s3 - Phys 263/Amath 261 Assignment 3 SOLUTIONS 1 The...

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Unformatted text preview: Phys 263/Amath 261 Assignment 3 SOLUTIONS 1. The equation of motion of the oscillator is ¨ x + 4 x = 0 (1) where ω 2 o = k/m = 4. The general solution, from class, is x = A sin( ω t + φ ) (2) which we can also write x = A sin ω o t cos φ + A cos ω t sin φ (3) substituting new constants A cos φ = B and A sin φ = C (4) we get x = B sin ω t + C cos ω t (5) with A = √ B 2 + C 2 and φ = arctan C B (6) (a) Solving equation 3 for initial conditions: x ( t = 0) = √ 3 = 0 + C cos(0) ⇒ C = √ 3 . (7) Differentiating equation 3 gives ˙ x = ω B cos ω t- ω C sin ω t (8) and solving for initial conditions gives- 2 = 2 B cos(0) + 0 ⇒ B =- 1 (9) Therefore x = √ 3 cos(2 t )- sin(2 t ) (10) (b) From equation 6 A = √ 1 + 3 = 2 (11) is the amplitude of oscillations. (c) The initial zero in the displacement happens when √ 3 cos(2 t ) = sin(2 t ) (12) √ 3 = tan(2 t ) (13) arctan √ 3 = 2 t (14) Or t = π/ 6. 1 2. (a) The potential integral over one period T is h U i time = 1 T Z T dtkA 2 sin 2 ( ω t + φ ) / 2 = kA 2 4 (15) and for the kinetic energy h K i time = 1 T Z T dtkA 2 cos 2 ( ω t + φ ) / 2 = kA 2 4 (16)...
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s3 - Phys 263/Amath 261 Assignment 3 SOLUTIONS 1 The...

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