### QS&BBQuarks, Spacetime, and the Big Bang

Lesson 5. The Big Mo, Force and Momentum

We’re all about modern particle physics and the collisions we create in order to see into the deepest depths of reality. But in order to get there we need forces to accelerate our beams to high momenta and a language to describe their energies. In addition, we learned long ago that the most intriguing feature of the particles we produce is: mass. All of these concepts have their roots in Newton’s notebooks: Force, acceleration, momentum, and mass.

But even though our subject is very contemporary, we’re stuck with the definitions and terminology from the 1600’s when our heroes emerged: Galileo, Kepler, Descartes, Huygens, Leibnitz, and Newton. While some of those fellows are important for their brilliant intuition (Galileo, Kepler, Descartes), only Isaac Newton, Gottfried Leibnitz and Christiaan Huygens wrote down mathematical relations which form the language and, yes the metaphors of our modern models of how things move. Typically, we divide mechanics into two parts:

Kinematics is the description of the motion of objects without regard to what caused them to move. If a ball is accelerating by some amount, the rules of kinematics will tell you how far it will go in a given time, how fast it’s going after a given distance: time, distance, speed, and acceleration are the parameters of kinematics. When you estimate how long it will take you to arrive at a destination, you’re doing kinematics. We tend to attribute the important ideas of kinematics to Galileo, but he didn’t have algebra and the actual mathematics of the kinematical rules came later with Newton and others. What we did in the lesson on Galileo and motion is sufficient for our purposes.

Dynamics is the study of forces—their causes and their consequences. According to Newton, forces create accelerations in objects by pushing or pulling. That’s how the ball in the previous paragraph got its acceleration—something pushed on it. This is a big subject, but we will only consider motions in one dimension, except for circular motion.

###### Goals of this lesson:
I’d like you to Understand:
How to calculate a force required to produce a particular acceleration in one dimension.
How to calculate the average change in momentum induced by the application of a force through a finite period of time.
How to calculate the centripetal force and acceleration for an object moving at a constant, circular speed.
I’d like you to Appreciate:
Forces create accelerations.
That if a momentum vector is changing in direction or magnitude, a force is the cause.
That an object moving in a curved path must have had a force applied to it perpendicular to the trajectory.
I’d like you to become Familiar With:
Newton’s third law
Newton’s Life

EDIT: 2019.08.21; 16:00

https://qstbb.pa.msu.edu/ed/lectures/L5_MechanicsMomentum19_8lt/

Notes:

• Here’s one way to insert notes to the copy editors…
• I’d set any new notes or instructions here and then below I’d add to the glossary sort of physics-y things that might seem odd in normal life, like specialized capitalization for example.
• I’ll set them off with horizontal lines and date/time-stamp them
• I’d hope at some point to start to leave repeated items off the glossary as we all get acquainted with one another but previous files will have older ones.

Glossary:

• Spacetime Diagram, not spacetime diagram
• Space Diagram, not space diagram
• Feynman Diagram, not Feynman diagram

# A Little Bit of Newton

I’m always saddened when I think of this young man’s childhood. Imagine a 3 year old abandoned by his newly-married mother and forced to live with grandparents who didn’t want him. That’s the beginning of the biggest scientific life of all: Isaac Newton’s. His childhood was a mixture of pain and some accidental fortunate associations. His father was a farmer in Woolsthorpe, not far from Nottingham and a little more than 100 miles north of London. Isaac senior died before tiny, premature Isaac was born. When he was 3 years old, his mother, Hannah—a semi-literate woman for whom the farm and manor was a big job—married the 63 year old Rector of North Witham who wanted nothing to do with a frail toddler who was then left in the care of his maternal grandparents and female cousins. Seven years later, Hannah, a widow yet again, returned with two young children in tow. While she had been away, young Isaac had been sent to school, a privilege that might not have happened had his father been alive. He’d have become a farmer. A bad one.

When he was 12 years old, he was sent to a Free Grammar School in Grantham where the emphasis was Latin (“Grammar,” after all), which was the language of intellectuals and in which he wrote his great works. He lived with the apothecary, another lucky break as that family indulged his precocious abilities with tools and crafts. In what must have been one of the most frivolous activities of not just his adolescent, but entire odd life, one night he constructed dozens of kites with firecrackers, which he flew over the town at night “…wonderfully affrighting all of the neighboring inhabitants for some time, and causing not a little discourse on market days…”

He was a loner then, and for most of his adult life. “His school fellows generally were not very affectionate toward him. He was commonly too cunning for them in everything. He who has most understanding is least regarded.” When he was 17 years old Hannah tried to turn him into a farmer, but he was a disaster. She gave up and sent him back to Grantham to prepare him for university…as the only outlet for his increasingly apparent, unusual mind.

He was sent to Trinity College at Cambridge University where he was enrolled as a “Sizar” which was essentially the role of servant to an upperclassman. He was 18 years old, older than most of the students and as a studious person, different from the mostly carefree student body. He made a single friend, a most unlikely event for the difficult Newton. One day he bumped into John Wickins while he was alone on a bench and they became roommates for 20 years until Wickins married and moved out. Wickins was Newton’s assistant as Newton began his life of changing the world.

1 / 3 Newtons's birthplace in Woolsthorpe, about 130 miles from London.
2 / 3 Notes in his hand from his day book.
3 / 3 Newton's rooms at Cambridge University, just above the doorway.
3 / 3 A page from _Principia_ showing notations.
3 / 3 The cover of his _Principia_ with some notations.
3 / 3 Portrait of Sir Isaac Newton around 1709 by the famous British portraitist, James Thornhill.
3 / 3 Newton's tomb in Westminster Abbey.

