Reading the content in the Work-Energy theorem

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 its 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 equals the change in the kinetic energy.  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:

$$\Delta \big({1 \over 2} \;  m_A v_A^2 \big) = \overrightarrow{F}_{net\;on\; A} \cdot  \overrightarrow{\Delta r}_A$$

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, ${1 \over 2} \;  m v^2$.
  2. A — The subscript "A" on everything tells us that this Work-Energy Theorem is about a single object, 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 -- the "directional amount of motion". One object interacting with others kept changing its momentum, but when we looked at the objects causing the change, we saw that they changed their momentum opposite to the first object, keeping the total momentum constant. Something similar will happen here, but it will be a bit more complicated.
  3. Work — The thing that changes an object's kinetic energy is the sum of the forces acting on it -- when they act in the same direction (along or opposite) to the direction the object is moving. That's what all the markers on the symbols on the right tell you in general. Here's explicitly what they mean.

    3.1 Net on A — The subscript on the force term on the right 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 add up 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:

    $$\Delta \big({1 \over 2} \;  m_A v_A^2 \big) = \big( F^{\parallel}_{B \rightarrow A} + F^{\parallel}_{C \rightarrow A} + F^{\parallel}_{D \rightarrow A} + ... \big)\Delta r$$

    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, I should hit it from behind in the same direction that it's going. If I want to slow it down, I should hit it from in front, opposite to the direction that it's going. Note that this is the reason that we DON'T put vector arrows 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, $\overrightarrow{F}$ and $Δ\overrightarrow{r}$.

    3.3 Displacement — The $Δr_A$ 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 by 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 up by integrating them.
  4. The dot product — In the algebra of vectors, the notation of dot product between two vectors is 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 vector and multiplying that component times the magnitude of the second vector.  For any two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$, it's written like this:

    $$\overrightarrow{A} \cdot \overrightarrow{B}   = AB cos{\theta}$$

    where $A$ and $B$ without vector arrows means the magnitude of the vector (positive), and $cos θ$ 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:

    $$\Delta \big({1 \over 2} \;  m_A v_A^2 \big) = \sum_{j = 1}^N \int  \overrightarrow{F}_{j \rightarrow A} \cdot \overrightarrow{dr}_A$$

    where the Sigma (Σ) simply tells you to sum over all the forces exerted by objects that act on A. We've replaced the $\Delta r$ by a $dr$ and added up lots of small changes to produce a large change in KE. This allows the forces to change along the path we're following.

    This complex expression is just is a way of including all the conceptual ideas that we have discussed above in a mathematically explicit form.

    When does this equation work? Since this equation follows by rigorous mathematical manipulations from Newton's second law, it holds whenever Newton's second law does -- basically always as long as you don't get down to the quantum scale in size (smaller than nanometers) or get up to the relativistic scale in speed (a significant fraction of the speed of light).

Joe Redish 10/30/11

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