None Reading_09

Table of Contents

Challenge

As a reminder, we have dealt with two systems so far in this course that look very different, but behave in very similar ways. Both the blue toy truck slowing down from a nonzero initial velocity and the tank draining through an open valve seemed to show behavior that could be described by a differential equation, in which the change in system outputs was dependent on how large the output was.

In Readings 7 and 8, we saw that building a physics-based model of the toy truck problem gave us insight into how the system's behavior might change if we changed its properties. Using Newton's second law, we found a model for the truck's coasting behavior as:

$$\dot{v} = - \frac{1}{\left( J_w + m_w r^2 + \frac{m_c r^2}{2} \right)}b v$$

What we did not do in reading 8 is discuss how changing the truck's parameters affects its behavior. We also did not evaluate the model we derived! Below, see a comparison between Euler integration for our physics-based model and data. Feel free to change parameters like wheel mass, damping coefficient, and car mass to see what effects these have on the truck's behavior. While changing these will make the model fit the data more poorly, they do tell us what would happen if we changed the truck's properties.

In [1]:
% clear variables
clear all

%load the data we will use for model validation
data = load('reading6data.txt');
td = data(:,1);
vd = data(:,2);

%set up a new time and velocity vector to use for simulation
dt = .01;
simtime = td(end);
tsim = 0:dt:simtime;
vsim = vd(1);

%define the measured parameters for the toy truck parts
mc = .05; %kg, truck mass
mw = .01; %kg, mass of one axle, including 2 wheels
rw = .005; %m, radius of wheels
Jw = 1/2*2*mw*rw^2;
b = 1.5e-6;

%use Euler integration to simulate our model
for k=2:length(tsim)
    %compute acceleration using model
    vdot = -b/(Jw+rw^2*(mw+mc/2)) * vsim(k-1);
    vsim(k) = vsim(k-1) + vdot*dt;
end

%compare model with data to evaluate
figure
plot(td,vd,'ks',tsim,vsim,'r')
xlabel('Time (s)')
ylabel('velocity (m/s)')
legend('data','simulation')

But this model came from Newton's laws, which would be difficult (not impossible, but difficult) to apply to the draining tank. The water in the tank is not a rigid body, so a free body diagram does not seem like the right tool. But imagine if there was a way to approach the tank-drain problem in the same way as the blue toy truck problem, using some "grand unifying quantity" that applies similarly to both systems. As it turns out, there is such an approach, and it is based on a concept that you are probably already familiar with from high school physics: Energy.

In this assignment, your challenge is to be able to develop a model for a system based explicitly on how it exchanges energy with its environment. Your assignment will be an experiment of sorts. We need to determine whether we have the tools we need to revisit reading 8, in which you derived a model for a mass-pulley system based on Newton's laws, and build the same model using an energy-based approach.

To prepare you for the assignment, we will proceed by example, and look at how to derive the model for the toy truck in a similar fashion. Before we begin, we will need to look at a few key concepts. The first one, the biggest, is energy.

Energy

In one of his famous lectures, now available online, Nobel Prize-winning physicist Richard Feynman stated:

"It is important to realize that in physics today, we have no knowledge of what energy is."

What does he mean? Well, the human-made concept of "energy" has a long and storied history. In essence, the concept was created to describe what scientists reliably observe when conducting experiments.

Let's proceed by example. If you rub your hands together, this involves "doing work." What is the result of this work? Your hands increase in temperature. If you place a ball on top of a hill, and let it roll down, it will have a predictable amount of velocity when it reaches the bottom of the hill. If I apply heat to a balloon full of air, it will expand. Energy is something "we" have come up with to link the relationships between the work done on a system and the heat a system exchanges with its environment. Specifically, based on repeated observation through the centuries, scientists accept the following about energy:

  • Energy is neither created nor destroyed.

