1 The simple harmonic oscillator is a good approximation for the vibrational dynamics of a diatomic molecule. The SHO is governed by Hooke's Law: E-km'mdtz Eq.'l Rearranging this we obtain the following differential equation (equation of motion): d'x kill +=0 d!2 m Eq. 2 We guessed (t) that x(t) = A sin(wt+d) was a solution for Eq. 2. Here, A is the maximum displacement of the oscillator, and is known as the angular frequency and has units of radians per second (rad s"). n is frequency in Hertz (Hz, with units 3'1). The term cl is the phase factor (in rad); it sets the value of x for i=0. a) By explicit differentiation, show that x(t) = A cos(wt+d) is also a solution to the differential equation above and derive an expression for in terms of k and m. b) For n = 10'14 s", A = 10'11 m (the largest change in the bond length for a typical molecule), and d = 0 radians, compute the (earliest) time, t*, at which x(t*) = A cos(wt*+d) = 0. 0) Convert wt* into radians. 0!) Graph x(t) = A cos(wt+d) for n = 10'14 s", d = O radians over the range t= 01.25x10'14 sec. You may do this analytically, but you may want, construct a table of ten evenly spaced times over the range, plot the points, and draw a smooth curve through them. Be sure to label your axes clearly and accurately. 2. Derive the SH0 potential energy, from Eq. 1 starting from and draw the potential energy vs. x