Q.To start a pendulum swinging you pull it to one side and release it. What kind of energy is transferred to the mass in doing this?

A particle in S.H.M. possesses kinetic energy everywhere except at the two extreme positions, and potential energy everywhere except at the mean position; the two interconvert continuously as the particle oscillates, but their SUM -- the total energy -- stays constant throughout the motion. Kinetic energy at displacement x is E_k = (1/2)momega^2*(A^2 - x^2) = (1/2)k(A^2 - x^2), which as a function of time varies as cos^2 of the phase. Potential energy, found as the work done against the restoring force to reach displacement x, is E_p = (1/2)kx^2 = (1/2)momega^2*x^2, varying with time as sin^2 of the phase.

Adding the two, the x-dependence cancels completely: E = E_k + E_p = (1/2)momega^2A^2 = (1/2)kA^2 = (1/2)mv_max^2, a constant independent of both position and time -- energy conservation in S.H.M. Written using frequency n = omega/(2pi), this is E = 2pi^2mn^2A^2 = 2pi^2m*A^2/T^2, showing total energy is directly proportional to mass, the square of amplitude, and the square of frequency (or force constant), and inversely proportional to the square of the period. At the mean position all the energy is kinetic; at the extremes all of it is potential; at x = +/-A/sqrt(2) the energy splits exactly evenly between the two forms; and at x = +/-A/2, the energy is 25% potential and 75% kinetic.