This empirical relationship, which we now accept as a "law," began with experiments. Its authority derives largely from the fact that no matter what we do, we can't seem to disprove it. Liebniz saw that slowing a mass down through friction produced heat, and posited that the amount of heat produced in the act of slowing the mass must be equal to the "energy" the mass posessed before beginning to decelerate. Later, during the industrial revolution, the development of steam engines and other thermodynamic systems led to experiments that tried to quantify how much "work" could be done by heating something up. All of these experiments led to a few key conclusions about how energy can be stored and/or transferred. If a system is taken from one "state" of energy to another, it must be because either heat was transferred between the system and its surroundings, because the mass inside the system changed, or because work was exchanged between the system and its environment. If the system was then brought back to its original energy state, the amount of work done on the system, and the amount of heat transferred to the environment would always balance out. This law, which we call "the conservation of energy," has been complemented with the following, empirical observation about how the energy "stored" in a system (also called its energy state) changes:

  • Work and Heat describe the only two ways we know of that a closed system (a system in which no mass crosses the system boundary) can exchange energy with its surroundings. If the mass contained in a system's boundaries changes, the system is called an "open system," and this change of mass can also result in a transport of energy if the material entering or leaving a system's boundary carries energy with it.

To understand how to use these observations about "energy," we must first be specific about what we mean by "energy stored within a system," as well as what we mean when we describe how a system exchanges that energy with its surroundings.

Energy stored within a system

When we talk about the energy "stored within" a system or element in ES103, also called the system's total energy, we will generally be talking about a system or component's:

  • Kinetic energy (energy due to motion)
  • Potential energy (capability of the system to do work on its surroundings)
  • Internal energy (energy stored at the molecular and atomic level. Think of a hot turkey having high internal energy that decreases as it cools off over time.)

You may or may not already be familiar with these categories. However, this list is not exhaustive. For example, internal energy can take many forms, and you will learn about some of these in later courses, especially those that deal with systems containing compressible materials like gases, or courses that deal with chemical reactions!

Energy transfer due to mass crossing a system boundary

When we defined energy in the preceding section, we said that mass crossing a system or element boundary means that the system is an "open system." We also said that this transport of mass could change the total energy inside the system's boundaries. This is because if material (that has mass) crosses a system boundary, that material can carry energy into or out of the system scope.

For the purposes of ES103, we will say that energy changes in a system due to mass transfer will be limited to effects associated with kinetic, potential, or internal energy carried into or out of a system by material that has mass. Recall from our earlier definition that internal energy is stored in material at the atomic or molecular level.

In ES103, we will think of internal energy as completely described by energy associated with temperature, called thermal energy. This is an assumption, because this isn't the only type of internal energy. However, it will work for us in this course.

Work

Work, in the thermodynamic sense, is what occurs when a system exerts forces on its surroundings whose effects can be measured. Work can be produced by a pressure applied in order to change a system's volume, or by a force that physically moves a rigid body, or by an electric field that moves a charged particle. Work is usually expressed as a capital $W$, and has SI units of $Nm$ or "Joules."

Most relevant to our current challenge is mechanical work, which is measurable if one knows the quantity of an applied force (or toque) and the distance (or angle) over which it was applied.

Mechanical Work via a Force

Mechanical work is the product of a force and a distance. Specifically, work is the integral of force applied along a path.

$$W_{a\rightarrow b} = \int _\vec{a}^{\vec{b}} \vec{F}\cdot d\vec{S}$$

Consider a situation in which you are pushing a block of wood around on a table. By modeling the interaction between the block and the table using dry friction, where $F_f = \mu m g$, we can imagine that if the block of wood is moving at all times, the friction force applied to resist us pushing it around is constant, and always opposing motion. Imagine that you could choose to push the block along two possible paths, $S_1$ or $S_2$. The destination is the same, but $S_1$ is much longer.

image-2.png

you might imagine that path $S_1$ would colloquially "take more work" to achieve, or "drain more of your energy." This is consistent with the formal definition of work!

