Question

1.4 The potential ergy of an object in simple harmonic about an equilibrium at x = 0 is given by U (1/2)mo2x2, while its kinetic energy is K (1/2)mi2 by definition. (a) Use these facts. aid general expressions for x(t) and X(t) in SHM. to show that the total mechanical energy is constant. Etot-K + U = (1/2) mao2A2. (b) Assume that the total energy of some object is given by Etot-(-) mx2 + (-) mu'x2 where a) 1s just a constant number. Show that if this energy is conserved, such that dEtot/dt 0, then the object must obey the equation of motion. x =-ω, which implies SHM. (Use the chain nile ofderivation)

Answer 1

E_total = 1/2 mx^2 + 1/2 mw^2 x^2 equations (1) let x = Acos (omega + phi) m the sol^n of this eu^n: then Diff this eu^n w middot r to t x = -A sin(omega + phi) times w x = -Aw sin(omega + phi) Now eu^n equation (1) E_total = 1/2 m {-aw sin (omega + d)}^2 + 1/2 mw^2{ a cos(omega + phi)}^2 E_total = 1/2 mw^2 A^2{sin^2(omega + phi) + cos^2(omega + phi)} E_total = 1/2 mw^2 A^2 Diff eu^n equations (1) w. r tr t d E_total/dt = 1/2 mx derivation xdouble derivation + 1/2 mw^2 2 x x derivation) 0 = (m x double derivation + m w^2 x)x derivation given ie E_total = (n + 1) mx(x + w^2x) = 0 m xdouble derivation notequalto 0, So x double derivation + w^2 n = 0 or x double derivation = -w^2 x This in the eu^n of SHM