ENERGY IN SIMPLE HARMONIC MOTION
We can learn even more about simple harmonic motion by using energy considerations. Take another look at the mass oscillating on the end of a spring in Fig. 13-1. We’ve already noted that the spring force is the only horizontal force on the body. The force exerted by an ideal spring is a conservative force, and the vertical forces do no work, so the total mechanical energy of the system is conserved. We will also assume that the mass of the spring itself is negligible.
The kinetic energy of the body is K = tmv2, and the potential energy of the spring is U = tlcx2, just as in Section 7-3. (It would be helpful to review that section.) There are no non conservative forces that do work, so the total mechanical energy E = K + U is conserved:
The total mechanical energy E is also directly related to the amplitude A of the motion. When the body reaches the point x = A, its maximum displacement from equilibrium, it momentarily stops as it turns back toward the equilibrium position . when x = A (or -A), v = O.At this point the energy is entirely potential, and E = fkA2• Because E is constant, this quantity equals E at any other point. Combining this expression with Eq. (13-20), we get
This agrees with Eq. (13-15), which showed that v oscillates between wA and + JA. Figure 13-10 shows the energy quantities E,K, and U at x = 0, x = ±A/2, and x = ±A. Figure 13-11′ a graphical display ofEq. (13-21); energy (kinetic, potential, and total) is plotted vertically and the coordinate x is plotted horizontally. The parabolic curve in Fig. 13-11a represents the potential energy U= kx2.The horizontal line represents the total mechanical energy E, which is constant and does not vary with x. This line intersect the potential-energy curve at x = -A and x = A, where the energy is entirely potential. At any value of x between -A and A, the vertical distance from the x-axis to the parabola is U; since E = K + U, the remaining vertical distance up to the horizontal line is K. Figure 13-11b shows both K and U as functions of x. As the body oscillates between -A and A, the energy is continuously transformed from potential to kinetic and back again.
Figure 13-1la shows the connection between the amplitude A and the correspond total mechanical energy E = ~kA2. If we tried to make x greater than A (or less -A U would be greater than E, and K would have to be negative. But K can never x can’t be greater than A or less than -A.