The curriculum at Cambridge was reliably old-fashioned: Aristotle, from top to bottom. But René Descartes from the Continent was all the rage and was read and secretly discussed by students. To Descartes, the world was mathematical in a manner beyond what Galileo might have imagined. It was he who blended geometry together with the brand new algebra so that equations could be presented as curves and curves, as equations. Of course we call our axes, “Cartesian” after their inventor.

Descartes was also a mechanist: meaning he believed that matter’s mechanical interactions caused all motion. Nothing spiritual, nothing occult, nothing random. God originally inserted overall motion into the universe, and apparently then abandoned his creation in order to pursue other interests? In any case, Descartes believed that original motion persisted as the universe formed and that God did not drop in and adjust things. Even the planets moved by being carried by vortices of invisible spheres. His world was predictable according to Laws (yes, capital L for Descartes’s purposes) and he proposed to start with them and draw observable conclusions—deductions. Mechanical modeling was his goal, and analytical mathematics was his tool. Very top-down, was Descartes: postulate the principle idea and draw conclusions deductively. Without that original cause, the conclusions can’t happen. Always: the causes of things had to come first.

Descartes was all about Why. Newton weaned us from Why to How.

The young, inquisitive Newton ate Descartes up. Here was an escape from Aristotle and a systematic and mathematical way out—right up Newton’s alley. But Descartes’ conclusions and his method were problematic for Newton and much of what he later wrote was in reaction to this unrequited disagreement.

But the Frenchman’s mathematics stuck and the neo-Platonists at Cambridge carried Descartes’ mathematics program forward. The leader of that movement was Isaac Barrow, the first “Lucasian Professor of Mathematics.” Isaac Newton was the second (Paul Dirac, whom we’ll meet later was 15th and Stephen Hawking was the 17th). He became a student of Barrow’s and eventually his beneficiary.

Disaster struck in London when in 1664 the Bubonic Plague flourished. Before it was over, 20% of the population was dead. By 1665 London emptied out (one imagines frightened servant staff remaining behind to protect their masters’ homes while the families fled). Cambridge University closed and the 23 year old Newton went home to Woolsthorpe where he remained to himself for more than a year. During that time he consumed all mathematics known at the time and went beyond. While at the farm, he essentially invented calculus, had his first ideas about gravity (the famous apple was to have fallen in his presence during this period), and reinvented the science of optics.

When he returned to Cambridge his mathematical skills exploded, surpassing those of all around him. In 1667 at the age of 25 he was named a Fellow (like an assistant professor) with a salary. In 1669, Barrow resigned the Lucasian Chair insisting on Newton’s appointment to this highest post in the college. (“Mr Newton, a fellow of our College, and very young … but of an extraordinary genius and proficiency in these things.”) One of the only duties of the Lucasian Chair was to offer a single course a year, which he dutifully fulfilled, but often lecturing to an empty classroom since nobody could understand him.

Among his discoveries during the plague years was that sunlight is composed of all colors, which was in conflict with the standard (Cartesian!) idea that white light was a color of its own and that the colors we see are mixtures of white and dark. He passed light through a prism and found that it spread out into a continuous spectrum of colors. This led him to experiment with light passing through glass and to grinding his own lenses for telescopes. Telescopes had become long and unwieldy and because glass lenses spread out the colors (“chromatic aberration”) precise images were difficult to achieve. Newton changed the design completely using mirrors rather than lenses. The result was images of higher quality with higher magnification in a compact instrument. His original six inch “Newtonian reflecting telescope” would magnify 40 times, which would have required a six foot long conventional telescope resulting in a poorer image to boot. All telescopes today are off-shoots of the Newtonian design. A replica of Newton’s original telescopes. Insert the Getty image: https://www.gettyimages.com/detail/news-photo/this-is-a-replica-of-the-first-reflecting-telescope-made-by-news-photo/90732747#this-is-a-replica-of-the-first-reflecting-telescope-made-by-sir-isaac-picture-id90732747

The reflecting telescope and his explanation constituted his first entry toward membership in the Royal Academy of Sciences. His report on how it worked led to the explication of his theory of light, which the Curator of Experiments, Robert Hooke, thought was stolen from his (incorrect) ideas. They became bitter enemies for life, one of a number of such vicious rivalries that Newton suffered into—and beyond—old age.

So galling was this dispute and the criticism that his theory of colors attracted from all over Europe, that Newton went into nearly complete isolation, vowing to keep the products of his research secret, rather than ever again suffer such public antagonism. He communicated almost exclusively through voluminous correspondence, much of which still exists. Newton didn’t suffer fools well, and even a legitimate dispute would send him into a towering rage, or stony silence. His response to Hooke was to write a book on optics, which he inadvertently destroyed when a fire from an alchemy experiment went out of control. (He rewrote it many years later.)

What we recognize as Newton’s enduring scientific work came as the result of a wager and we’ll pick up that story when we study his law of Gravitation and the first-ever attempt at a scientific cosmology in our lesson on Gravitation. Everything he discovered about mechanics and gravitation is all contained in one enormous, ponderously written, thickly Latin, often revised book.

## The Book

Newton was in his 40’s when he basically sequestered himself in his rooms working on his alchemy and intensely strange religious researches. In 1684 three of his colleagues (the famous London architect Christopher Wren, the scientist Edmund Halley—of eventual comet-fame)—, and his arch nemesis Robert Hooke) were trying hard to figure out the shape of a planetary orbit if the force of gravity varied like the inverse square of the distance from the Sun. Hooke claimed in his obnoxious way that he knew the answer, but he would not produce a calculation—because he couldn’t: he had no mathematical training. He prospered entirely on his remarkable instincts as an experimenter.

