K is T / . What is the potential energy ( U ) of a particle executing simple harmonic motion? In Chapter , we have seen that the concept of potential energy is possible only for conservative forces. The spring force F = –kx is a conservative force, with associated potential energy U = k x ( .
) Hence the potential energy of a particle executing simple harmonic motion is, U ( x ) = x k cos ( + ) = k A ( . ) Thus, the potential energy of a particle executing simple harmonic motion is also periodic, with period T / , being zero at the mean position and maximum at the extreme displacements. It follows from Eqs. ( .
) and ( . ) that the total energy, E , of the system is, E = U + K cos ( + ) + sin ( + ) = k A k A cos ( + ) + sin ( + ) = k A Using the familiar trigonometric identity, the value of the expression in the brackets is unity. Thus, E = k A ( . ) The total mechanical energy of a harmonic oscillator is thus independent of time as expected for motion under any conservative force.
The time and displacement dependence of the potential and kinetic energies of a linear simple harmonic oscillator are shown in Fig. . . Observe that both kinetic energy and potential energy in SHM are seen to be always positive in Fig.
. . Kinetic energy can, of course, be never negative, since it is proportional to the square of speed. Potential energy is positive by choice of the undermined constant in potential energy.
Both kinetic energy and potential energy peak twice during each period of SHM. For x = , the energy is kinetic; at the extremes x = ± A , it is all potential energy. In the course of motion between these limits, kinetic energy increases at the expense of potential energy or vice-versa. u Example .
A block whose mass is kg is fastened to a spring. The spring has