SMU PHYSICS 1303: Introduction to Mechanics

Home , Sliding (motion)

Stephen Sekula1

1Southern Methodist University Dallas, TX, USA

SPRING, 2019

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 1 Outline

Conservation of Energy

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 2 Conservation of Energy

Conservation of Energy

NASA, “Hipnos” by Molinos de Viento and available under Creative Commons from Flickr S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 3 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We will understand the mathematical description of energy conservation. I We will explore the implications of the conservation of energy.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 4 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We will understand the mathematical description of energy conservation. I We will explore the implications of the conservation of energy.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 4 Conservation of Energy A Principle of Nature: Energy is Conserved

It is a remarkable fact of the universe: that this quantity which we call “energy” is, in a system where nothing can leave and nothing can enter the system, conserved. I That is to say, if we wall off a system of matter such that nothing can get in and nothing can get out, and if we can account for all the ways in which energy can be distributed inside the system, we observe that energy changes form but is neither created nor destroyed. I Figuring this out for the ﬁrst time took immense effort. Consider the experiments of the French chemists (and husband and wife) Lavoisier and Paulze, whose careful and quantitative experiments with chemical transformations showed that the mass of matter before and after chemical reactions was the same — that something was conserved. I Ultimately it would be realized that not mass, but in fact energy, is what nature conserves.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 5 Conservation of Energy A Principle of Nature: Energy is Conserved

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 5 Conservation of Energy The Conservation of Mechanical Energy

Mechanical Energy is deﬁned explicitly as the sum of kinetic energy and potential energy. Note that mechanical energy excludes situations where a non-conservative force acts. Considering only conservative forces, it is observed that mechanical energy is conserved.

Let us deﬁne the total mechanical energy in a given moment of time as:

Emec = K + U The conservation of mechanical energy then requires that the following be true at some initial time, ti , and some ﬁnal time, tf > ti :

Emec,f = Emec,i −→ Kf + Uf = Ki + Ui

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 6 Conservation of Energy The Conservation of Mechanical Energy

Let us deﬁne the total mechanical energy in a given moment of time as:

Emec = K + U The conservation of mechanical energy then requires that the following be true at some initial time, ti , and some ﬁnal time, tf > ti :

Emec,f = Emec,i −→ Kf + Uf = Ki + Ui

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 6 Conservation of Energy The Conservation of Mechanical Energy

Let us deﬁne the total mechanical energy in a given moment of time as:

Emec = K + U The conservation of mechanical energy then requires that the following be true at some initial time, ti , and some ﬁnal time, tf > ti :

Emec,f = Emec,i −→ Kf + Uf = Ki + Ui

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 6 Conservation of Energy The Conservation of Mechanical Energy

Let us deﬁne the total mechanical energy in a given moment of time as:

Emec = K + U The conservation of mechanical energy then requires that the following be true at some initial time, ti , and some ﬁnal time, tf > ti :

Emec,f = Emec,i −→ Kf + Uf = Ki + Ui

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 6 Conservation of Energy Conservative Forces and the Conservation of Mechanical Energy We almost derived this conservation earlier, when considering changes in kinetic energy and work done by a conservative force. We know from our earlier exploration of, for instance, gravity that

∆K = Kf − Ki = W where W is the work done by a conservative force. We also saw that:

∆U = −W It’s a small leap from those observations to the conservation of mechanical energy, once we deﬁne Emec = K + U:

∆K = W = −∆U −→ Kf − Ki = −(Uf − Ui ) −→ Kf + Uf = Ki + Ui

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 7 Conservation of Energy Conservative Forces and the Conservation of Mechanical Energy We almost derived this conservation earlier, when considering changes in kinetic energy and work done by a conservative force. We know from our earlier exploration of, for instance, gravity that

∆K = Kf − Ki = W where W is the work done by a conservative force. We also saw that:

∆U = −W It’s a small leap from those observations to the conservation of mechanical energy, once we deﬁne Emec = K + U:

∆K = W = −∆U −→ Kf − Ki = −(Uf − Ui ) −→ Kf + Uf = Ki + Ui

∆K = Kf − Ki = W where W is the work done by a conservative force. We also saw that:

