### Work and Energy

The work done by a force on a particle (or a system of particles) equals the energy transferred to the system by the force. For example, a spring which does work on an attached particle either increases or decreases its kinetic energy. The force of a hand pulling on the particle does work on the “spring plus mass” system and changes the total energy of the system: the particle’s kinetic energy plus the energy in the spring.

In fact, the defining formula for the work done on a particle which moves from point 1 to 2, \(~\textrm{W} =\int_{1}^{2} \vec{\textrm{F}}.\vec{\textrm{dr}}~\), is nothing other than the change in kinetic energy along its trajectory \(~ \textrm{KE}(2) – \textrm{KE}(1)~\) re-expressed in terms of the force using Newton’s second law \(~\vec{\textrm{F}}~=~\textrm{m}\frac{\textrm{d}\vec{\textrm{v}}}{\textrm{dt}}~\) (This equivalence is called the “work-energy theorem”).

###### Text and Video resources:

Section 14-1 in Feynman’s lectures on physics has an excellent discussion of the relation between the concept of work in physics and the colloquial use of the term.

This video from Khan academy has an introductory algebra based discussion of the work energy theorem.

This lecture video by Prof Shankar of Yale University has an accesible discussion of energy, work, and the work-energy theorem.

This short video by Dr. Peter Dourmashkin of MIT has a succinct discussion of the work-energy theorem.

The equivalence of work done and energy transferred is discussed in section 1.1.3 of these lecture notes from Caltech, in Chapter 13 of Feynman’s lectures and in this Wikipedia article.

A force is called conservative if the quantity of work it does on a particle which moves between any two points A and B is independent of the particle’s trajectory from A to B. This implies that a conservative force contributes zero work when a particle returns to its initial position (B=A): there is always a path from A to A for which the work done is manifestly zero – the particle just stays at A! (The second statement implies the first and can also be used as the definition.)

It follows that the contribution of a conservative force to the change in kinetic energy is the same for every trajectory from A to B (see here). As explained nicely by Feynman, if the work done by gravity in going around a full loop is not zero, the extra kinetic energy generated by going around could be used to obtain perpetual motion!

Here’s an explicit example: if gravity is conservative, a block which shoots up a straight frictionless ramp, stops, and then glides back down on a fancy “spiral” ramp, will end up with the same kinetic energy. (The normal forces of the ramps do no work. Can you explain why?)

###### Text and video resources:

The equivalence of the two definitions is explained clearly in this Wikipedia article, which also gives a succinct mathematical description of these conditions.

This video by Prof. Walter Lewin (filmed at MIT) has a clear and accessible explanation of conservative and non-conservative forces.

This short video by Dr. Peter Dourmashkin of MIT has a succinct discussion of conservative forces.

Section 14-3 of Feynman’s lectures also has a nice discussion . Section 14-2 discusses “forces of constraint” such as the normal forces of the ramp in the example above.

This short video by Dr. Peter Dourmashkin of MIT has a discussion of non-conservative forces.

First, some background:

A key feature of the quantity we call “energy” is conservation – its total value is unchanged by every natural process we are familiar with. For example, the kinetic energy of a block sliding on a rough surface decreases with time, but we expect careful observation to show the missing energy pop up as “heat” – in molecular vibrations within the ground, the block, the surrounding air, in electromagnetic radiation, etc. Not an easy accounting to actually perform!

Another essential aspect of energy is that it is a property of the “state of the system” at a given instant. Mathematically speaking, our formulas for energy are functions of variables used to describe the “appearance” of the system at a particular moment in time (for example, the instantaneous positions and velocities of its particles) – as opposed to depending on the system’s history. **The fact that there exists a physical quantity which is both a “function of state” and conserved is an incredible simplification for modelling physical phenomena**.

**Can we write down a formula for such an unchanging quantity in the simpler situation where a conservative force acts on a particle?**

Yes! The particle’s kinetic energy alone won’t cut it since the force accelerates the particle. However, if the force is conservative, we can define a quantity – the potential energy (\(\textrm{U}(\vec{\textrm{r}})\)) – associated with each possible position, whose change as the particle moves exactly compensates the change in kinetic energy. We just set,

\[ U(\vec{r}) ~-~ U(0) ~=~ -[KE(\vec{r}) ~-~ KE(0)] = ~-\int_{0}^{\vec{r}}~ \vec{F}.\vec{dx} \].