Mechanical Work via a Torque

In an analagous sense to the definition of mechanical work via a force, mechanical work is produced when a torque is applied to an object which moves through some angle. Formally, we can say:

$$W_{\theta_1 \rightarrow \theta_2} = \int_{\theta_1}^{\theta_2} T d\theta$$

Because torque has units of $Nm$, we find that in SI units, with radians as our measure of angle, mechanical work has the same units as it does for translation: $\frac{kgm^2}{s^2}$, which is equivalent to $Nm$ or Joules.

In our toy truck problem, we have viscous damping from the bearing that does work on the truck to slow it down. This work is produced by a torque that depends on the angular velocity of the bearing, as we learned in Reading 7.

Heat

When people think of "heat," they usually think of temperature. Temperature is certainly a measure of the intensity of heat, but the way that we measure the quantity of heat is in terms of how much energy it takes to raise a closed system's internal temperature. Because not all systems' temperatures can be changed with the same ease, temperature itself is not enough to describe the magnitude of "heat." For example, we could imagine that while a candle will burn our fingers almost immediately, it could not be expected to raise a large pot of water to its boiling point just as quickly.

In general, raising temperature is accomplished by either transferring heat from a hot object to a cold object, or by doing work that is converted to heat, such as when you rub your hands together to warm them up. In most of thermodynamics, the quantity of heat is denoted with the letter $Q$. Heat, in base SI units, does not have units of calories-- like work, and like measures of kinetic and potential energy, it has units of "Joules," or $Nm$.

Translational and Rotational Kinetic Energy in a Rigid Body

When a rigid body is moving, we say that it "stores" translational kinetic energy, which it can then exchange with its surroundings. The translational kinetic energy in a rigid body is:

$$E_k = \frac{1}{2}mv^2$$

Where $v$ is the scalar magnitude of its overall velocity. In SI units, this energy has units of Joules, or $\frac{kgm^2}{s^2}$.

When a rigid body rotates, it stores rotational kinetic energy. Rotational kinetic energy, which also has units of Joules, is given by:

$$E_k = \frac{1}{2}J\Omega^2$$

Where $\Omega$ is the magnitude of the angular velocity of the object (in $\frac{rad}{s}$ for SI units) about its center of mass, and $J$ is the mass moment of inertia of the object about its center of mass (in $kgm^2$ in SI units). Once again, note that I used a capital Omega ($\Omega$) rather than a lowercase omega ($\omega$) to describe angular velocity here-- there is no real distinction. Both are often used, depending on your field.

In our toy truck problem, the truck stores kinetic energy translationally for the truck body, which does not rotate, and both translationally and rotationally for the axles, because they both rotate about their CG and translate with the truck.

In modeling the truck as an Energetic Physical System, we can employ the fact that energy is always conserved to help us develop our model for the truck's motion. The law of conservation of energy, which states that energy is neither created nor destroyed, has another name: the "First Law of Thermodynamics."

The First Law of Thermodynamics: The conservation of Energy

The first law of thermodynamics, also called "the conservation of energy," states that energy is never created nor destroyed... merely exchanged between a system and its surroundings. It is this law that tells us that perpetual motion machines cannot exist, as they produce work without the input of energy from outside of the system.

Because energy can only be transferred through either work, mass transfer, or heat transfer across a system boundary, we can draw the relationship between a system of interest and its surroundings using the following diagram:

image-4.png

You may notice that this looks like our "system scope" diagram, but it deals with energy only. We represent internal (stored) energy $E$, work done on the system from the external environment $W_{in}$, work done by the system on its external environment $W_{out}$, heat applied to the system from the external environment $Q_{in}$, heat applied by the system to its external environment $Q_{out}$ explicitly. We also explicitly call out terms $E_{m}$, which represent energy transfer due to mass crossing the system boundary (either into our out of the system).