∆U = −W It’s a small leap from those observations to the conservation of mechanical energy, once we deﬁne Emec = K + U:

∆K = W = −∆U −→ Kf − Ki = −(Uf − Ui ) −→ Kf + Uf = Ki + Ui

∆K = Kf − Ki = W where W is the work done by a conservative force. We also saw that:

∆U = −W It’s a small leap from those observations to the conservation of mechanical energy, once we deﬁne Emec = K + U:

∆K = W = −∆U −→ Kf − Ki = −(Uf − Ui ) −→ Kf + Uf = Ki + Ui

∆K = Kf − Ki = W where W is the work done by a conservative force. We also saw that:

∆U = −W It’s a small leap from those observations to the conservation of mechanical energy, once we deﬁne Emec = K + U:

∆K = W = −∆U −→ Kf − Ki = −(Uf − Ui ) −→ Kf + Uf = Ki + Ui

∆K = Kf − Ki = W where W is the work done by a conservative force. We also saw that:

∆U = −W It’s a small leap from those observations to the conservation of mechanical energy, once we deﬁne Emec = K + U:

∆K = W = −∆U −→ Kf − Ki = −(Uf − Ui ) −→ Kf + Uf = Ki + Ui

∆K = Kf − Ki = W where W is the work done by a conservative force. We also saw that:

∆U = −W It’s a small leap from those observations to the conservation of mechanical energy, once we deﬁne Emec = K + U:

∆K = W = −∆U −→ Kf − Ki = −(Uf − Ui ) −→ Kf + Uf = Ki + Ui

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 7 Conservation of Energy The Implications of this Observation About Mechanical Energy There are some deep implications from these relationships: I If a system can be isolated in such a way that only conservative forces can act, then this relationship absolutely holds I Practically speaking, how can one do this? I Friction can be immensely reduced to negligible levels by using chemicals that reduce friction coefficients (e.g. lubricants such as grease, oil, graphite, etc.) I Drag can be immensely reduced by spending effort on reducing Cd through aerodynamic engineering, or by reducing the cross-sectional area of the object moving in a fluid, etc. You can also remove the fluid (e.g. operate in “vacuum”). I Once you have established these conditions, then the total mechanical energy of any state (configuration) of the isolated system is equal to the total mechanical energy of any other state (configuration) of the system. Let’s look at an interesting example of how mechanical energy is conserved, even while it is shared between kinetic and potential energy “ buckets” during complex motion: the chaotic pendulum.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 8 Conservation of Energy The Implications of this Observation About Mechanical Energy There are some deep implications from these relationships: I If a system can be isolated in such a way that only conservative forces can act, then this relationship absolutely holds I Practically speaking, how can one do this? I Friction can be immensely reduced to negligible levels by using chemicals that reduce friction coefficients (e.g. lubricants such as grease, oil, graphite, etc.) I Drag can be immensely reduced by spending effort on reducing Cd through aerodynamic engineering, or by reducing the cross-sectional area of the object moving in a fluid, etc. You can also remove the fluid (e.g. operate in “vacuum”). I Once you have established these conditions, then the total mechanical energy of any state (configuration) of the isolated system is equal to the total mechanical energy of any other state (configuration) of the system. Let’s look at an interesting example of how mechanical energy is conserved, even while it is shared between kinetic and potential energy “ buckets” during complex motion: the chaotic pendulum.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 8 Conservation of Energy Another deep implication of energy concepts: energy and forces

Recall the relationship between potential energy and work done by a force, Fx , in one-dimension:

∆U = −W = −Fx ∆x

Let us consider not large changes in potential energy or displacement (e.g. ∆U), but inﬁnitesimal changes in these quantities. In that case ∆U → dU and ∆x → dx and: Z Z U = dU = − Fx dx

If we now take the derivative with respect to x of both sides of the above, we “undo” the integral on the right-hand side (remember: the integral is the “anti-derivative,” so the derivative of the anti-derivative is just the integrand, Fx !) and we learn something truly deep.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 9 Conservation of Energy Another deep implication of energy concepts: energy and forces