But, a little of Robert Hooke went a long way and Wren and Halley got tired of listening to him so they deputized Halley to go ask Newton. He showed up unannounced at Newton’s messy room and asked him. Immediately came the recluse’s famous response: “An ellipse.” “Why?” asked Halley. “Because I have calculated it.” But, typical of the paranoid Newton, he’d not told anyone.

He’d worked out the mathematical rules for the motion of the planets and kept it a secret! While Halley waited, Newton searched but could not find his calculation and Halley left empty-handed. A few days later, Halley received nine pages from Newton that showed: if the force on a planet varies like the inverse square of the distance from the center of the orbit at the Sun, then the orbit’s shape must be a conic (a parabola, ellipse, circle, or hyperbola). And he showed that if the orbit is an ellipse, that the force of attraction must be an inverse-square. This pamphlet became known as, De Motu Corporum in Gyrum (On the Motion of Revolving Bodies). De Motu, as it’s known, was a summary of the first book of his eventually triumphant work. The figure at the side shows a page of De Motu in Newton’s hand that he later prepared for his correspondent-friend, John Locke. This electrified Cambridge and London and set Hooke’s teeth on edge as he’d guessed some of the same conclusions and again insisted that Newton had stolen his ideas.

Halley realized what Newton had done and implored him repeatedly to write it all. Newton finally agreed and went into one of the historically most intense periods of concentration ever embarked on by anyone. For two years he worked night and day, forgetting to eat, wandering around Cambridge without regard to his surroundings. Thousands of pages of manuscript littering his quarters mingled with days’ worth of uneaten food. Two years! Eventually he emerged with the first book of what was to be three volumes of Philosophicæ Naturalis Principia Mathematica, or the Mathematical Principles of Natural Philosophy affectionately known ever after as The “Principia.” It was all there in Latin. His laws of motion and gravitation, but also of fluids and the strengths of materials and his own description of the scientific method. He’d pestered scientists and astronomers from around Britain for data on the planets and the tides. A portion of *De Motu in Newton’s hand in 1684 written for* John Locke who was an early reader of Principia.

He’d made measurements of motion in his own lab. He let his alchemy furnace go out forever as he worked solely on his “system of the world.” The arguments were mathematical and constituted the first workable model of motion, kinematics and dynamics, and also gravity (later). He continued to hide his calculus, preferring to speak in terms of limits and extrapolations using geometrical constructions, surely backed up by his own private calculus based calculations. Principia went through three editions after the original 1686 start, often with him revising his last chapter, which was more philosophical, but also with successive furious deletions of the names of rivals.

Halley had persuaded the Royal Society to act as the publisher of Principia and Newton dedicated it so. But the coffers of the Society were dry when it came time to print as they had used up their entire accounts in a lavishly illustrated two volume, History of Fishes.

So Halley took a deep breath and paid for the initial publication himself. This of course led to his active interest in encouraging Newton to push the book off at booksellers and libraries himself. Never was there a more generous gift to science than Halley's unselfish gesture. And for a book that only a few people in the world could read, but a book that quite possibly initiated the Enlightenment and people's relationship to our universe.

# Impulse: Getting Going

In a conventional physics course, Newton’s Three laws of mechanics would be a major focus of study. But in QS&BB we don’t need that depth but we do need two important concepts…the ones that underpinned his system: the concepts of mass and momentum. Let’s move:

In the previous lesson we laid out the rules that govern an object’s motion, including whether it moves at a constant speed or accelerates. What we conveniently avoided was what causes motion—on that score Galileo had nothing to say. (Remember that Descartes insisted on the cause of things before beginning to work on the outcomes? He explicitly denigrated Galileo’s work because Galileo explicitly avoided speculating on why an object would accelerate—its cause.) Of course Aristotle had something to say about everything and as we saw in the last lesson he insisted that unnatural motion is not for free, that one always needs to apply a contact force to start something to move and then to keep it going. Natural motion is for-free for him. No explanation of how, just that weird desire that heavy, falling objects have for the earth. There’s so much that’s wrong with both of these ideas!

One of the many ways that Isaac Newton got into the textbooks was to argue with Aristotle: “No.” Constant motion is free. It's only accelerated motion that requires payment in the form of a force. Further, while Aristotle simply declared what his rules were, Newton built the first-ever mathematical model describing them. Remarkably, his model has functioned for four centuries and still forms the basis of mechanical and civil engineering—even NASA projects.

To start something moving from rest? Apply a force. To speed up or slow down something already moving? Apply a force! To cause something to deviate from a straight line? Yes. Another force. To keep something moving at a constant speed? No (net) force required, thank you.

Here’s what he said:

Whenever there’s a change of velocity, a force is at work—forces are responsible for acceleration.

These are Newton’s conclusion, but let’s start slowly and sneak up on this idea. Impulsively.

## Impulse

Newton: To get something up to speed, you need to whack it or shove it—either a sharp collision or a steady push increases the speed of an object. Push harder? More speed. Push longer? Again, more speed. And as you know from any sport involving a collision, something that’s moving fast can in turn exert a bigger force than something that’s moving slowly.

So let’s codify that everyday notion into a model of forces and motion. Let’s imagine a force, F that pushes during some time interval, $\Delta t$. A whack means that $\Delta t$ is small (like a golf club hitting a ball) while a steady shove (like a rugby scrum) means that the force is slowly applied so $\Delta t$ is larger.