Or:

\[ U(\vec{r}) ~=~ U(0) ~-~ \int_{0}^{\vec{r}}~ \vec{F}.\vec{dx} \]

Now comes the punchline: only for a conservative force is the RHS a function of a position, rather than the entire history of the particle! \( \int_{0}^{\vec{\textrm{r}}}~ \vec{\textrm{F}}.\vec{\textrm{dx}} \) is dependent only on \( \vec{\textrm{r}} \) for a conservative force (the initial position 0 and the value of U there, U(0), are chosen arbitrarily and then held fixed). We have arrived at a conserved quantity which is a function of the instantaneous state (specified by \(\vec{\textrm{r}}\), \(\vec{\textrm{v}}\)), the Energy!

\[ \textrm{Energy} = \frac{1}{2} m \overrightarrow{v} ^2 + U(\overrightarrow{r}) \]

Text and Video resources –

Feynman has an excellent discussion of the concept of energy in section 4-1 of his lectures. He has discussions of potential energy in sections 13-1, 14-3 and 13-3.

This short video by Dr. Peter Dourmashkin of MIT explains the potential energy of a spring.

This video by Prof Matt Anderson discusses the concept of gravitational potential energy and derives a formula for it.

This short video by Dr. Peter Dourmashkin of MIT has explains gravitational potential energy near the earth’s surface.

This video by Prof. Chakrabarty of MIT introduces the concept of potential energy.

This question imagines the following situation: an “agent” applies a force to a particle in a gravitational field and displaces it from point 1 to point 2 while keeping its kinetic energy unchanged throughout. How much work does this agent do? Well, in order to keep the kinetic energy unchanged, the amount of work the agent does must be the negative of the work done by gravity, since change in kinetic energy equals total work done. Next, recall that the difference in potential energy \( \textrm{U}(2) ~ – ~\textrm{U}(1) \) is precisely (by definition) the negative of the work done by gravity during the displacement from 1 to 2. It follows that the work done by the external agent equals \(\textrm{U}(2)~ – ~\textrm{U}(1) \).

Here’s an explicit picture: When you raise or lower a ball at a constant pace, the amount of work your muscles do on the ball equals the change in the ball’s gravitational potential energy.

Text and video resources:

This video by Prof Matt Anderson relates the work done by an external agent to the change in potential energy.

Yes. The law of conservation of energy has been confirmed in empirical observations of all manner of natural processes. This principle is baked into the formulas for the fundamental laws of nature known to us: those governing electric and magnetic fields, the nuclear forces, electrons, quarks etc.

Text and video resources –

Feynman has a highly accessible discussion about how non-conservative forces like friction and air resistance can emerge from microscopic phenomena involving only conservative forces in section 14-4 of his lectures.

Consider a system made up of many particles exerting forces on each other – you can imagine a bunch of particles connected by springs, or a collection of masses attracting each other gravitationally. **By definition, the work done on this system equals the change in its energy (= the sum of each particle’s kinetic energy plus the total potential energy associated with all internal forces) caused by external forces acting on it.**

Using Newton’s 2nd law the change in the system’s total energy can be written in terms of the external forces acting on each particle:

Work = Change in Energy = \(\sum\limits_{i}~\int\vec{\textrm{F}}^{\textrm{ext}}_i.\textrm{d}\vec{\textrm{r}}_i~\). Here \(\vec{\textrm{r}}_i\) is the position of the i’th particle and \(\vec{\textrm{F}}_i\) is the external force acting on it. (See also this answer.)

**Note**: In mechanics class you often deal with rigid bodies (def: the distance between every pair of particles in the body is fixed). If you make the usual assumption that the internal energy depends only on the distances between particles, then it never changes and the work done by external forces equals the change in the kinetic energy of the rigid body.

###### Text and video resources –

This video by Prof. Purwar of Stony Brook University derives the work-energy relation for a general system of particles.