Recall the relationship between potential energy and work done by a force, Fx , in one-dimension:

∆U = −W = −Fx ∆x

Let us consider not large changes in potential energy or displacement (e.g. ∆U), but inﬁnitesimal changes in these quantities. In that case ∆U → dU and ∆x → dx and: Z Z U = dU = − Fx dx

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 9 Conservation of Energy Another deep implication of energy concepts: energy and forces

Recall the relationship between potential energy and work done by a force, Fx , in one-dimension:

∆U = −W = −Fx ∆x

Let us consider not large changes in potential energy or displacement (e.g. ∆U), but inﬁnitesimal changes in these quantities. In that case ∆U → dU and ∆x → dx and: Z Z U = dU = − Fx dx

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 9 Conservation of Energy Another deep implication of energy concepts: energy and forces

Recall the relationship between potential energy and work done by a force, Fx , in one-dimension:

∆U = −W = −Fx ∆x

Let us consider not large changes in potential energy or displacement (e.g. ∆U), but inﬁnitesimal changes in these quantities. In that case ∆U → dU and ∆x → dx and: Z Z U = dU = − Fx dx

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 9 Conservation of Energy Another deep implication of energy concepts: energy and forces

Recall the relationship between potential energy and work done by a force, Fx , in one-dimension:

∆U = −W = −Fx ∆x

Let us consider not large changes in potential energy or displacement (e.g. ∆U), but inﬁnitesimal changes in these quantities. In that case ∆U → dU and ∆x → dx and: Z Z U = dU = − Fx dx

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 9 Conservation of Energy Another deep implication of energy concepts: energy and forces

Recall the relationship between potential energy and work done by a force, Fx , in one-dimension:

∆U = −W = −Fx ∆x

Let us consider not large changes in potential energy or displacement (e.g. ∆U), but inﬁnitesimal changes in these quantities. In that case ∆U → dU and ∆x → dx and: Z Z U = dU = − Fx dx

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 9 Conservation of Energy Another deep implication of energy concepts: energy and forces

Recall the relationship between potential energy and work done by a force, Fx , in one-dimension:

∆U = −W = −Fx ∆x

Let us consider not large changes in potential energy or displacement (e.g. ∆U), but inﬁnitesimal changes in these quantities. In that case ∆U → dU and ∆x → dx and: Z Z U = dU = − Fx dx

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 9 Conservation of Energy Another deep implication of energy concepts: energy and forces If we now take the derivative with respect to x of both sides of the above, we “undo” the integral on the right-hand side (remember: the integral is the “anti-derivative,” so the derivative of the anti-derivative is just the integrand, Fx !) and we learn something truly deep.

dU − = F dx x That is to say, the force exerted is equal to the negative of the change in potential energy with respect to displacement. Force and Changes in Potential Energy can be directly related to one another.

You can see from the above why it was almost inevitable that mechanics would be reformulated in terms of energy concepts, rather than force concepts. Energy, being a scalar, is easier to work with; Force, being a vector, can be determined from energy by the action of the derivative.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 10 Conservation of Energy Another deep implication of energy concepts: energy and forces If we now take the derivative with respect to x of both sides of the above, we “undo” the integral on the right-hand side (remember: the integral is the “anti-derivative,” so the derivative of the anti-derivative is just the integrand, Fx !) and we learn something truly deep.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 10 Conservation of Energy A Glimpse of the Future: Classical Mechanics and its Energy Foundation The force associated with a conservative actor (e.g. gravity) in general is a vector, and that vector can, in general, be determined directly from the potential energy as a function of coordinates, U(x, y, z) ≡ U(~r):

dU dU dU dU F~ = − = − ˆi − ˆj − kˆ d~r dx dy dz

As a sneak preview of both energy-based classical mechanics, and vector calculus, one can deﬁne the following vector: d d d ∇~ ≡ , , dx dy dz which is the gradient operator. It can tell us how potential energy changes with spatial location. Thus we learn that force is the negative of the gradient of potential energy, which is represented as:

F~ = −∇~ U From a scalar arises a vector. S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 11 Conservation of Energy A Glimpse of the Future: Classical Mechanics and its Energy Foundation The force associated with a conservative actor (e.g. gravity) in general is a vector, and that vector can, in general, be determined directly from the potential energy as a function of coordinates, U(x, y, z) ≡ U(~r):

dU dU dU dU F~ = − = − ˆi − ˆj − kˆ d~r dx dy dz

F~ = −∇~ U From a scalar arises a vector. S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 11 Conservation of Energy Hamilton, Lagrange, and the Lagrange-Hamilton Formalism for Mechanics

Complex problems are difﬁcult to handle with the way we have learned forces so far in this course. The methodology developed by Lagrange and Hamilton much more readily allows for the solution of complex problems. Note that Kinetic Energy depends on the time-derivative of position:

Joseph-Louis dx 1 dx 2 v = ≡ x˙ −→ K = m = K (x˙ ) Lagrange dt 2 dt (1736-1813) while potential energy depends on position along, U = U(x). What Lagrange and Hamilton realized was that if one deﬁnes what is called the Lagrangian of a system, L = K (x˙ ) − U(x), all equations of motion for any system can be determined by solving this equation: dL d dL − = 0 William Rowan dx dt dx˙ Hamilton

(1805-1865)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 12 Conservation of Energy Hamilton, Lagrange, and the Lagrange-Hamilton Formalism for Mechanics

(1805-1865)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 12 Conservation of Energy Hamilton, Lagrange, and the Lagrange-Hamilton Formalism for Mechanics

(1805-1865)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 12 Conservation of Energy Hamilton, Lagrange, and the Lagrange-Hamilton Formalism for Mechanics

(1805-1865)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 12 Conservation of Energy Hamilton, Lagrange, and the Lagrange-Hamilton Formalism for Mechanics

(1805-1865)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 12 Conservation of Energy Hamilton, Lagrange, and the Lagrange-Hamilton Formalism for Mechanics

(1805-1865)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 12 Conservation of Energy Lessons from the Conservation of Mechanical Energy

From the simple equation, Emec = K (v) + U(x) (where v is velocity and x is location, such that kinetic energy depends on the rate of change of position and potential energy depends on position alone), we learn a few things about motion: I If there is a place in x, x = x0 where K (v) = 0 (implying v = 0), such that 0 Emec = U(x ) is what is known as a turning point of the motion; that is, the object can proceed no further, and motion reverses as energy is then depleted from potential and returned to kinetic. I Think of the example of throwing a ball up into the air. The ball rises, slows, stops, and reverses. The turning point of this motion is when Emec = mgh, where h is the maximum height of the ball. At that point, we know that K = 0 → v = 0.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 13 Conservation of Energy Lessons from the Conservation of Mechanical Energy

From the simple equation, Emec = K (v) + U(x) (where v is velocity and x is location, such that kinetic energy depends on the rate of change of position and potential energy depends on position alone), we learn a few other things about motion: I Since force is given by the negative of the change in potential energy with displacement (F = −dU/dx), we learn something about key points in the motion as regards forces:

I If there are locations in position such that U(x) = Emec, and dU/dx = 0, then there can be no kinetic energy and no change in the state of motion (no net forces) — this is known as neutral equlibrium. I If there are places in x, x = x0 where dU/dx = 0, but where even a slight displacement from x0 puts an object in a place where dU/dx < 0, then F > 0 and acceleration will further displace the object from x0. These are points of unstable equilibrium. I Finally, if there are places in x (again, denoted x = x0) where even a slight displacement from x0 puts it in a location where dU/dx > 0, then F < 0 and that force restores it back toward x0. These are places of stable equlibrium.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 14 Conservation of Energy Lessons from the Conservation of Mechanical Energy

From the simple equation, Emec = K (v) + U(x) (where v is velocity and x is location, such that kinetic energy depends on the rate of change of position and potential energy depends on position alone), we learn a few other things about motion: I Since force is given by the negative of the change in potential energy with displacement (F = −dU/dx), we learn something about key points in the motion as regards forces:

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 14 Conservation of Energy Lessons from the Conservation of Mechanical Energy

From the simple equation, Emec = K (v) + U(x) (where v is velocity and x is location, such that kinetic energy depends on the rate of change of position and potential energy depends on position alone), we learn a few other things about motion: I Since force is given by the negative of the change in potential energy with displacement (F = −dU/dx), we learn something about key points in the motion as regards forces:

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 14 Conservation of Energy Lessons from the Conservation of Mechanical Energy

From the simple equation, Emec = K (v) + U(x) (where v is velocity and x is location, such that kinetic energy depends on the rate of change of position and potential energy depends on position alone), we learn a few other things about motion: I Since force is given by the negative of the change in potential energy with displacement (F = −dU/dx), we learn something about key points in the motion as regards forces:

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 14 Conservation of Energy Lessons from the Conservation of Mechanical Energy

From the simple equation, Emec = K (v) + U(x) (where v is velocity and x is location, such that kinetic energy depends on the rate of change of position and potential energy depends on position alone), we learn a few other things about motion: I Since force is given by the negative of the change in potential energy with displacement (F = −dU/dx), we learn something about key points in the motion as regards forces:

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 14 Conservation of Energy External Forces

We’ve so far been thinking about closed and isolated systems without non-conservative forces, or even the possibility for any additional force to reach into the system and change it. But what if there is the possibility of such a force, one that can do work on some part of the system? In that case, additional changes in kinetic and potential energy are possible, and in general if such a force can do work, W ,

W = ∆K + ∆U

when W = 0 (no work done by any external force), then we recover the conservation of mechanical energy. But when W 6= 0 we have that:

W = ∆Emec

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 15 Conservation of Energy External Forces

W = ∆K + ∆U

when W = 0 (no work done by any external force), then we recover the conservation of mechanical energy. But when W 6= 0 we have that:

W = ∆Emec

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 15 Conservation of Energy External Forces

W = ∆K + ∆U

when W = 0 (no work done by any external force), then we recover the conservation of mechanical energy. But when W 6= 0 we have that:

W = ∆Emec

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 15 Conservation of Energy External Forces

W = ∆K + ∆U

when W = 0 (no work done by any external force), then we recover the conservation of mechanical energy. But when W 6= 0 we have that:

W = ∆Emec

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 15 Conservation of Energy Enter Non-Conservative Forces: Friction as an Example Just as we might imagine some external force putting more mechanical energy into a closed system by acting on one or more of its parts, we can imagine a force sapping energy from closed system. Non-conservative forces do just that. Consider a block of mass, m, pulled along a surface from an initial speed, v0. I I pull it by on a string attached to the block (force, F). Friction acts to oppose the motion (force fk ). I do work, and friction does work. If the block is accelerating during 2 2 v −v0 this process, then F − fk = ma. Equations of motion tell us that a = 2∆x . I We learn that: ! (v 2 − v 2) 1 ∆K F − f = m 0 = (K − K ) = k 2∆x ∆x f i ∆x I If the block were sliding down a ramp, then it might start with some initial gravitational potential energy and later have a different gravitational potential energy, so in general:

F∆x − fk ∆x = ∆K + ∆U = ∆Emec

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 16 Conservation of Energy Enter Non-Conservative Forces: Friction as an Example Just as we might imagine some external force putting more mechanical energy into a closed system by acting on one or more of its parts, we can imagine a force sapping energy from closed system. Non-conservative forces do just that. Consider a block of mass, m, pulled along a surface from an initial speed, v0. I I pull it by on a string attached to the block (force, F). Friction acts to oppose the motion (force fk ). I do work, and friction does work. If the block is accelerating during 2 2 v −v0 this process, then F − fk = ma. Equations of motion tell us that a = 2∆x . I We learn that: ! (v 2 − v 2) 1 ∆K F − f = m 0 = (K − K ) = k 2∆x ∆x f i ∆x I If the block were sliding down a ramp, then it might start with some initial gravitational potential energy and later have a different gravitational potential energy, so in general:

F∆x − fk ∆x = ∆K + ∆U = ∆Emec

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 16 Conservation of Energy Where does the energy go? Other forms of energy and friction

We see that friction takes energy out of the block/gravity system. Especially if fk > F, then ∆Emec < 0 and the total mechanical energy of the system declines over time. Where is that energy going?