Here’s what we know from experience: applying a force ($F$) to something for a time interval ($\Delta t$) results in the speed increasing in proportion, ($\Delta v$). We have the beginnings of a model:

Increase $F$, $\Delta t$, or both on the left-hand side, and the speed goes up on the right-hand side. The quantity on the left side is called the Impulse. It’s the sports-quantity. Any game involving a ball involves impulse and exceptional athletes can control both the $F$ and the $\Delta t$. The quantity on the right implies that the speed changed and of course if the speed changed, then the object accelerated.

We need to refine the model. But first some units and language:

May the force be with you by so many different names

We will use forces many times in QS&BB which is a common quantity in our lives because of “weight.” But if you’re from the United States, when you step on the bathroom scale it reports back to you your weight in the Imperial measurement system (or “customary measures system”) as too many Pounds, lbs. In a minute, you’ll see why that’s confusing when comparing to the rest of the world where the bathroom scale would report kilograms.

In any case, the unit of force in the International System of Units (SI), which includes the older “metric system” or MKS (Meter, Kilogram, Second) is the Newton, N. If you go to the gas station in Berlin, you’ll fill your tires to a pressure measured in Pascals, which is Newtons per square meter. Of course up the street from me, our Michigan gas station reports pounds per square inch for my pressure.

Let’s get a feel right now: use the conversion-engine here to convert 100 pounds of force to Newtons.

 Conversion. 2019.09.10; 17:48; convert pounds <—> newtons 

Pressing forward (see what I did there?). Let me ask you: suppose I apply a force of $F=100$ pounds for 60 seconds to a Volkswagen and you apply a force of $F=100$ pounds for 60 seconds to a little red wagon. We both begin our efforts at point A: The same force applied to two red objects, of very different sizes.

Will the resulting $\Delta v$ at point B be the same for both vehicles? Of course not. The little red wagon will gain more speed than the Volkswagen (regardless of its color). So Eq.\eqref{impulse1} is not the whole story. What’s missing is the reluctance that any object has to being accelerated, which has a name: inertia.

Inertia is the resistance that an object has to being accelerated.

# Newton’s Mass

Mass is a toughy and we’ll see over and over how complicated it is—to the current day. Here’s how he defined it in the Principia:

“Mass is the quantity of matter arising from its density and bulk conjointly.”

There you go. Useful? …no?

Wait. Doesn’t that seem circular?

Glad you asked. You might think so, yes. You know that density is an object’s mass divided by its volume and so to define mass…using mass…does seem so. I think it’s probably hard to invent your own science out of almost nothing! So word-explanations can seem a little strange. (There’s also some reason to think that “density” for Newton referred to specific gravity, but let’s not follow that.) You’re correct to be a little unsatisfied with this definition! But we can forgive him since “mass” becomes almost a theme in QS&BB. It’s really complicated!

In any case, you do perfectly well—at least in solving engineering homework or building bridges—to accept the idea that mass is the amount of the “stuff” in an object and that it's also the quantitative measure of an object's reluctance to be coaxed into changing its motion.

Here begins our love-hate relationship with mass. This much works:

Inertia is an object's resistance to being accelerated and the measure of an object’s inertia is its mass.

This is the most profound and at the same time, the most mundane idea that Newton had! It’s with us today as a primary focus of particle physics research:

What is the nature of Mass?

At the deep level of elementary particles, mass confuses us, perhaps in a different way from how it confuses engineering freshmen. We think that mass may actually not be an actual property of object, but rather a result of an object’s interaction with a spooky field that sprang into existence just after the birth of the Universe. Now, in the 21st century, we’ve got a whole new set of neuroses about this subject, as understanding it occupies almost the entire Particle Physics community. So, Mass has been a problematic subject since its beginning in Newton’s hands.

And then, there are units. The MKS unit for mass is the kilogram, kg and we’ll use it almost universally. So remembering the bathroom scale discussion above, you can now see that US bathroom scales report a force while for everyone else in the world, their scales report people’s masses, kg. Their scales will go to Mars without difficulty. We’ll come back to this.

With this “feeling” for pushing on big and little things, we can state the impulse proportionality as a full-fledged model—an equation.

# The “Quantity of Motion”: Momentum

We just developed a sense that our hand-built, car-pushing formula, Eq.\eqref{impulse1} has to depend on speed and mass and so we’ll just add it in on the right-hand side to get:

🖋

Being more explicit, when fleshed out impulse is

So since the mass of the Volkswagen is much bigger than the little red wagon, the same force applied through the same time results in a smaller speed change for the former, rather than the latter. Think of it this way: the numerical product of $F\Delta t$ (which was the same for your push as it was for my push) is shared by $m$ and $\Delta v$ so more m leaves smaller $\Delta v$ and of course, a smaller m means more is left for $\Delta v$. This collecting $m$ and $v$ together proves to be useful.

Let’s calculate the impulse:

showing that the common amount of impulse is distributed between the mass and the change of speed of the two vehicles.

Let’s empty my garage and apply the same force for the same time to Janet’s red VW beetle and my BMW. For reasons that will become apparent later, we don’t want to just walk away from pound units, we want to run! So let me make the conversion for us from insanity to sanity. First the weight-mass conversion (from US to MKS):

And, we can convert the impulse between the two unit systems:

Another units-alert: Notice that the units of impulse of N-s correspond nicely to the left hand side of the defining equation, a force times a time. But the right hand side of the equation shows a different way to represent the same thing, but now it looks like a mass times a velocity, or kg-m/s. So the impulse is also, 27,000 kg-m/s.

