A marble moves along the x-axis. The potential-energy function is shown in Fig. E7.36. (a) At which of the labeled x-coordinates is the force on the marble zero? (b) Which of the labeled x-coordinates is a position of stable equilibrium? (c) Which of the labeled x-coordinates is a position of unstable equilibrium?

Potential Energy:

The required energy for the movement of a particle from one position to another position is called potential energy. The slope of the potential energy-distance curve gives the force on the particle.

(a)

The force on one atom is given as:

$F\left(x\right)=-\frac{dU}{dx}$

The zero force occurs at the zero slope of the given potential energy-position graph. The slope of the curve is zero at points b and d, so the force on the marble is zero for x coordinates of point b and point d.

Therefore, the force on marble is zero at x coordinates of point b and point d.

(b)

The x coordinate of point b is at stable equilibrium because the marble uses potential energy to move away from point b.

Therefore, the x coordinate for point b is at a stable equilibrium.

(c)

The potential energy of the marble decreases as marble moves away from point d. The decrease in the potential energy of marble shows that the kinetic energy of marble increases as marble moves away from point d, which makes the marble unstable.

Therefore, the x coordinate corresponding to point d is a position of unstable equilibrium.