These lecture notes by Profs Peraire and Widnall of MIT discuss the work-energy theorem for a rigid body.

### Gravitation

Yes, because the quantity of work it does on a particle which moves between two points is the same for every trajectory connecting them. For example, you can check, using Newton’s second law, that if you launch a ball straight up from some point A on the moon, its kinetic energy will be unchanged when it comes back down to A. This must be so if gravity is conservative for there is another path from A to A along which the work done is manifestly zero: the ball just sits at A!

###### Text and video resources:

Section 13-2 of Feynman’s lectures on physics gives a clear derivation of the conservative nature of gravity.

See this stack exchange answer for a thought experiment concerning the work done by gravity, and this answer for a mathematical demonstration (using vector calculus) that gravity is conservative.

It doesn’t have to be! Notice that the definition of potential energy references an arbitrary point (0) and an arbitrary value for the potential energy at that point – \(\textrm{U}(0)\). The potential energy at any other point \(\vec{\textrm{r}}\) is “built up” by subtracting from \(\textrm{U}(0)\) the work done by gravity on the particle when it travels from 0 to \(\vec{\textrm{r}}\). This guarantees that the energy \( \frac{1}{2}\textrm{m}\textrm{v}^2 + \textrm{U}(\vec{\textrm{r}})\) is conserved during motion, irrespective of the choices made for 0 and \(\textrm{U}(0)\). (Changing the value of \(\textrm{U}(0)\) amounts to shifting the potential energy everywhere by the same constant).

If you choose 0 to be a point at infinity and U(0) to be zero then the potential energy will be negative at every point a finite distance from the origin. This is the usual convention. However, you could also choose: \(\textrm{U}(\textrm{Highest point on Mt. Everest}) ~=~ 2\) !

Here is another perspective on why a shift in the value of the potential energy by the same value everywhere is physically irrelevant: We can express force in terms of potential energy by inverting the defining relation \( \textrm{U}(\vec{\textrm{r}}) = \textrm{U}(0) -\int_{0}^{\vec{\textrm{r}}} \vec{\textrm{F}}.\textrm{d}\vec{\textrm{x}}\). To do so, undo the integral by taking the “derivative” of both sides (really the gradient). We get, \( \vec{\textrm{F}} = – \vec{\nabla}\textrm{U} \) (If you’re unfamiliar with the gradient, replace it with the regular derivative \(\frac{\textrm{d}}{\textrm{dr}}\) in your mind and drop the vector arrows everywhere). The derivative (gradient) of a function is unchanged when it is shifted by a constant. Thus shifting the potential by the same value everywhere leaves the force on the particle, and hence its motion, unchanged.

Text and video resources:

This video by Dr. Peter Dourmashkin of MIT discusses the arbitrary choice of a reference state in the definition of potential energy.

This video by Dr. Peter Dourmashkin of MIT derives the potential energy of gravity (with the reference point chosen to be at infinity).

### Momentum

Yes, it is a law that the momentum of an isolated system is conserved. By isolated we mean that no external forces act and also that no component of the system “leaves it”. This law is true regardless of the type of forces the components exert on each other, say during a collision.

It is important to keep track of every component of the system. For example, if two balloons collide in a vacuum (no external force of air resistance) but some air leaks out of one of them during the collision, the total momentum of the balloons themselves will not be conserved – some momentum is carried away by the air molecules! In some complex situations you may even have to keep track of things like emitted radiation, since electric and magnetic fields can carry momentum.

###### Text and video resources:

Section 10-2 of Feynman’s lectures has a clear derivation of the conservation of momentum from Newton’s 3rd law (“action equals reaction”). Section 18-1 an excellent discussion on the center of mass in the context of the motion of a system of particles,

This short video by Dr. Peter Dourmashkin of MIT discusses conservation of momentum for 1-D collisions.

This video by Dr. Elliot Schneider discusses the relation between Newton’s 3rd law and conservation of momentum.

This video by Dr. Elliot Schneider discusses the relation between conservation laws (like that for momentum) and symmetries (aka Noether’s theorem).

This lecture by Prof Shankar of Yale University explains the concepts of center of mass and conservation of momentum.