Those are two equivalent ways to report the measurement of an impulse. Why’s that useful for us? Areas and multiplication and our model:

Isn’t this just our area-approach to formulas of the “form” $A=a \times b​$ (where $F\Delta t$ is $A$ and $m$ and $v$ are $a$ and $b$) at work here? Our wagon/VW example is perfect for this. On the left side of Eq. \eqref{carimpulse} is the product of our forces and the times…they were the same for both of us, so for this situation:

Those last two terms can be an area in a “mass-speed space.” Maybe you can already begin to form the image in your mind of this as areas. No? Let me help you: The areas are both the same with the Red area corresponding to my wife’s red VW and the Black area corresponding to my black BMW (cars are supposed to be black). The key shows how many kg-m/s are in each square.

We’re forming the “area” of mass times speed, twice: once for the BMW and once for the VW. Here, each mass is fixed. Likewise the area is fixed and the same for each (the force times the time interval, $F\Delta t$). And we can then construct the two rectangles and you can see that these areas are equal (roughly count the squares in each and you can convince yourself). Now we’ve learned the answer: we see that the VW will reach a speed of about 20 m/s after 10 seconds while the BMW will lag behind at around 14 m/s.

An Example: Fore!

The Question: You all know about the game of golf—you take a metal weapon; strike a small, white ball that’s just sitting there, minding its own business; and cause it to leap into the air. A regulation golf ball has a mass of 0.05 kg and a pro golfer can regularly create an exit velocity of about 70 m/s (which is a little more than 150 mph). If the club-head is in contact with the ball for a half a millisecond, $0.5 \times 10^{-3}$ s, then what’s the force that such a golfer applies to the ball?

This is a question about impulse and so we have everything we need to use the impulse model and calculate $F$.

$F\Delta t = mv - mv_0 = \Delta(mv) = mv$ The initial $mv_0$ is zero, since in golf the ball is sitting still…or you’re breaking the rules. So the difference in $\Delta mv$ is simply the final $mv$. Rearranging,

Putting in the numbers that we know:

That’s about 1600 pounds, or just under a ton of force.

 Conversion. 2019.09.10; 18:27; convert pounds <—> newtons 

🖋 The area is the amount of the impulse with each box having the indicated impulse amount in N-s. The key shows how many kg-m/s there are in each square in this impulse area picture.

 Interactive: 5.1; 2019.09.06; 15:19 

## Momentum

Newton’s second good idea was the concept of “momentum” which he called the “quantity of motion”—a nice description, I think. The idea that a moving object possessed something—some quality—was pretty hard to ignore. But, nobody could figure out how to describe it for 2000 years before him. Aristotle just denied it: “No,” a moving object doesn’t possess any quality. Galileo vaguely said “yes,” there is something “in” a moving body that he called impetio. Kepler seemed to say “yes.” Descartes definitely said “yes.” Newton agreed with his 17th century predecessors but made the idea useful.

🖋

What he concluded was that the “quantity of motion” is momentum. Keeping with tradition by using the symbol “p” as its nickname, momentum is:

With this now we can continue the manipulation of Eq.\eqref{fullimpulse} and restate it one more time in terms of momentum:

We’re going to find that momentum is the most important quantity in our particle physics story.

We’ll use it over and over in different guises.

the sports quantity

Equation \eqref{impulse4} works in three ways: either you know the force and you use the formula to calculate the change in momentum in a given time. Or, you know the change in momentum and you use it to calculate the total force in a given time, or you adjust the time for a given force.

Now you’ve gotten the formula that governs all sports involving whacking one thing with another, like baseball, golf, tennis, soccer, or football. Think about what you almost always want to do: you want to make the ball go faster after you hit it. That means, you want the change in the momentum to be the highest possible. So, Eq. \eqref{impulse4} tells you how: you hit the ball as hard as you can (that’s a large $F$) and you get “good contact” (which means you hit the part of the bat or racquet or club where you can touch the ball as long as possible…which is the largest $\Delta t$).

This also explains how airbags and bumper-crumple zones in automobiles work. There, you know what the change in momentum is:

That’s the change where the final velocity is zero (the car stops). The initial velocity is fixed and so $v_{before}$ has to be divided up between the force and the time in $F\Delta t$ where the force is applied, say to your bumper. High force is not good for the occupants inside the car, so this leads to the design goal of spreading out the time—large $\Delta t$ so that the force will be smaller. This is the same reason that you bend your legs when you jump off a table and hit the floor!

## Momentum Is a Vector

Because velocity is a vector, momentum is also. In the next lesson when we consider collisions, which is where momentum shines in QS&BB, the direction-part of the momentum vector will play a crucial role. For that matter, since momentum and force are vectors the actual general statement about Impulse is:

Here’s a buzz-word: “state.” If you give me the position and the momentum of any object—I know everything I need to know to predict where it will be at any subsequent time. In Newton’s world, which is good enough for us for a while:

An object’s position and its momentum are called its “state.”

# Newton’s Famous Three laws

Newton’s momentum and mass are at the heart of his three laws of motion. Let’s go through them in words, and then one of them in more detail algebraically.

Newton’s 1st law of motion says that anything that’s moving at a constant speed (which could be zero) will continue in that way unless a force acts on it. That’s a statement about inertia—resistance to acceleration. (Newton inherited this idea from Galileo, Descartes, Isaac Beeckman, and Marin Mersenne, but gave it a quantitative meaning.)

Newton’s 2nd law of motion says that momentum is changed when a force acts on an object for a duration of time. Or, you might have learned it as a defining statement about “force” namely:

Force is equal to the rate of change of momentum.

Newton’s 3rd law of motion is subtle, and solely due to Newton’s ingenuity. It says that if you push on something—anything and with any amount of force—that object—will push back with exactly the same force. We’ll think harder about the 3rd law when we talk about collisions.