Formally, we can write the change in energy stored inside the system from some "initial state" 1 to some "later state" 2 using the following equation, which is a statement of the first law:

$$ E_2 - E_1 = \Delta E = \delta W_{in} - \delta W_{out} + \delta Q_{in} - \delta Q_{out} + \delta E_{m,in} - \delta E_{m,out}$$

This says that if the balance of energy transfer between our system and its environment can be interpreted as the environment "doing work" on, contributing energy via mass transfer to, or "transferring heat to" our system of interest, it will increase the amount of energy stored inside our system if our system did not do work on its environment, contribute energy via mass transfer to, or exchange heat with its environment.

In the opposite scenario, if the balance of energy transfer between our system and its environment can be interpreted as our system "doing work" on its surroundings or "transmitting heat to" its surroundings, we say that the amount of energy stored in our system goes down.

If we consider that bringing our system from "energy state 1" to "energy state 2" requires time (which it always will), we can say that the rate of change in energy in our system can be written as:

$$ \frac{E_2 - E_1}{t_2-t_1} = \frac{\Delta E }{\delta t}= \frac{\delta W_{in}}{\delta t} - \frac{\delta W_{out}}{\delta t} + \frac{\delta Q_{in}}{\delta t} - \frac{\delta Q_{out}}{\delta t}+ \frac{\delta E_{m,in}}{\delta t} - \frac{\delta E_{m,out}}{\delta t}$$

Shrinking $\delta t$ to an infinitesimal value leads to the derivative of our original first law equation, which is a totally equivalent statement about the conservation of energy:

$$ \dot{E} = \dot{W}_{in} - \dot{W}_{out} + \dot{Q}_{in} - \dot{Q}_{out} + \dot{E}_{m,in} - \dot{E}_{m,out}$$

The derivative of energy (and thus, the derivatives of power and heat) have units of $\frac{J}{s} = \frac{kgm^2}{s^3}$, which are commonly called "Watts" in SI units. There is also a very special name for the time-derivative of energy, and it is power.

Power

Power is the time derivative of energy. In SI units, power has units of "Watts," or Joules per second. Power can take many forms, including thermal, mechanical, and electrical.

Energetic Diagrams

In ES103, an energetic diagram will be defined as a diagram that explicitly shows how we will model energy that is exchanged between the system (or a subsystem or an element) and its environment. "How we will model" is important here. In drawing certain arrows that represent energy transfer, we will often make assumptions about how this energy is transferred and with what. Because these will often be simplifications, it is important to keep in mind that the energetic diagram we end up with is not "truth" by default. It is a model.

Arrows pointing in to the system should reflect outside influence from the system's surroundings that would increase its energy in the absence of other factors. These influences could be in the form of either heat, energy associated with mass transfer, or work as a function of time. Arrows pointing out of the system should reflect work, energy associated with mass transfer, or heat that a system transfers to its environment. In the absence of other factors, an outward-pointing arrow would decrease the amount of energy stored within a system (internal energy, such as kinetic, potential, thermal, chemical, etc). Note that we draw the $\dot{E}_m$ terms with their arrows physically crossing the system boundary. This is to show explicitly that these arrows represent material going through our system boundary.

If we draw the energetic diagram to represent the first law's time derivative, arrows represent a flow of power between a system and its surroundings, which can either be a known subsystem that is also being modeled, or a system's unmodeled, external surroundings ('the universe,' for instance). An example for an arbitrary system is shown below:

image-5.png

Disciplined Process for Model Construction for a dynamic system using the first law of thermodynamics

After completing steps 1 and 2 of our overall process for model construction, you will have a list of independent and dependent variables. To use the conservation of energy to relate the two sets of variables and obtain your final model, you can:

  1. Draw an energetic diagram for each element in your model scope, crossing off any terms you are assuming to be insignificant. When you do this, justify each assumption clearly in writing. Making an assumption is always a deliberate risk; the model evaluation step of our overall disciplined process for ES103 will guide you in determining the quality of the assumptions you make.
  2. Write the time-derivative of the first law of thermodynamics for each element to describe how the element transfers energy between itself and its surroundings. Take care to substitute what you know about the energy, work, and heat terms for each element into the equation.
  3. Write equations that indicate how each element in your scope exchanges energy between itself and other elements in the scope, or between itself and the overall system's surroundings. Use physical connections between elements in your system to guide you here.
  4. Using algebraic manipulation, combine your equations from steps 2 and 3 to obtain a final system model.
  5. Check your final system model for internal validity, including the signs and units of each term, and the overall form of the equation.

