The Path Independence of Work and Kinetic Energy

 

(The following assumes familiarity with high school calculus and physics.) 

 

Newtonian mechanics is the study of motion, force, and energy—which, in turn, forms the basis of the rest of physics. Acceleration, speed, velocity, force, power, and energy have great practical utility and meaning. Comprehending them is arguably more important than knowing how to calculate them—solving calculational problems is largely a means to that end. If understanding physics means understanding these phenomena and how they have fit in with each other, it’s imperative that students intelligently (and actively) distinguish between them and their variants: energy vs. power; contact force vs. net force; net work vs. “internal work”( e.g., the energy expenditure of the muscles of a weightlifter)—etc. In proving the path independence of kinetic and potential energy, the following thought experiments will expound these relationships in ways other treatments don’t.  

      

 

Gravitational potential energy is “stored” work against gravity, equal to weight multiplied by height. If you put WORK into lifting a rock up to smash something on the ground, you put the initial work into opposite direction, but gravity will give you energy back—in the form of an equal amount of KE. Net force equals the integral of Fdr: Mdv/dt multiplied by Vdt, which comes out to be ½ MV^2. KE is path independent, meaning the object’s mechanical energy and the net work required to achieve it are the same regardless of how you acquired that KE: You could subject it to high acceleration in the beginning, decelerate it to a stop, then accelerate it in the opposite direction until it has half the KE it had at its peak; or put it on a curved path against gravity to the same point moving at a constant medium speed the whole way until you reach the former KE--etc. Two objects with the same KE must be able to transfer an equal amount of mechanical energy to another object/system; otherwise, energy isn’t conserved.  

 

Since PE is stored net work, the KE it’s converted into is the same regardless of path, making PE path independent, as well. The infinite number of ways (involving all manner of forces and velocities) to achieve the same KE and PE makes them of great use, but it can also obfuscate the distinctions between the variants of motion, force, and energy and their nuanced relationships. Demonstrating their path independence remedies potential confusion and makes for an excellent lesson in these other mechanical phenomena. An example of non-net work or internal work, as I call it, would be the work done BY THE MUSCLES when lifting a weight straight overhead, measured by the integral of CONTACT force multiplied by distance or energy expenditure, with contact force being measured by stress-strain analysis. Both net and contact force can vary throughout a given lift and from one repetition to another. With an overhead barbell press, the net work would be the change in potential energy, or weight multiplied by distance. Since the weight could have various values of KE and net and contact force throughout the lift, beginners might question the path-independence of gravitational PE and its basis on NET work. 

 

Physics pedagogy 101: Don’t confuse velocity with acceleration (including simply CHANGE IN VELOCITY and acceleration).

 

A non-accelerating or decelerating barbell can be moving quickly. Acceleration and, therefore, net force are independent of velocity. One hundred Newtons of force decelerates a ten pound ball moving ten thousand miles per hour as much as one moving one hundred miles per hour. Though the muscles can still be expending energy, the fast but decelerating barbell is moving mostly on momentum, and the net work is negative. Like whether you can pull on a rope at full force, muscles, like the rope, require optimal tension for the lifter to pull at full force (this would be “internal force” measured by contact force). This occurs when the bar moves a fraction of an inch per second. A lifter that’s more powerful/explosive than strong can put in more force into the bar in this time interval than he’s equally strong but less powerful counterpart (power is measured by energy output per time/force times velocity, strength by force). Average contact/internal force is higher when the lifter restrains to move slowly, while peak contact/internal forces are highest when the lifter does not. Newton’s third law says that the force is determined by the LIFTER, not the WEIGHT.

 

But, so long as the bar stops at the same height, they’ve done the same amount of NET work. How? When the lifter drops the bar and it freefalls to its initial height, the KE is the same regardless of path to the top, and energy must be conserved. Thus, if the bar moves in a straight path, work is path independent. 

 

But what about if an object moves up a hill, on a zig-zag, or along a curved path?

 

Whenever the path is perpendicular, the work against gravity is zero; if it’s down, it’s negative. 

 

Now for the hill. Imagine an upright triangle. If someone walks up the hypotenuse, the work is greatest, and equal to carrying it straight up, when the hypotenuse equals opposite; when the ratio between opposite and hypotenuse approaches zero, force approaches zero; if opposite is half of hypotenuse, the force is half. Therefore, W(sin) is force. But while force becomes less and less as hypotenuse grows proportionally larger, the DISTANCE you fight against gravity increases in exact proporiton. After all, hypotneuse is 1/(sin). In other words, when moving up the hypotenuse, force is always (sin) times that moving straight up and distance is always 1/(sin) times opposite. Thus, so long as the distance traveled upwards is commensurate, work is, as well. The same would apply going downwards: If you drop the ball straight down the summit, you get a proportionally bigger force over a proportionally smaller distance than if you rolled it down the slope. 

 

If the path is curved, even when it turns, the distance will always approach a straight line as the change in position approaches zero. Throughout the tiny change in distance, so long as the path is against gravity, you get the situation like that above, where the sins cancel. The cancellation means you don’t need to know what the angles are. You then could cut this path up into an arbitrarily large number of intervals, calculate the change in work, and the total work approaches the PE as the intervals approach infinity. But since the calculation will always equal PE, all you need is weight times height. 

 

When an object moves horizontally against resistance without accelerating, the force is MV(cosign)/dt, but since there isn’t an increase in SPEED, the force is either an “internal force,” or simply a shift in direction and doesn’t count as net work or work at all. When an object moves in a circle, for example, centripetal force is always balanced by centrifugal force. Two objects with the same KE must be able to transfer an equal amount of energy to another system, such as a spring.     

 

 

 

 

Outside of science, force, power, and energy are used interchangeably, and it’s very easy to confuse them. The example with the barbell (along with the reference to the isokinetic machine) illustrates the differences between velocity and acceleration, net force and contact force, and energy EXERTION and energy OUTPUT. There may or may not be a power output when someone is demonstrating that they are powerFUL, but they may well still be exerting a lot of energy.  

 

Physics courses tend to assign problems that deal in net force and work, and, over time, it becomes easy to forget about contact force and internal energy exertion and the like, which are just as important (maybe throwing in some stress-strain analysis problems might help?). A pic or video of an athlete exerting force on a bending barbell held back by ropes can illustrate this. If the lifter is using a isokinetic (slowly moving) exercise machine and he’s beat-red and screaming at the top of his lungs to exert maximal force, this can imbed the lesson into the mind of the student—especially if the athlete performs it in class and prompts a security or, even better, a police response. Neither exercise nor physics must be civilized activities.