Explain why the points$\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathit{A}$ and$\mathit{x}\mathbf{}\mathbf{=}\mathbf{-}\mathbf{}\mathit{A}$ in Fig. 7.23b are called turning points. How are the values of$\mathit{E}$ and$\mathit{U}$ related at a turning point?

The turning point is described as a point in which a change in the situation mainly occurs. In the turning point, the changes in the values of one particular point also occur.

According to the graph, at x = +A , the slope of the function$U_x$ is positive. The magnitude of the force also becomes$F_x<0$ , and the direction of the force is in the displacement’s opposite direction. Hence, if a particle moves in the +x direction, then it will travel in the -x direction.

According to the graph, at x = -A , the slope of the function is negative ,$U_x$. The magnitude of the force also becomes$F_x>0$ , and the direction of the force is at the displacement’s opposite direction. Hence, if a particle moves in the -x direction, then it will travel in the +x direction.

Thus, the points x = +A and x = -A are called turning points as at this point, the velocity of a particle changes.

The points x = + A and x = - A have the maximum displacement, and it occurs in the opposite direction. Moreover, when there is a maximum displacement, the kinetic energy also becomes zero.

The equation of the total energy is expressed as:

E = P +K

Here, E is the total energy, P is the potential, K and is the kinetic energy.

Substitute 0 for K and U for P in the above equation.

E = P + 0

E = U

Thus, the values of E and U becomes equal at a turning point.