Challenge Continued

Let's see how we might use the conservation of energy to build a model of our toy truck. Remember that because we used Newton's laws of motion to derive our model before, we should get the exact same model for our truck if we use the first law of thermodynamics since both approaches should be consistent with our physical understanding of the world (this is what a "law" of nature implies).

Using Conservation of Energy in Model Construction

Let's think about how we might apply the first law of thermodynamics to help us develop our model. We know (from observation, over hundreds of years of experiments) that energy is neither created nor destroyed, and we know that at the beginning of our model's "existence," we chose to give the truck an initial velocity, without much care for how that velocity was produced. But let's think about that for a second. In order to speed the truck up, something had to do work on it to transfer energy to it. Once it stops, it will have no internal kinetic energy left... so we already know that it must have transferred all of the energy transmitted to it to speed it up back to its environment as either work or heat.

To get a little more quantitative with this discussion, let's use the concept of an Energetic Diagram, which explicitly accounts for energy only.

Model Scoping using energetic diagrams

If we would like to draw an energetic diagram for our toy truck, we will first stick with our overall system definition. We should ask: If the truck is at some initial energy state, how will it exchange that energy with its environment?

We know that because there is a fixed amount of mass in the truck, no energy will "escape" the system boundaries due to mass transfer, so any energy exchange will either be because the truck "did work" on its environment, or because heat was transferred to its environment. Here, the truck's "environment" could be considered to be the surface upon which it rolls, and the air around it.

Mechanical work is the product of a force and a distance, so if we neglect air resistance (because for now we don't know how to model that piece), we can say that the truck actually does no work on its surroundings, since there is no measurable slip between the tires and the ground. At any moment, the point of contact between the tires and the ground is stationary, so the product of force times distance due to this contact is zero.

That means that all of the energy the truck transfers to its surroundings must be transferred as heat. With this simplification, we can represent our toy truck model with the following energetic diagram.

image-5.png

As you can see, the energetic diagram shows us that when we look at the whole scope of our system, we should be able to account for all of the change in the system's energy by accounting for the heat that the system exchanges with its environment. Note that this is implicitly a model construction step as well as a scoping step. If the truck's tires were to slip, or if its bumper fell off, this diagram may be invalid, because our modeling assumptions about the truck's mass and the tires' rolling without slip would be violated.

But because this diagram tells us very little about how that heat is transferred between the truck and its environment, we need to break the system up into smaller chunks, like we did when applying Newton's laws.

In Reading 07, we decided that there were three main subsystems of the toy truck that were important to us: The wheels, the bearings, and the body of the truck. In Reading 07, we broke the scope down to include each bearing and wheel as a separate piece, but in the end, we considered the contributions of the front and rear wheels as equal, and built our model for only half of the truck by leveraging symmetry. This means that a three-element scope makes sense. I have reproduced our "broken out" scope as an energetic diagram below, still considering only half of the truck using the symmetry argument.

image-8.png

If we consider each of these pieces individually, we can actually develop an energetic diagram that includes power flow (energy transfer over time) between each of these subsystems or "elements" that represent the "whole" toy truck system.

Before we do that, though, let's look at a first-law analysis for each individual element in our toy truck model.

Model Construction

Armed with our overall disciplined process for model construction and our new knowledge of how to construct a model using conservation of energy, we can begin our process of model construction with lists of key independent and dependent variables to help us keep track of what quantities we should see in our final equation. As in Reading 08, we should be expecting a differential equation relating truck acceleration $\dot{v}$ to truck velocity $v$.

As always, we will start with a list of independent variables. We will recycle this list from Reading 08.

