Question

In this problem we will explore the energy of a mass on a spring.

(a) Write down the kinetic energy (as a function of time) of a mass m oscillating on a spring with a spring constant k according to the equation

x(t) = A cos(ωt + φ). Give your answer in terms of m, ω, φ, A, and t.

(b) Now write down the elastic potential energy as a function of time for the same mass. Give your answer in terms of k, ω, A, φ, and t.

(c) Now write down the total energy, remember that ω = ?k/m, and find the total energy in terms of just k and A , showing that the energy is constant in time.

Answer 1

a(t) = A cos (cotto I Kimetic energy = Imre +(+)= dx = -Aw sin (60%+4) ate Kifis im A² w² sin (wtto) b) Elastic potential energy nerg = kx² chergy KAcos? (wt +9) c) Total energy = 1 mv² + 1 Ro? = A (k) sün?w0t+) + k Aʼces? lut+q) = uka? ( sin”(cut +40+ cos? (wt+8)) E is independent of time.

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