Let’s consider what’s going on in one of those problems where kinetic energy is not conserved. Imagine two blocks that collide and then stick together – kinetic energy cannot be conserved if momentum is (can you argue why?). Total energy is conserved, so where did the missing energy go? When the blocks collide you imagine that they “compress” or “crumple” a bit – some kinetic energy gets transformed into the potential energy of internal forces involved in the crumpling. You also expect that some of the kinetic energy gets transferred into vibrations of the material and into heat. Textbook problems do not expect you to keep track of this complex internal transfer of energy!

But what about momentum? The point is that unlike energy, the initial momentum cannot be “hidden away” in internal degrees of freedom – it always shows up in the collective (“center of mass”) motion of the system. Perhaps a helpful intuition is that, unlike energy, momentum is a vector, it encodes the directed linear motion of the body and this cannot be conserved by “confining” it internally.

Indeed in the (commonly encountered) case of a system of particles with inter-particle obeying Newton’s third law, the principle of conservation of momentum can be deduced in a few lines. However, conservation of momentum also holds in more complex situations where the 3rd law cannot be directly applied to inter-particle forces (see this answer).

###### Text and video resources:

Section 10-4 of Feynman’s lectures has an engaging discussion along these lines of energy and momentum conservation in collisions. Section 10-2 deduces the conservation of momentum from Newton’s 3rd law.

This short video by Dr. Peter Dourmashkin of MIT discusses the behavior of kinetic energy in collisions.

This video by Dr. Elliot Schneider discusses the relationship of Newton’s 3rd law with the conservation of momentum.

Yes, it is a law that the total angular momentum of a system is unchanged so long as the total torque exerted on the system by external forces is zero. During a collision, angular momentum may be transferred from one component of the system to another, but so long as you keep track of every component, and external agents don’t apply a torque, angular momentum will be conserved.

**Note**: Both torque and angular momentum – \( \vec{\textrm{r}} \times \vec{\textrm{F}}\) and \( \vec{\textrm{r}} \times \vec{\textrm{p}}\) – are defined relative to a choice of origin (the “tail” of the vector \(\vec{\textrm{r}}\) is anchored there). For example, imagine a rigid stick placed along the “x-axis” with its center at the origin. Apply a force at both ends – equal magnitudes and pointed up. The torque and angular momentum about the origin will remain zero (do you see why?). However the torque about an “off-center point” on the x-axis will be non-zero and the angular momentum about it will increase.

###### Text and video resources:

This video by Dr. Peter Dourmashkin of MIT defines the angular momentum of a point particle.

This video by Dr. Peter Dourmashkin of MIT discusses the relationship between angular momentum vectors defined wrt different points.

This video by Dr. Elliot Schneider discusses the relation between conservation laws (like that for angular momentum) and symmetries (aka Noether’s theorem).

This video by Dr. Peter Dourmashkin of MIT discusses the angular momentum of a rigid body that is both translating and rotating.

Section 18- 2 of Feynman’s lectures motivates the expression \( \vec{\textrm{r}}\times \vec{\textrm{F}}\) for torque. Sections 18-3 and 18-4 contain a succint discussion of angular momentum and its conservation.

Conservation of momentum is a general principle which holds for a physical system with no external force on it. In the case of a system of matter particles exerting forces on each other, conservation of momentum follows if all inter-particle forces obey the 3rd law. However, it could happen that the “stuff” – the “field” – responsible for mediating these forces, for example the electro-magnetic field, may itself carry away momentum. With the “field” treated as a component of the system, momentum conservation is recovered. However in this case the matter particles by themselves don’t conserve momentum and the forces exerted by them on each other don’t obey Newton’s third law.

In most situations encountered in mechanics class, the momentum carried away by a field can be ignored. However, this effect shows up, for example, when dealing with magnetic forces between a pair of moving charged particles.

###### Text and video resources:

This video by Dr. Elliot Schneider discusses the relationship between conservation of momentum and Newton’s 3rd law.

This stack-exchange answer has an example involving the magnetic forces exerted by a pair of moving charged particles.

I have taught physics at levels ranging from introductory classical mechanics to advanced graduate quantum mechanics, along with calculus and linear algebra.