Independent variables:

  • time
  • half of the truck chassis (body) inertia/mass $\frac{m_c}{2}$
  • wheel/axle mass $m_w$
  • wheel/axle moment of inertia $J_w$
  • bearing damping constant $b$
  • Wheel radius $r$

Next, we'll look at a list of relevant dependent variables in our model. This list will usually be short-- what are we trying to get an equation for?

Dependent variables:

  • truck acceleration $a = \dot{v} = \ddot{x}$
  • truck velocity $v = \dot{x}$

Because we have numerical integration as a tool, we can focus our efforts on finding an equation for the dependent variable representing the highest derivative in our system. In this case, that's the acceleration of the truck body, $a = \dot{v} = \ddot{x}$.

To relate independent to dependent variables using conservation of energy, we will follow the disciplined process outlined earlier in this notebook.

First Law analysis of the toy truck wheel

A general energetic diagram for one of the truck's wheels is presented below.

image-8.png

At this point, all we can do is write the time-derivative of the first law based on the diagram above.

$$\dot{E}_w = \dot{W}_{in,w}-\dot{W}_{out,w} + \dot{Q}_{in,w}-\dot{Q}_{out,w} + \dot{E}_{m,in,w} - \dot{E}_{m,out,w}$$

How can we fill these terms in? Let's start with what we know. For the wheels, whose masses don't change, we can write the following expression for the energy "stored" in them as a combination of rotational kinetic and translational kinetic energy. We know that the wheels (along with the truck body) are translating with a speed $v$ at any moment, and that the wheels are rotating with an angular velocity $\Omega$ at any moment. therefore, the kinetic energy stored in each wheel can be written:

$$E_w = (\frac{1}{2}m_w v^2 + \frac{1}{2}J_w \Omega^2)$$

Note that I am using a capital $\Omega$ instead of a lower-case $\omega$ for angular velocity of the wheel here, just to avoid confusion between angular velocity $\omega$ and the "wheel" subscript $w$.

Each wheel has a mass $m_w$ and rotational inertia $J_w$. Both have the same velocity. This means that the derivative of the energy stored in the wheel, which must be motivated by either heat transfer or work, is a restatement of the first law's time derivative:

$$\dot{E}_w = (m_w v \dot{v} + J_w \Omega \dot{\Omega}) = \dot{W}_{in,w}-\dot{W}_{out,w} + \dot{Q}_{in,w}-\dot{Q}_{out,w} + \dot{E}_{m,in,w} - \dot{E}_{m,out,w}$$

If you're unsure about this derivative, note that it requires the use of the chain rule, which is where the $\dot{v}v$ and $\dot{\Omega}\Omega$ terms came from.

By observation during our experiments, we can't measure any difference between the temperature of the wheels and the environment, or between the temperatures of any of the truck's components. We also can say that the very small temperature changes of the wheels that may occur are not measureable, and are outside the scope of our system definition. So we will ignore heat transfer and any terms associated with internal energy. We know that the wheel's mass doesn't change appreciably, so no energy changes due to mass transfer will occur. We also know, based on our overall scoping diagram, that no external work is being applied to the truck, but we know that the inertia of the wheels causes a sustained torque to be applied between them and the bearing as the wheels rotate. This is a source of work, and we will say that this is work out of the wheels, since it causes their stored energy to decrease. This allows us to write a simplified version of the first law for one of the truck's wheels as:

$ -(m_w v \dot{v} + J_w \Omega \dot{\Omega}) = \dot{W}_{out,w}$ (EQN 1)

This says that the wheel transfers energy to the bearing, which decreases its internal, stored kinetic energy. This can be represented by the following simplified energetic diagram.

image-4.png

First Law analysis of the toy truck's chassis

A general energetic diagram for one half of the truck's body is shown below:

image-4.png

Once again, our starting point is writing the time-derivative of the first law based on the diagram above.

$$\dot{E}_c = \dot{W}_{in,c}-\dot{W}_{out,c} + \dot{Q}_{in,c}-\dot{Q}_{out,c}+ \dot{E}_{m,in,c}- \dot{E}_{m,out,c}$$

