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Reading the content in the Work-Energy theorem (2012)

last edited by 7 years, 4 months ago

Prerequisites

The Work-Energy Theorem (WEθ) arose from our looking at the vector form of Newton's second law and asking, "N2 is an equation that tells us what makes an object change it's velocity -- a vector. This involves direction. Suppose I don't care about direction and want to know what it is that changes the object's speed, not paying attention to direction?" We found that the answer was to look at the part of N2 that is along (or against) the direction of the object's motion, ignoring parts of the force that are perpendicular to the motion. When we multiplied by the object's displacement, Δr, we got the WEθ.

 The Work-Energy Theorem may be written quite simply:  Δ(KE) = W -- the work W equals the change in the kinetic energy KE.  But as is often the case with the way we use equations in physics, there is a lot hidden in the symbology. To make sense of what the equation is saying, we have to unpack a lot of hidden meaning and unstated assumptions. Let's do this by opening up each symbol and see what meaning is hidden. To do that, let's start with a more explicit form of the theorem:

This is our first elaboration of the simple "change in KE = work" idea.  This immediately contains two qualitative ideas.

 1. KE -- The quantity we use to measure an object's "directionless amount of motion" is the kinetic energy, ½mv2.   The focus here is on understanding how the amount of kinetic energy changes. We can represent that using a bar graph which shows the kinetic energy increasing if there is positive work done in this situation.

 2.A -- The subscript "A" on everything tells us that this Work-Energy Theorem is about a single object (A), in the same way as Newton's 2nd law is about a single object. (No wonder, since we derived this theorem by manipulating N2.)  Note that at this stage, we don't have even the concept of "conservation of energy" hidden in this equation.  This equation is about what changes the kinetic energy of a single object. To get to a conservation law, we'll have to look at the objects that interact with the one we are considering here and see what happens to them.

This is somewhat analogous to what happened when we looked at momentum conservation. One object will change its momentum if it collides with another object (lets call that object B), but when we looked at object B, we saw that it changed its momentum opposite to the first object.  So the total momentum of both objects remains constant. Something similar will happen here.

 We can represent the focus on a single object by drawing a system schema. We indicate our focus by drawing a dotted line around that object, but we see that there are interactions that cross the dotted line. These correspond to the forces that are acting on the object of interest. In our energy analysis, these interactions will yield potential energies -- that we will sometimes associate with our object of interest and sometimes with the interacting pair.

3. Work -- The thing that changes an object's kinetic energy are the sum of the forces acting on it -- more specifically the component of the forces that acts in the same direction (or opposite) to the direction the object is moving.

The equation contains a lot of important subscripts and superscripts (on top right of the symbol) that you can "read" and that help you make sense of the equation.  Here's explicitly what they mean.

3.1 Net on A -- The subscript on the force terms in the equation above "net on A" means the same as what "net" meant for us in Reading the content in Newton's 2nd law.  We look at all the objects that interact with our object "A" and sum the forces acting on A. (See object egotism.) This means that if we have objects B, C, ... acting on A, that the "net" on the right side is actually hiding the following:

All of the forces that appear in Newton's second law for object A potentially appear in the Work-Energy Theorem for object A.

3.2 Parallel -- The two parallel lines we have put as a superscript on the forces mean that it is only the component of the force that is parallel to the direction of motion that can change the KE.  (Note that here "parallel" doesn't say anything about direction.  It could be positive -- in the same direction as the motion, or negative -- in the opposite direction as the motion.) We already know this intuitively. If I want to speed up a rolling bowling ball, we should hit it from behind in the same direction that it's going.  If we want to slow it down, we should hit it from in front, opposite to the direction that its going. Note that this is the reason that we DON'T put vectors on the forces -- since we are only considering one component of them.

 To see how to get the component of the force along the direction of motion, take a look at the figure at the right.  In it, we have decomposed a particular force vector, shown in solid red, into two component vectors, shown in dashed red -- one along the displacement (shown in blue) and one perpendicular to it. (The sum of the two dashed red vectors equals the solid red vector, so we can replace the solid one by the two dashed ones.) The component of F along Δr is clearly F cos(θ) where θ is the angle between the two vectors. Therefore   Work done by a force F = F Δr cos(θ)   where θ is the angle between the two vectors, F and Δr.

3.3 Displacement -- The ΔrA term in the work on the right says that the amount of work done by a force pushing or pulling on an object is proportional to how far it pushes or pulls it. (Note: This is NOT how long -- that is, for how much time it pushes or pulls it.  If you multiply the force my the time it tells you how much the momentum changes.) This hides an assumption: that the force doesn't change during the time that the object moves through this displacement. If the force is a function of position or is changing as the object moves, you have to take your displacements as tiny steps -- tiny enough that the force can be treated as constant during those steps. And then you have to add those steps us by integrating them.

4. The dot product -- In the algebra of vectors, the notation of dot product between two vectors in introduced. It is a multiplication between two vectors to give a scalar and it is obtained by taking the component of one vector along a second and multiplying that component times the magnitude of the second vector.  For any two vectors A and B, it's written like this:

where A and B without vectors means the magnitude of the vector (positive) and cos(theta) is the cosine of the angle between the two vectors. (This will handle the sign correctly if the vectors point in opposite directions.) This is just what we need here, but we will tend to focus on the physical meaning of "the part of the force in the direction of motion" since it emphasizes what's happening. If you want to read more about how the math works, read the dot product page.  Note the interesting point that it doesn't matter which vector you start with to project on the other.

These last two items means that in reading more advanced texts, you might find the Work-Energy Theorem for a single object written in the more mathematically intimidating form:

where the Sigma (the large first symbol on the right of the equals sign) simply tells you to sum over all the forces exerted by objects that act on A. This just is a way of including all the conceptual ideas that we have discussed above in a mathematically explicit form.

Follow-ons:

Joe Redish 10/30/11

Wolfgang Losert 11/17/12

Vashti Sawtelle 11/16/12