2. Having a simple harmonic oscillator described by: \ddot{x}(t) + \omega^2 x(t) = 0, \\ knowing that x(0) = 0 and \dot{x}(0) = 10, derive the formula and calculate \\ the total energy of the system. How to comment the result in view of the \\ principle of conservation of total mechanical energy?

          2. Having a simple harmonic oscillator described by: \ddot{x}(t) + \omega^2 x(t) = 0, \\ knowing that x(0) = 0 and \dot{x}(0) = 10, derive the formula and calculate \\ the total energy of the system. How to comment the result in view of the \\ principle of conservation of total mechanical energy?

2. Having a simple harmonic oscillator described by: ẍ(t) + ω^2 x(t) = 0,
knowing that x(0) = 0 and ẋ(0) = 10, derive the formula and calculate
the total energy of the system. How to comment the result in view of the
principle of conservation of total mechanical energy?

Added by Sarah F.

The Chemistry Maths Book

Erich Steiner 2nd Edition

Chapter 12

Instant Answer

Solved by Expert David Morabito

Step 1

When t = 0, x(0) = A cos(ϕ) = 0. Since the cosine function is zero when the argument is π/2, 3π/2, etc., we can conclude that ϕ = π/2 or ϕ = 3π/2. Differentiating x(t) with respect to t, we get v(t) = -Aω sin(ωt + ϕ). When t = 0, v(0) = -Aω sin(ϕ) = 1. Since Show more…

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Having a simple harmonic oscillator described by x(t) = A cos(Ï‰t + Ï†), where A is the amplitude, Ï‰ is the angular frequency, and Ï† is the phase constant. Knowing that x(0) = 0 and v(0) = 1, derive the formula and calculate the total energy of the system. How do you interpret the result in view of the principle of conservation of total mechanical energy?