m = 1 kg
[tex]\displaystyle{ A = \frac{\sqrt{2}}{4} \ m }[/tex]
E = 4 J
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a) [tex]\displaystyle{ \omega }[/tex] = ?
b) x = ? v = ? (Ec = Ep)
c) F = ?
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a)
[tex]\displaystyle{ E = \frac{K \cdot A^{2}}{2} \rightarrow K = \frac{2E}{A^{2} }[/tex]
[tex]\displaystyle{ K = \frac{2 \cdot 4}{\frac{2}{16}}= \frac{8}{1} \cdot \frac{16}{2} = 64 \ N/m }[/tex]
[tex]\displaystyle{ K = m \cdot \omega ^{2} \rightarrow \omega = \sqrt{\frac{K}{m}} }[/tex]
[tex]\displaystyle{ \omega = \sqrt{64} = 8 \ rad/s }[/tex]
b)
Ec = Ep (1)
E = Ec + Ep (2)
(1) + (2) ⇒ E = 2 × Ec
[tex]\displaystyle{ Ec = \frac{mv^{2}}{2} }[/tex]
[tex]\displaystyle{ E = mv^{2} }[/tex]
[tex]\displaystyle{ v = \sqrt{\frac{E}{m}} = \sqrt{4} = 2 \ m/s }[/tex]
[tex]\displaystyle{ x = A \cdot sin (\omega t + \phi _{0}) }[/tex]
[tex]\displaystyle{ x = \frac{\sqrt{2}}{4} \cdot sin (8t + \phi _{0}) }[/tex]
c)
[tex]\displaystyle{ Ep = \frac{E}{2} }[/tex] (3)
[tex]\displaystyle{ Ep = \frac{K \cdot x^{2}}{2} }[/tex] (4)
[tex]\displaystyle{ (3) + (4) \rightarrow \frac{K \cdot x^{2}}{2} = \frac{E}{2} }[/tex]
[tex]\displaystyle{ E = x^{2} \cdot K \rightarrow x = \sqrt{\frac{E}{K}} }[/tex]
[tex]\displaystyle{ x = \sqrt{\frac{4}{64}} = \sqrt{\frac{1}{16}} }[/tex]
[tex]\displaystyle{ x = \frac{1}{4} = 0,25 \ m }[/tex]
[tex]\displaystyle{ F = -K \cdot x = -64 \cdot \frac{1}{4} }[/tex]
[tex]\displaystyle{ F = -\frac{64}{4} = -16 \ N }[/tex]
