The Simple harmonic oscillator: A particle of mass m constrainedto move in the x-direction only is subject to a force F(x) = −kx,where k is a constant. Show that the equation of motion can bewritten in the form d^2x/dt2 + ω^2ox = 0, where ω^2o = k/m . (a)Show by direct substitution that the expression x = A cos ω0t + Bsin ω0t where A and B are constants, is a solution and explain thephysical significance of the quantity ω0. Show that an alternativesolution may be expressed as x = xmax cos(ω0t + φ) where xmax isthe amplitude and φ is the phase constant of oscillation. (b) Findin terms of m and ω0 the change in the potential energy U(x) − U(0)of the particle as it moves from the origin. Explain the physicalsignificance of the sign in your result. (c) The potential energyis subject to an arbitrary additive constant; it is convenient totake U(0) = 0. The kinetic energy T(x) = 1/2mv^2. Show bydifferentiation that the particle’s total energy (E = T +V ) isconstant. Express E as a function of the particle’s (a) maximumdisplacement xmax and (b) maximum velocity vmax.