Again, we try to fill in this equation using what we know. The body of the truck stores kinetic energy. Because it is not rotating, it only stores translational kinetic energy. Because it is at room temperature and we have not included any height changes of the truck in our scope, kinetic energy represents total energy. Therefore, we can write an expression for the energy stored in the truck's body at any moment as:

$$E_c = \frac{1}{2} \frac{m_c}{2} v^2$$

Recall that we use $\frac{m_c}{2}$ for the chassis mass to remind us that we are only considering half of the truck. Taking the derivative of this quantity, which shows us how it changes over time, and writing the first law equation's time derivative, we get:

$$\dot{E}_c = \frac{1}{2}m_c v \dot{v} = \dot{W}_{in,c}-\dot{W}_{out,c} + \dot{Q}_{in,c} - \dot{Q}_{out,c}+ \dot{E}_{m,in,c}- \dot{E}_{m,out,c}$$

Like the axle, the car body is at the same temperature as the environment and the other components in our system. Like the axle, we can say that the temperature of the car body is relatively constant. So we will make the assumption that no heat transfer is involved in this first law balance, and no thermal energy is stored within the boundaries. The chassis also has a mass that doesn't change. Finally, we know that because nothing is acting to increase the car's kinetic energy during our experiments, we can say that the car's change in energy over time is negative, and thus the car body is doing work on the bearing. This leads to the following simplified first law equation:

$ - \frac{1}{2}m_c v \dot{v} = \dot{W}_{out,c}$ (EQN 2)

And the following simplified energetic diagram:

image-4.png

First Law analysis of the toy truck's bearings

A general energetic diagram of the toy truck's bearings for the front wheel/axle is shown below.

image-5.png

Thinking back to our modeling choices from Reading 7, we said that the bearing had no significant mass, because at least in the case of the toy truck, it really just exists as the lubricated interface between the rotating axle and the translating truck body. If we make the assumption that the bearing is "massless," then we can say that it has no capacity to store kinetic, potential, or even thermal energy. Therefore, we can say definitively that under our assumptions, the first law analysis for the bearing looks like:

$$\dot{E}_b = 0 = \dot{W}_{in,b} - \dot{W}_{out,b} + \dot{Q}_{in,b} - \dot{Q}_{out,b}+ \dot{E}_{m,in,b}- \dot{E}_{m,out,b}$$

We can also say that because the bearing has no capacity to store its own energy, that all of the power that is involved in spinning the bearing is the result of work done on the bearing. This means that the work done on the bearing must be balanced by the production of heat which leaves the bearing and exits our model's scope.

Is this heat going "directly into the atmosphere" to warm up the room? Not necessarily. Technically it probably warms the truck body a little, and warms the axles a little. It probably warms the room even less, but by the first law, we know that the friction in the bearing must produce heat at a rate equal to the rate at which work is done on the bearing itself. This tracks with the "rubbing your hands" together example. With that said, we can write the first law for the bearing as:

$$\dot{W}_{in,b} = \dot{Q}_{out,b}$$

This results in the following simplified energetic diagram:

image-5.png

Further, we know from Reading 07 that the torque produced by the bearing is proportional to the difference in angular velocity of the inner and outer races. This means that we can actually write a mathematical expression for the rate of work done on the bearing as:

$$\dot{W}_{in,b} = T \Omega = b\Omega^2$$

This allows us to write an even more specific expression for the first law applied to the bearing as:

$\dot{W}_{in,b} = \dot{Q}_{out,b} = b\Omega ^2$ (EQN 3)

Modeling energy transfer between elements in the toy truck energetic model

With the bearing, the wheel, and half of the chassis taken care of, we have completed steps 1 and 2 of our disciplined process. Now the task is to write equations that relate the flows of energies between elements to each other, and relate the flow of energy into and/or out of the system boundary.

