Simple Work Formula is Parabolic with Respects to Time?

[Desiderus]

Hello all. I'm having trouble understanding why exactly it is that the simple work formula is defined both as change in energy and force times distance, and I'm hoping that it can be made clear to me.

Allow me to clarify. In physics, the word work has two definitions. The first, what I call the "conceptual definition", is a measure of change in energy of an object over a distance with a force acting constantly on an object. The second, what I call the "mathemetical definition", is :

W = F x d

with "W" being work in joules
"F" being Force in Newtons
and "d" being distance in meters

My problem lies in reconciling these two definitions. To me, they seem inconsistent. A force necessarily involves acceleration, which means non-linear (parabolic) increases in distance with respect to time. This, of course, means that work, being a product of what we shall assume is a constant force and a non-linear distance, is also non-linear with respect to time.

Now, if energy is being applied through a force of constant magnitude, and we are to assume that the object is indeed moving and the force is indeed parallel to the object being acted upon, why is it that, per the work formula, energy does not flow into an object at a constant (linear) rate with respect to time? If a force is measured in Newtons, and thus kilogram meters per second per second, then isn't a certain amount of energy flowing into the object per second (per any time interval, for that matter)? And isn't it the same amount of energy being transferred from second to second (or time interval to time interval, so long as the time intervals being compared are the same length)? Clearly not, though I cannot for the life of me see why. I have been laboring under what is perhaps the delusion that the magnitude of a force in Newtons describes how much energy the force in question is applying per second.

If the work formula were instead the momentum formula (p = m x v) this would make perfect sense to me. A certain speed times a certain mass yields a certain energy. If, force example, a certain amount of energy would make an object of one kilogram move at a rate of 10 meters per second, it would seem very natural to me that the same amount of energy would move an object of two kilograms at a rate of 5 meters per second. Energy proportional to both mass and velocity seems perfectly sensical, so where do we get all this stuff about force times distances and (1/2)mv^2?

 

[russ_watters]

An applied force need not result in acceleration. It can be balanced by something like aerodynamic drag.

Where did you get that "conceptual definition"? It isn't right, or at least, is very badly worded. There is no "change in energy of an object..." in many cases where work is performed, such as sliding a box across a floor at constant speed.

If a force is measured in Newtons, and thus kilogram meters per second per second, then isn't a certain amount of energy flowing into the object per second (per any time interval, for that matter)?

Those units are not the units of energy, they are the units of force due to acceleration. Not all forces cause acceleration, so those units (the conversion of units via f=ma) does not always apply. If you apply a static force to a box sitting on the ground and it doesn't move, the f in Newtons is just f in Newtons.

Anyway, you are not alone in your discomfort about kinetic energy. The easiest way to deal with it is to compare potential and kinetic energy with each other and with work. Ie,

Work = force * distance
Potential Energy = Weight * height

As you can see, the above two equations are the same: Weight is a force and height is a distance.

Now if energy is conserved, dropping a weight must result in the potential energy being converted to kinetic energy. And to do that, the weight has to accelerate at a constant rate, covering more distance (in a square function) for each next interval of time.

 

[astrokreng]

I understand your confusion and frustration with reconciling the two definitions of work. However, it is important to note that the simple work formula, W = F x d, is a simplified version of the more complex work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. This theorem takes into account the non-linear relationship between force, distance, and time.

To better understand this, let's break down the components of the work-energy theorem. The first part, work, is defined as the force applied to an object multiplied by the distance it moves in the direction of the force. This is the mathematical definition you mentioned. However, this definition only takes into account the work done by a constant force. In reality, most forces are not constant and can vary over time. This is where the concept of energy comes in.

Energy is the ability to do work. In the case of a non-constant force, the work done on an object is equal to the change in its kinetic energy. Kinetic energy is defined as 1/2 x mass x velocity squared (KE = 1/2mv^2). This is where the parabolic relationship with time comes into play. As an object's velocity changes, its kinetic energy also changes. This means that the work done on the object is not constant, but rather increases or decreases as the object's velocity changes.

To address your example with force and energy, it is important to note that while force does play a role in determining an object's kinetic energy, it is not the only factor. Mass and velocity also play a significant role. So while a certain amount of energy may move a one-kilogram object at a rate of 10 meters per second, it would not necessarily move a two-kilogram object at a rate of 5 meters per second. This is because the two-kilogram object has more mass, which requires more energy to move at the same velocity.

In summary, the simple work formula is a simplified version of the more complex work-energy theorem, which takes into account the non-linear relationship between force, distance, and time. While it may seem inconsistent at first glance, it is important to understand the underlying principles of energy and how it relates to work. I hope this helps clarify things for you.