10.2.2 Energy Bar Charts

If an object of mass $m$ interacts with the earth (or other astronomical body) and there are no other forces, the energy principle for the object $+$ earth system, Equation 10.3, is

Δ K + Δ U_{G} = (K_{f} - K_{i}) + (U_{Gf} - U_{Gi}) = 0

(10.10)

We can rewrite this as

K_{i} + U_{Gi} = K_{f} + U_{Gf}

(10.11)

The quantity $E_{mech} = K_{tot} + U_{int}$ , the total macroscopic kinetic and potential energy, is called the mechanical energy of the system. Equation 10.11 is telling us that—in this situation—the mechanical energy does not change as the object undergoes vertical motion. Whatever initial mechanical energy the system had before the vertical motion, it has exactly the same mechanical energy after the motion. Kinetic energy may be transformed into potential energy during the motion, or vice versa, but their sum remains unchanged.

A quantity that is unchanged during an interaction is said to be conserved, and Equation 10.11 is our first statement of the law of conservation of energy. We’ll explore energy conservation thoroughly later in this chapter, but we’re already beginning to see the power of thinking about mechanical systems in terms of energy.

Equation 10.11, which is really just energy accounting, can be represented graphically with an energy bar chart. For example, FIGURE 10.5 is a bar chart showing how energy is transformed when a ball is tossed straight up. Kinetic energy is gradually transformed into potential energy as the ball rises, then potential energy is transformed into kinetic energy as it falls, but the combined height of the bars does not change. That is, the mechanical energy of the ball $+$ earth system is conserved.

The figure illustrates the energy bar chart for a ball tossed upward in the air from a horizontal surface. Five instances of the position of the ball and its respective bar chart are shown.

FIGURE 10.5 Full Alternative Text

NOTE Most bar charts have no numbers. Their purpose is to think about the relative changes—what’s increasing, what’s decreasing, and what remains constant; there’s no significance to how tall a bar is.

EXAMPLE 10.2 Dropping a watermelon

A 5.0 kg watermelon is dropped from a third-story balcony, 11 m above the street. Unfortunately, the water department forgot to replace the cover on a manhole, and the watermelon falls into a 3.0-m-deep hole. How fast is the watermelon going when it hits bottom?

MODEL Let the system consist of both the earth and the watermelon, which we model as a particle. Assume that air resistance is negligible. There are no external forces, and the motion is vertical, so the system’s mechanical energy is conserved.

VISUALIZE FIGURE 10.6 shows both a before-and-after pictorial representation and an energy bar chart. Initially the system has gravitational potential energy but no kinetic energy. Potential energy is transformed into kinetic energy as the watermelon falls. Our choice of the $y$ -axis origin has placed the zero of potential energy at ground level, so the potential energy is negative when the watermelon reaches the bottom of the hole. Even so, the combined height of the two bars has not changed.

The figure illustrates the initial and final positions of a watermelon in a System and its energy bar chart.

FIGURE 10.6 Full Alternative Text

SOLVE The energy principle for the watermelon $+$ earth system, written as a conservation statement, is

K_{i} + U_{Gi} = 0 + m g y_{0} = K_{f} + U_{Gf} = \frac{1}{2} m v_{1}^{2} + m g y_{1}

Solving for the impact speed, we find

\begin{matrix} v_{1} & = & \sqrt{2 g (y_{0} - y_{1})} \\ = & \sqrt{2 (9.80 {m/s}^{2}) (11.0 m - (- 3.0 m))} \\ = & 17 m / s \end{matrix}

REVIEW A speed of $17 m/s \approx 35 mph$ seems reasonable for the watermelon after falling $\approx 4$ stories. In thinking about this problem, you might be concerned that, once below ground level, potential energy continues being transformed into kinetic energy even though the potential energy is “less than none.” Keep in mind that the actual value of $U$ is not relevant because we can place the zero of potential energy anywhere we wish, so a negative potential energy is just a number with no implication that it’s “less than none.” There’s no “storehouse” of potential energy that might run dry. As long as the interaction acts, potential energy can continue being transformed into kinetic energy.

10.2.2 Energy Bar Charts

FIGURE 10.5 Energy bar charts for a ball tossed into the air.

EXAMPLE 10.2 Dropping a watermelon

FIGURE 10.6 Pictorial representation and energy bar chart of the watermelon $+$ earth system.