AP Physics 1: Energy Bar Charts

Mastering the flow of energy is critical to solving complex problems in AP Physics 1, but tracking multiple forms of energy across different states can feel chaotic. An energy bar chart—often called an LOL diagram (LOL standing for "Levels Of energy")—provides a powerful visual and systematic framework to organize this chaos. It transforms the abstract principle of energy conservation into a concrete, step-by-step problem-solving tool, allowing you to clearly see energy storage, transfer, and transformation at a glance before you write a single equation.

The Core Framework: What is an Energy Bar Chart?

An energy bar chart is a qualitative visual model that represents how energy is stored within a defined system at different moments in time and how it is transferred into or out of that system. The chart uses bars of varying heights to represent the amounts of different categories of energy. The standard "LOL" format divides the chart into three distinct columns: the initial state, any work done by external forces, and the final state. This structure directly mirrors the Conservation of Energy equation, $E_i+W=E_f$, where $E_i$ is the initial total energy of the system, $W$ is the work done on the system by external forces (forces from objects outside your defined system), and $E_f$ is the final total energy.

The primary energy categories you will represent are:

• Kinetic Energy ($K$): Energy of motion, given by $K=\frac{1}{2}m{v}^2$.
• Gravitational Potential Energy ($U_g$): Energy stored due to height in a gravitational field, given by $U_g=mgh$, where $h$ is measured from a chosen zero reference point.
• Elastic/Spring Potential Energy ($U_s$): Energy stored in a stretched or compressed spring, given by $U_s=\frac{1}{2}k{x}^2$.
• Internal/Thermal Energy ($E_{int}$): Energy associated with friction, air resistance, or deformation. An increase in this bar signifies energy "lost" from the mechanical system, often represented as $\Delta E_{th}$.

Constructing Your Chart: A Step-by-Step Guide

Building an accurate energy bar chart is a systematic process that forces you to clarify your analysis before doing math.

Step 1: Define Your System. This is the most crucial step. The system is the collection of objects whose energy you are tracking. Everything else is part of the external environment. For example, if you are analyzing a block sliding down an incline, your system could be just the block (where the incline's normal force and friction are external) or the block + the Earth + the incline (making gravity an internal force and friction possibly internal if the incline is part of the system). Your choice determines what energy forms appear in the bars and what counts as external work ($W$).

Step 2: Identify Initial and Final States. Clearly decide on the two distinct snapshots in time you are comparing. Is it a ball at the top of its flight versus at the ground? A spring compressed versus at its relaxed length? Be specific about the position and speed of objects at these moments.

Step 3: Sketch the Bars for the Initial State ($E_i$). In the first column, draw bars only for the types of energy present in the system at the initial moment. If an object is at rest at a height, you would draw a bar for $U_g$ but none for $K$. The combined height of all bars in this column represents the system's total initial energy.

Step 4: Analyze the Work Column ($W$). In the middle column, you account for energy transferred across the system boundary. This column is not for energy storage; it's for energy flow.

• If an external force does positive work on the system (e.g., a hand pushing a cart), you draw an upward arrow labeled "$W$" coming from the baseline. This adds energy to the system.
• If external forces do negative work (e.g., kinetic friction acting on your system-block), you draw a downward arrow labeled "$W$." This removes energy from the system.
• If no net external work is done, this column is left blank.

Step 5: Sketch the Bars for the Final State ($E_f$). In the third column, draw bars for all energy forms present in the system at the final moment. The total height of the final bars must equal the combined height of the initial bars plus or minus the work arrow. This visual "bookkeeping" is the essence of conservation of energy.

From Chart to Equation: Solving Problems

The power of the LOL diagram is that it directly scaffolds writing the correct conservation of energy equation. Once your chart is drawn, you translate it into mathematics by writing an term for each bar and arrow.

Example: A 2 kg box slides from rest down a 5-meter-long, 30° frictionless incline. Let the system be the box + the Earth. Use an energy bar chart to find its speed at the bottom.

1. System: Box + Earth. This makes gravity an internal force, so $U_g$ appears in our bars, and there is no external work from gravity.
2. States: Initial = box at top of incline ($v_i=0$). Final = box at bottom ($h_f=0$).
3. Chart: Initial column has only a $U_g$ bar. Work column is blank (normal force is perpendicular to motion, so no work; no friction). Final column has only a $K$ bar.
4. Translation: The chart shows $U_{g,i}=K_f$.
5. Equation: $$mg{h}_i=\frac{1}{2}m{v}_f^2$$

Here, $h_i=(5\text{m})\sin (30^{\circ })=2.5\text{m}$. Solving: $$(2\text{kg})(9.8{\text{m/s}}^2)(2.5\text{m})=\frac{1}{2}(2\text{kg})v_f^2$$ $$v_f=\sqrt{2(9.8)(2.5)}\approx 7.0\text{m/s}$$

For scenarios with friction, the work done by friction (an external force if the surface is outside your system) would appear as a negative $W$ arrow. That energy is transferred out of the box-Earth system, often appearing as an increase in thermal energy in the surface and box, which you could choose to model as an $E_{int}$ bar in the final state if you included those objects in your system.

Common Pitfalls

1. Inconsistent System Definition: The most common error is to change your defined system mid-analysis. If you define the system as just the ball, then gravitational potential energy ($U_g$) does not get its own bar because the Earth is external. Instead, the work done by gravity would appear in the $W$ column. Conversely, if the system is ball + Earth, then $U_g$ appears as a bar and gravity does no external work. Decide on your system first and stick to it.

1. Misplacing Energy in the Work Column: Remember, the middle column is strictly for energy transfer (work), not for energy storage. Do not draw bars for kinetic or potential energy in this column. Only an upward or downward arrow representing the net work done by forces from the external environment belongs here.

1. Ignoring the Zero Reference for $U_g$: The height $h$ in $U_g=mgh$ must be measured from a consistent zero point that you choose. Your chart's $U_g$ bar is meaningful only relative to this point. For example, if you set $h=0$ at the bottom of a ramp, an object halfway up has a $U_g$ bar. If you set $h=0$ at the starting point, that same halfway object would have a negative $h$—this is algebraically correct but difficult to represent visually on a bar chart. It's simplest to choose your zero at the lowest point in the problem.

1. Forgetting Internal Energy from Friction: When non-conservative forces like friction act within your system (e.g., system = block + rough surface), they convert mechanical energy ($K$ and $U$) into internal/thermal energy ($E_{int}$). This results in a bar for $E_{int}$ in the final state column, not a $W$ arrow. Failing to account for this energy transformation will make your final energy bars incorrectly tall.

Summary

• An energy bar chart (LOL diagram) is a visual tool that organizes the application of the work-energy theorem and conservation of energy principle by separating initial energy, external work, and final energy into three distinct columns.
• The chart's construction begins with the critical step of defining your system, which determines what energy forms are stored internally and what interactions count as external work.
• The bars represent stored energy ($K$, $U_g$, $U_s$, $E_{int}$), while the middle column uses an arrow to represent energy transfer via work ($W$) done by net external forces.
• The completed chart provides a direct, visual equation: the total height from the initial bars plus/minus the work arrow must equal the total height of the final bars, which you then translate into a solvable algebraic equation.
• Consistently applying this framework helps you avoid common errors like misidentifying work, forgetting energy transformations due to friction, or applying an inconsistent zero point for potential energy.