# Newton’s Second law

Students in a physics class become intimately familiar with the 2nd law, which is all about momentum and how to change it. A simple arrangement of the \eqref{impulsevector} relation yields the real mathematical definition of Newton’s 2nd law:

Maybe you’ve maybe seen this equation but written in a different way. Inserting back the definition of momentum, $p=mv$ (in one dimension, so we’ll drop the vector notation):

🖋

Let’s assume that the mass of the object doesn’t change, so we can push the $\Delta$ operation through the mass and let it act directly on the velocity and get

and what’s $\Delta v / \Delta t$? That’s nothing more than acceleration, so we finally arrive at the T-shirt version of the famous Newton’s Second law of motion:

Notice that if you isolate the acceleration by dividing both sides by $m$,

you get the force divided by the mass. There’s the “inertia” nature of mass making its appearance: the larger the mass (so that $\frac{1}{m}$ is a small number) the harder it is to accelerate for a given force (the Volkswagen)—$a$ is small. On the other hand, if that force is now applied to a light object (so that $\frac{1}{m}$ is a larger number) then the acceleration will be greater (that’s the wagon).

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Stare at Eq.\eqref{second1}: if you think about it, there are three ways for $\vec{p}$ to change:

1. As we’ve just seen, if the speed changes, the velocity vector changes, so then the momentum vector changes. That’s easy and what you’d expect.

2. If the mass changes, then the momentum changes.

3. If the velocity vector changes, but the speed stays the same? Ah. That’s interesting and we’ll look at that carefully in section Circular Motion below.

#1 makes sense in our everyday lives. If you change the speed, the velocity will change. #2 is less obvious but think about pinching the opening of a balloon that you’ve blown up. The balloon’s mass consists of the stretchy material of the balloon plus the mass of the air inside. Held out in front of you there is no net force and it’s at rest in your hand. Puncture it with a pin and the air rushes out fast and so the mass of the air inside the balloon rapidly decreases. The consequence of that decrease in mass, a $\Delta m$ in a time $\Delta t$ all by itself results in a change in the momentum of the balloon! That’s how rockets work. For #3 we’ll wait a bit.

## Weight

I’ve been toying with you over units so let’s clean this up. Since we’re Earth-bound we tend to mix up the units of weight and mass at the grocery store. You’ll appreciate that in a bit.

A gram is a unit of mass, (1000 grams in a kilogram) while an ounce is a unit of force (16 ounces in a pound). A kilogram still a measure of mass and a pound is a measure of weight. But we get away with using both systems since we tend to buy things and compare them on Earth. If we had a Mars colony, well then there would be trouble. If you put that 5 Earth-pound bag of Gold Bond flour at Kroger on your bathroom scale which you transported to Mars, it would read back about 2 pounds. But in each location, the bag of flour would still have a mass of 2.27 kg. How can that be?

Weight is a special kind of force—still subject to being the “$F$” in Newton’s 2nd law, but called out as a unique characteristic of massive objects on the Earth.

What Galileo showed was that the acceleration due to gravity near the surface of the Earth is a particular value—he presumed it was a constant. If the ground disappeared beneath your feet, then because you have a mass, you’d start to fall towards the Earth’s center with that acceleration of g. But happily the ground pushes back and you are stable on the surface. (It’s a little confusing since you’re not moving but you’re still accelerated) So from Newton’s Second law, when we have a mass and we have an acceleration, we can calculate a force and we define that particular force of attraction by the Earth as the the object’s weight.

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Let’s call it w and we can write it out:

Notice that this is just Newton’s 2nd law, but with a particular acceleration, $g$ which leads to a particular force, $w$.

Weight is the force that a planet exerts on an object on its surface.

We can measure an object’s weight by making use of the fact that the Earth pushes back with a force that’s the same value as the weight. When you “weigh” something, probably a spring is doing the pushing-back. There’s one in your bathroom scale which is calibrated in the U.S. to read that push-back in pounds. In France, that reading would be in kilograms (but it’s still a spring, so really measuring weight and converting to mass)! Basically everyone (else) in the world deals in mass terminology. We’ll see how the Earth does this in a bit when we get to Newton’s other law, that of Gravitation.

By the way, the acceleration due to Mars gravity, its “little $g_M,$” is about 4 m/s$^2$, to compare with our “little $g_M,$ of about 10 m/s$^2$. That explains why that bag of flour appears to be light using your Earth-calibrated scale.

I have bad news: unfortunately in the English system, the unit of mass is “slugs.” Now I have good news. We’ll not use slugs for anything in QS&BB.

Wait. Seriously. Slugs.

Wish you’d not asked. I confess I don’t know the origin of the “slug.” One of life’s mysteries.

So we can collect our units appropriate to Newton’s Second law in this table below. In the last column, the standard abbreviations are shown as well.

English - MKS
acceleration ft/s$^2$ m/s$^2$ 1 ft/s$^2$ = 0.305 m/s$^2$
$g =$ 32 ft/s$^2$ $g = 9.8$ m/s$^2$
mass slugs kilograms 1 slug = 14.59 kg
on earth mass of 1 slug = weight of 32.2 lbs mass of 1 kg = weight of 9.8 N
force pounds (lb) Newtons (N) 1 lb = 4.45 N

An Example: weight

The Question: If I weigh 200 pounds on Earth, what is my mass in slugs (okay. I lied. one slugs reference.) and in kilograms?

