Exercise:
Which of the following differential s describe a simple harmonic motion? For those that do derive a formal expression for the period for those that do not characterise the motion. abcliste abc utu_cosomega t^ abc vt Asinomega^ t abc xit+k^ ddotxit abc dot rt-beta^ rt abc psitC_cosalpha t-C_sinalpha t abc ddot yt-delta^ y^t abc dvxtB^ x abcliste

Solution:
abcliste abc No simple harmonic motion. The t^ in the cosine function causes the time erval between two consecutive maxima to become shorter and shorter. abc Simple harmonic motion with angular frequency omega^. The period is thus T fracpiomega^ In order to avoid unnecessary confusion it is strongly advised not to name the constant in the sine function omega^! abc Simple harmonic motion. The differential can be written in the standard form ddotxit -frack^xit This is the characteristic differential for a simple harmonic motion with angular frequency omega/k. It follows for the period T fracpiomegapi k abc No simple harmonic motion. The differential states that the first and not the second derivative of the function rt is a constant times the function itself. The solution is given by rt r_ e^-beta^ t as can readily be verified. abc Simple harmonic motion with angular frequency alpha i.e. with period T fracpialpha The phase shift and the amplitude of the superposition of the two oscillations can be found using a phasor diagram. abc No simple harmonic motion. The square on the right hand side makes the derivative negative no matter whether the displacement yt is positive or negative. As soon as the displacement is negative the negative acceleration drives the system away from the equlibrium and it will never return. abc No simple harmonic motion missing minus sign. The solution to this differential is xt x_ e^B t as can readily be verified. abcliste