Bar Chart Illustrations

One tool which can be utilized to express an understanding of the work-energy theorem is a bar chart. A work-energy bar chart represents the amount of energy possessed by an object by means of a vertical bar. The length of the bar is representative of the amount of energy present, with a longer bar representing a greater amount of energy. In a work-energy bar chart, a bar is constructed for each form of energy. Consistent with the work-energy relationship discussed in this lesson, the sum of all forms of initial energy plus the work done on the object by external forces equals the sum of all forms of final energy.

In a work-energy bar chart, a bar is used to represent the amount of each term in the above equation. Consequently, the sum of the bar heights for the initial condition (initial energy + external work) must equal the sum of the bar heights for the final condition (final energy). Since the potential energy comes in two forms - the elastic potential energy stored in springs (PE_spring) and the gravitational potential energy (PE_grav) - the above equation is rewritten as

In this portion of Lesson 2, we will investigate the use and meaning of work-energy bar charts and make an effort to apply this understanding to a variety of motions involving energy changes or energy transformations.

Procedure for Constructing Bar Charts

The following procedure might be useful for constructing work-energy bar charts:

• analyze the initial and final states of the object in order to make decisions about the presence or absence of the different forms of energy
• analyze the forces acting upon the object during the motion to determine if external forces are doing work and whether the work (if present) is positive or negative
• construct bars on the chart to illustrate the presence and absence of the various forms of energy for the initial and final state of the object; the exact height of the individual bars is not important; what is important is that the sum of the heights on the left of the chart is balanced by the sum of the heights on the right of the chart

The above procedure is illustrated below.

Consider a ball falling from the top of a pillar to the ground below; ignore air resistance. The initial state is the ball at rest on top of the pillar and the final state is the ball just prior to striking the ground. Given this motion and the identification of the initial and final state of the ball, decisions can be made about the presence and absence of each form of energy. Since there is no motion at the top of the pillar, there is no initial kinetic energy. Since the ball is elevated above the ground while on top of the pillar, there is an initial gravitational potential energy (PE_grav). There are no springs involved; thus, there is neither initial nor final elastic potential energy (PE_spring). In the final condition (just prior to striking the ground), the ball is moving. Thus, there is a final kinetic energy. And finally, the ball is no longer elevated above the ground so there is no final gravitational potential energy. The ball falls under the influence of gravity (an internal force) alone. Thus there are no external forces present nor doing work. The diagram at the right summarizes this analysis.

To complete the bar chart, an arbitrarily decided height for each bar is decided upon and a bar is constructed for each form of energy. As mentioned before, it is not important exactly how high each bar is. It is only important that the sum of the bar heights on the left balance the sum of the bar heights on the right.

 

Observe that this work-energy bar chart reveals that

• there is no kinetic and elastic potential energy in the initial state
• there is no gravitational and elastic potential energy in the final state
• there is no work done by external forces
• the sum of the heights on the right (5 units) equals the sum of the heights on the left (5 units)

It is not important as to how high the two bars are in the above bar chart. If the bars were 4 units high instead of 5 units high, then it would be an equally acceptable bar chart. The decision about bar height is entirely arbitrary.

The Example of a Skidding Car

Now we will repeat the process for a car which skids from a high speed to a stop across level ground with its brakes applied. The initial state is the car traveling at a high speed and the final state is the car at rest. Initially, the car has kinetic energy (since it is moving) but does not have gravitational potential energy (since the height is zero) or elastic potential energy (since there are no springs). In the final state of the car, there is neither kinetic energy (since the car is at rest) nor potential energy (since there is no height nor springs). The force of friction between the tires of the skidding car and the road does work on the car. Friction is an external force. Friction does negative work since its direction is opposite the direction of the car's motion.

 

Now that the analysis is complete, the bar chart can be constructed. The chart must be consistent with the above analysis.

 

Observe that the bar for work is a downward bar. This is consistent with the fact that the work done by friction is negative work. Whenever negative work is done by external forces, the W_ext bar will be a downward bar. Note also that the sum of the bar height on the left side (+5 plus -5) is the same as the sum of the bar heights on the right side of the chart. One final comment is in order: even though the height of all bars on the left equals the height of all bars on the right, energy is not conserved. The bar chart includes both energy and work on the left side of the chart. If work is done by external forces, then the only reason that the sum of the bar heights are equal on both sides is that the W_ext makes up for the difference between the initial and final amounts of total mechanical energy.