In order to convert to kilograms, we can do conversions within the correct quantities using the table above. So let’s do $w_{\text{me }}$(English)$\to$ $w_{\text{me }}$ (metric):

Now calculate the mass in kilograms like before:

We can check with our conversion engine and see that we got the right answer:

 Conversion. 2019.09.11; 11:54; convert kg <—> N 

 Conversion. 2019.09.11; 11:55; convert lb <—> N 

Let’s go for a ride:

An Example: Biking at a constant acceleration

The Question: Let’s imagine a bicyclist maintaining a constant acceleration of 2 m/s$^2$ which results in the subsequent increase in speed proportional to time and consequent quadratic increase in distance covered. (That acceleration is doable, but not normal.) Let’s not dwell on it, but we now all know that my mass is 90.7 kg. How much force do I have to apply to the ground through the pedals and the tires in order to keep up that constant acceleration? What fraction of my weight is this force?

This is a simple application of the popular form of Newton’s Second law, Eq.\eqref{second2}

To compare to my weight, we again can use the same formula with an important difference (we’ll call my weight w) and I’ll approximate the acceleration due to gravity, which is $g = 9.8$ m/s$^2$, as $g \approx 10$ m/s$^2$.

So the force that my legs would have to continuously apply to the pedals, and in turn to the ground through the friction between the tires and the road is about 20% ($\approx$ 180/900) of my weight.

How much sustained force is this? Well, suppose we have a stationary bike hooked up to a pulley and a bag with five bowling balls. The force required to keep that bag ‘o balls aloft—forever—is the amount of force that I’d have to sustain—forever—to maintain that acceleration on the road. Now is that sensible? After 10 seconds of this acceleration I’d be traveling at 20 m/s, ( 2 m/s$^2 \times 10$ seconds) which is about 45 mph. So obviously, that’s too fast to imagine pedaling a bicycle for 10 seconds. Rather, if it were possible for me to exert 181 N, after about a couple of seconds, I’d be moving around 10 mph and surely at that point I’d stop trying to accelerate and apply just enough force to maintain that speed.

 Conversion. 2019.09.07; 13:21; convert lbs <—> kg 

 Interactive: 5.2; 2019.09.06; 18:06 curb weight of VW 3000 pounds 

I know you’re asking, “Where is an apple question?”

 Interactive: 5.2.1; 2019.09.07; 13:22 apple weight 

 Conversion. 2019.09.07; 13:26; convert lbs <—> N 

An Example: more apples

The Question: In the apple example in the last lesson you calculated that the speed that an apple would attain if it was dropped 1 meter would be 4.4 m/s. You did that, right? Let’s be more realistic. Look at the figure for our new situation. The apple at A is dropped onto the carpet and bruises as it flattens out at B, slowing it down to a stop. The carpet applies a force to it which would be pointing up. (You can see the damage in the inset.) If it takes 0.090 s (90 milliseconds) for the apple to stop on the carpet, what is the average force during that time ($\Delta t$) that the carpet applies to it in order to bring it to rest? The mass of an apple here is 0.1 kg. Look at the squishing of the apple as it quickly slows to a stop.

The change in velocity is of course the velocity that the apple has at the end—just before it hits—minus the velocity it had at the beginning—which is zero, since it was just dropped. so $\Delta v = v-v_0=v$. Putting the numbers:

which is about half of the force of gravity. The apple would probably not bruise very much. Go try it.

 Interactive: 5.3; 2019.09.06; 18:07 apple on hard surface, MC 0.009 s 

 Conversion. 2019.09.07; 13:57; convert lbs <—> N 

 Interactive: 5.4; 2019.09.06; 18:08 Earl bike pedaling 

# Finally, Nothing: Forces in Balance

We’ve been dealing with Newton’s 2nd law as if there’s only one object and one force but what it really says is more general (but where the mass remains constant):

Here are three example situations: The left figure imagines a single force with no friction, like pulling a sled on smooth ice. The middle figure assumes that Mo is pulling the sled across grass, so there’s friction. The right figure imagines that Ossie is sitting on the sled which makes the friction more and adds to the overall mass. Notice the relative lengths of the two friction forces, with and without the added Ossie-load.

The simplest example: Mo is pulling his vintage wood & cast-iron Mickey Mouse Flexible Flyer sled across a frozen pond. He’s a strong guy and he can apply a pretty steady force, which we’ll call $F_M$. In that case, while he pulls, he’s working out the acceleration in his head:

He pulls with $F_M$, the sled has mass $m_S$, and the sled accelerates at $a_S$. Mo, while book-brainy is not too everyday-smart.

While he enjoys sledding in the winter, it’s too cold during that time so he prefers to do his sledding during the summer. In order to get to his favorite hill, he has to drag his sled across the grass. Carefully calibrating his force again, he pulls at the same value of $F_M$ but doesn’t achieve the same enthusiastic acceleration as on the ice because of the friction which is responsible for a force that points in the opposite direction, $F_{F}$. Now his acceleration is

So clearly the sled’s acceleration is less than in the cold months. The value of the frictional force, $F_F$, depends on how hard the sled is pressed against the grass and scales just about evenly with the mass of the sled.

Mo’s friend Ossie prefers to not walk and so he rides on the sled and so the mass that Mo needs to work against is that of the sled plus Ossie, $m_S+m_O$. Ossie presses the sled into the grass more so its frictional force, $F_{FO}$, is also more and it just balances Mo’s steady, determined force, $F_M$, so $F_M=F_{FO}$, or:

So when the forces balance, there's no acceleration. When there’s no acceleration, the speed doesn’t change. So can Mo still get to his warm hill? Why yes, if he got the sled up to some speed (say Ossie doesn’t get on until it’s moving) then it will continue at that speed, which is another way to say that it does not accelerate. Which is another way to say Newton’s 1st law is at work.