Looking at how all of the components in our model of the truck interact, we can draw an energetic diagram that shows how power flows between elements by drawing an energetic model of the whole toy truck model again, but now including the simplified first law analyses for each component. This is shown below.

image-3.png

We know that overall, the system only exchanges energy with the environment in the form of heat being transferred from the bearing to the outside world. The only element in the system that transfers heat this way is the bearing, so the work coming out of the wheels and the body (since these aren't 'powered' and their energy goes down over time) must be done on the bearing, since this is the only thing they're connected to.

Looking at this diagram also allows us to be more specific about the first law analysis of the bearing, because we can see that the power applied to the bearing is the sum of the powers that flow out of the truck and the body to provide the total work applied to the bearing. We can write the following relationships:

$\dot{W}_{out,w} +\dot{W}_{out,c}=\dot{W}_{in,b}$ (EQN 4)

$\dot{Q}_{out,b} = \dot{Q}_{out}$ (EQN 5)

This concludes our set of equations for steps 2 and 3 of our disciplined process.

Using Algebraic Manipulation to obtain a final model

Writing out the full list of equations we have obtained from steps 1-3 of our disciplined process, we obtain the following list of relationships:

  1. $ -(m_w v \dot{v} + J_w \Omega \dot{\Omega}) = \dot{W}_{out,w}$
  2. $ - \frac{1}{2}m_c v \dot{v} = \dot{W}_{out,c}$
  3. $\dot{W}_{in,b} = \dot{Q}_{out,b} = b\Omega ^2$
  4. $\dot{W}_{out,w} +\dot{W}_{out,c}=\dot{W}_{in,b}$
  5. $\dot{Q}_{out,b} = \dot{Q}_{out}$

In this list, we see the following terms that are neither independent nor dependent variables (unknowns):

  1. $\dot{W}_{out,w}$
  2. $\dot{W}_{out,c}$
  3. $\dot{W}_{in,b}$
  4. $\dot{Q}_{out,b}$

With more equations than unknowns, we are in a good position to attempt to eliminate our unknowns.

Writing the first law equation for the bearing and substituting Eqn 1 and Eqn 2 and Eqn 3 into Eqn 4, we get:

$$\dot{W}_{in,b} = \dot{Q}_{out,b} = b\Omega ^2 = - \frac{m_c}{2} v \dot{v} -(m_w v \dot{v} + J_w \Omega \dot{\Omega}) $$

Now, remembering that because the wheels roll without slip, $-\Omega r_w = v$ and $-\dot{\Omega} r_w = \dot{v}$, where $r_w$ is the radius of the wheels, we can make another substitution to get all of the terms in the equation in terms of our desired output from our input-output model of the system.

$$ b\frac{v ^2}{r_w^2} = - \frac{m_c}{2} v \dot{v} -(m_w v \dot{v} + J_w \frac{v}{r} \frac{\dot{v}}{r_w}) $$

This allows us to cancel an extra power of $v$ from the equation, which leaves us with:

$$ b \frac{v}{r_w^2} = - \frac{m_c}{2} \dot{v} -(m_w \dot{v} + J_w \frac{\dot{v}}{r_w^2}) $$

When we clean this up and rearrange terms to obtain a differential equation relating velocity with acceleration, we get the exact same model we got using Newton's laws:

$$ \dot{v} = - \frac{1}{(\frac{m_c}{2}+m_w)r_w^2+J_w} bv $$

Completing step 5 in our disciplined process is easy, since we have already verified units, signs, and form for this model in Reading 08.

This equation started off as a statement about the conservation of energy over time by relating the flow of power into and out of each "piece" of our system. Curiously, this statement is equivalent to a statement about the influence of forces on the masses in our system via Newton's second law!

This process shows that using the first law of thermodynamics can give us an equivalent system model to what we got using Newton's laws. Because writing kinetic and/or potential energy for a system's elements is much more tractable over different types of systems than using Newton's laws, this will be our ticket towards developing a unified approach to constructing models for physical systems of all different types.

Assignment

To practice using the first law of thermodynamics to construct a model for a physical system, you will construct a model for your coasting pulley system from Reading 07.