ \\ \text{Egalam ecuatiile:}\\ \\[/tex]


[tex]\displaystyle \frac {x_1^2}{A^2}+\frac{v_1^2}{w^2\times A^2}=\frac {x_2^2}{A^2}+\frac{v_2^2}{w^2\times A^2},\text{Scapam de } A^2\\ \\ x_1^2+\frac{v_1^2}{w^2}=x_2^2+\frac{v_2^2}{w^2}\\ \\ w^2\times x_1^2+v_1^2=w^2\times x_2^2+v_2^2\\ \\ w^2\times x_1^2-w^2\times x_2^2=v_2^2-v_1^2\\ \\ w^2\times(x_1^2-x_2^2)=v_2^2-v_1^2\\ \\ w=\sqrt\frac{v_2^2-v_1^2}{x_1^2-x_2^2}\\ \\ \\[/tex]


[tex]\displaystyle \text{Am aflat w, iar din formula: }\\ \\ T=\frac{2\times\pi}{w}\\ \\ \text{Aflam perioada, iar din formula:}\\ \\ V=\frac 1T\\ \\ \text{Aflam frecventa}\\ \\ \\[/tex]


[tex]\displaystyle \text{Pentru amplitudine, lucarm cu formula (1):}\\ \\ \frac {x_1^2}{A^2}+\frac{v_1^2}{w^2\times A^2}=1\\ \\ A^2=x_1^2+\frac{v_1^2}{w^2}\\ \\ A=\sqrt{x_1^2+\frac{v_1^2}{w^2}}\\ \\ \\ \text{Calcule:}\\ \\ w=\sqrt\frac{4^2-5^2}{0,02^2-0,03^2}\approx 134,164\frac{rad}s\\ \\ A=\sqrt{0,02^2+\frac{5^2}{134,164^2}}\approx 0,0423m\\ \\ T=\frac{2\times 3,14}{134,164}\approx 0,0468s\\ \\ V=\frac 1{0,0468}\approx 21,36Hz[/tex]