Question 7.17: Motion and the energy of a spring-mass system. An unstretche...

The free body diagram in Fig. 7.14a shows the gravitational and elastic forces that act on m when the string is cut. These are conservative forces with total potential energy   V(x)=-m g x  +  \frac{1}{2} k x^2,  wherein V(0) = 0. The kinetic energy is   K=\frac{1}{2} m \dot{x}^2.  Since the total energy initially is zero, the energy  principle (7.73) gives

K+V=E \text {, a constant. }                  (7.73)

\frac{1}{2} m \dot{x}^2  +  \frac{1}{2} k x^2  –  m g x=0.               (7.85a)

Because this is the first integral of the equation of motion for which the path variable is x, differentiation of (7.85a) with respect to x (or t) yields the familiar equation of motion:

m \ddot{x}  +  k x=m g .                (7.85b)

The motion is a gravity induced, free vibration of a simple harmonic oscillator. The general solution of this equation is

x=\frac{m g}{k}  +  A \sin p t  +  B \cos p t,                 (7.85c)

in which   p=\sqrt{k / m}.  The initial circumstances  x(0) = 0 and   \dot{x}(0)=0  require A = 0 and   B=-m g / k  in (7.85c), so the motion of the mass is described by

x(t)=\frac{m g}{k}(1  –  \cos p t)                  (7.85d)

It is seen from (7.85b) that   x_S \equiv m g / k  is the static equilibrium displacement. Hence, (7.85d) shows that the load oscillates about the equilibrium state with  circular frequency   p=\sqrt{k / m}   and amplitude equal to  x_S . The reader may readily confirm that dV / dx = 0 at  x_S,  and hence   V_{\min }=V\left(x_S\right)=-\frac{1}{2} k x_S^2.   As a  consequence,   K_{\max }  +  V_{\min }=E=0   yields the maximum speed  v_{\max }=p x_S.  It is simpler, however, to note from (7.85d) that   \dot{x}(t)=p x_S \sin p t,  hence  v_{\max }=p x_S.

Now let us consider the energy curve. Due to the static displacement, the curve described by the energy equation (7.85a) in terms of x and   \dot{x}  is an ellipse whose center is shifted a distance   x_S  along the x-axis. To see this, introduce the coordinate transformation   z=x  –  x_S,  which describes the motion of the load relative to its equilibrium position. Usc of this relation in (7.85a) and (7.85b) yields the corresponding transformed equations:

\left(\frac{\dot{z}}{p_{x_S}}\right)^2  +  \left(\frac{z}{x_S}\right)^2=1           (7.85e)

\ddot{z}  +  p^2 z=0 .                 (7.85f)

Clearly, the energy equation (7.85e) is an ellipse centered at the origin in the phase plane of z and  \dot{z}:  hence, the motion is periodic with circular frequency p  and symmetric amplitude   z_{\max }=x_S.  The energy curve is the graph of(7.85e) shown in Fig. 7.15. Both the original and transformed variables are indicated. The geometry characterizes the motion of a simple harmonic oscillator described by (7.85f) whose solution is ju st the transform ation of (7.85d) given by   z=-x_S \cos p t .