Careful measurement will teach you that it goes into heating the block and surface (energy is transferred to atoms, causing them to vibrate in materials and thus heating the materials) and sound (which is also a kind of thermal energy, having to do with the motion of microscopic components of material). So we might represent the total energy of the block/gravity/surface system by accounting for this energy:

Etotal = Emec + Ethermal = K + U + Ethermal

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 17 Conservation of Energy Where does the energy go? Other forms of energy and friction

We see that friction takes energy out of the block/gravity system. Especially if fk > F, then ∆Emec < 0 and the total mechanical energy of the system declines over time. Where is that energy going?

Etotal = Emec + Ethermal = K + U + Ethermal

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 17 Conservation of Energy Where does the energy go? Other forms of energy and friction

We see that friction takes energy out of the block/gravity system. Especially if fk > F, then ∆Emec < 0 and the total mechanical energy of the system declines over time. Where is that energy going?

Etotal = Emec + Ethermal = K + U + Ethermal

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 17 Conservation of Energy The General Conservation of Energy The most general equation we could write down would actually be the following. If work, W , is done by an external force on a system that can contain energy in kinetic, potential, thermal, or other kinds of “internal” energy (e.g. chemical energy stored in your muscles/cells; or atomic/molecular energy stored their vibration or rotation), then:

W = ∆Emec + ∆Ethermal + ∆Einternal = ∆Etotal For a closed and isolated system on which no external forces can act, then it will be true that: 0 = ∆Emec + ∆Ethermal + ∆Einternal That is that total energy does not change, and thus is conserved. This has been the observation of countless experiments, still ongoing today. It has not ever been observed to be violated, and is taken as a law of nature. In principle, the energy available at the beginning of the universe has, across the entire universe, remained unchanged since the beginning of time. It may be redistributed internally within the cosmos, but it is neither going up nor down; merely changing internal form.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 18 Conservation of Energy The General Conservation of Energy The most general equation we could write down would actually be the following. If work, W , is done by an external force on a system that can contain energy in kinetic, potential, thermal, or other kinds of “internal” energy (e.g. chemical energy stored in your muscles/cells; or atomic/molecular energy stored their vibration or rotation), then:

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 18 Conservation of Energy But. . . why is energy conserved? The answer to this question would have to wait for a brilliant young mathematical physicist named Emmy Noether. Her most famous work, summarized in what are known as Noether’s Theorems, noted that when a mathematical theory of nature has a special property known as a continuous symmetry, then it will have an associated conserved quantity.

The known mathematical theory of nature — the laws of mechanics and also electricity and magnetism — is invariant against shifts in time. This implies, via Noether’s Theorems, that an internal quantity which we know as “energy” is also conserved.

Emmy Noether (1882-1935)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 19 Conservation of Energy But. . . why is energy conserved? The answer to this question would have to wait for a brilliant young mathematical physicist named Emmy Noether. Her most famous work, summarized in what are known as Noether’s Theorems, noted that when a mathematical theory of nature has a special property known as a continuous symmetry, then it will have an associated conserved quantity.

Emmy Noether (1882-1935)

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 19 Conservation of Energy Key Ideas The key ideas that we have explored in this section of the course are as follows: I We have come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We have explored the mathematical description of energy conservation. I We have explored the implications of the conservation of energy.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 20 Conservation of Energy Key Ideas The key ideas that we have explored in this section of the course are as follows: I We have come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We have explored the mathematical description of energy conservation. I We have explored the implications of the conservation of energy.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

Jacques-Louis David. “Portrait of

Monsieur de Lavoisier and his Wife,

chemist Marie-Anne Pierrette

Paulze”. Available under Creative

Commons from Flickr.

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 20 References References I

S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 21