You already have an intuitive feel for this force-balancing act. Because, tug of war:

An Example: Tug of war

The Question: Let’s suppose that Myrtle and Bernie are pulling on a rope toward the west. Myrtle pulls with a force of 50 pounds while Bernie pulls with a force of 70 pounds. This game of tug of war is only among three (former) friends and Chester is on the other side pulling directly to the east. What minimum force must Chester exert in order to win the game?

You know the answer: the eastern-pulling team exerts a total of $50 + 70 = 120$ pounds of force and so Chester must pull with any amount over $120$ pounds in the other direction. In symbols, our Newton 2nd law statement says that for balance:

So $F_C=121$ pounds wins. Or $F_C=120.5$ pounds. Or $F_C=120.0005$ pounds. Notice that the negative signs signify that I’ve designated east as the positive direction, so vectors are implied but not explicitly designated.

 Interactive: 5.5; 2019.09.06; 18:09 tug of war 

# Circular Motion

In Eq.\eqref{second1} the relationship of force to the vector momentum is subtle, but you use it and experience it every day. Suppose Hazel and Ira are a passengers in a car going around a curve. When they enter the curve, they’re moving north. When they emerge from the curve, they’re pointing west. Hazel is driving and watching the speedometer carefully in order to make sure that they remain at the same speed through the whole path. So did their speed change? No! Did their velocity change? Yes! Because, the direction of their speed changed. This is shown in the figure: A car going around a curve at constant speed: at A, its speedometer reads 60 mph, at C, 60 mph, and as it’s going through the curve, also 60 mph. During the period when it’s going around the curve (when the steering wheel is turned), for a brief time it traces out a segment of the dashed circle with its center shown at x.

Let’s think in terms of momentum and this diagram shows the momentum vectors at A, B, and C: An abstraction of our above car-curve-cartoon showing the momentum vectors: all of equal length but pointing in the direction of the velocities, which is the direction of the car.

Let’s recite the series of events that follow from going around this curve: They both remain in the car, so somehow they are both “forced” to deviate from straight-line motion along with their vehicle. A variety of phenomena cause that to happen: Hazel’s smart and is wearing her seatbelt. That keeps her attached to the car as it turns. Nobody can tell Ira what to do, so he never wears a seatbelt and slides across the seat and is pressed up against his door and that keeps him moving in a circle. Each of these phenomena constitute forces that act in a direction pointing to the inside of the circle that their car is moving—the point labeled x in the figure. Let’s state some obvious, but subtle things about this simple act:

• Their speed didn’t change.

• But, their direction changed, so their velocity changed.

• If their velocity changed then their momentum changed (from the definition of momentum, $\vec{p}=m\vec{v}$.

• If their momentum changed, there had to be an inward force applied to both of them. (from the 2nd law).

These various forces all caused them to go in the same circle as the car (Second law) while they are trying desperately to continue to go straight (First law!). (In fact, Ira had an apple in his hand and when the car turned left, he let go and the apple kept going straight north, out of the open window.)

The forces are:

• The seatbelt pulled on Hazel;
• Ira experienced a frictional force between his jeans and the seat but it but was not enough to keep him from sliding;
• so he slid until he pressed against the door and then the door applied a force;
• and the tires of the car also have a frictional force to the pavement and that points “in” as well…in fact, that’s the only way that a car ever turns a corner.

Each of these forces points “in” toward x, the center of the circle and they all do the same job: pull the objects towards that center. These kind of forcing-in-a-circle forces have a special name:

A force that causes motion to deviate from a straight line is called “centripetal force.”

Another pretty good idea of Newton’s (and Christiaan Huygens’).

## Newton’s Reasoning

The essence of circular motion can be visualized in this cartoon:

where a figure is twirling a ball attached to a rope in a circle. Let’s ignore gravity for a second and concentrate on the motion in the plane of the rope and ball…and his fist. You know by now what would happen if he let go of the rope. Without the rope pulling in towards the center, there is no horizontal force and according to Newton’s First law, the ball would go straight. (That’s a sling-shot.) So the rope is causing the ball to deviate from a straight line. Can you agree that the rope is doing the same thing that the seatbelt and tire friction do?

Here’s how Newton explained this. The next figure shows the view of the ball from over head. In order to go in a circle the ball needs to move from point A to point B, and on to C, D, and E and then all over again. A ball going in a circle from overhead at successive times, labeled (a), (b), (c), and (d). The motion is uniform, so the speed around the circle is unchanging which is reflected in the fact that the momentum vector lengths are all the same.

Since the speed is constant, the momentum vector has a constant length, but because the motion’s direction is around a circle (“not straight”!), the momentum vector is tangent to the circle at all points around the path. Newton’s brilliance was to explain this using his three laws and some geometry. Let’s deconstruct this a bit:

The ball starts at A in (a) and needs to go to B. Newton reasoned that the ball would “like” to go straight, on to point B’ (like Ira’s apple) but that the rope tugs it back to point B in (c). So the ball goes a little, gets tugged back, goes a little further, gets tugged back, and so on. These little tugs were in his mind acting all around the circle, which in the limit of being infinitesimally spaced create a continuous, circular trajectory. This notion of “infinitesimal” was kin to the habit of mind he was developing in the invention of calculus.

But let’s carry this further. Since the momentum at, say A, is different from the momentum at, say C (because the direction is different), even though the magnitudes are the same, there is still a changing momentum, a non-zero $\Delta \vec{p}$. If there’s a change in momentum of any kind, there’s a force:

Let’s see what he found.

This figure shows two strategically placed points on the circle “here” and “there” and the corresponding momenta of the ball associated with each point, $\vec{p}{\text{here}}$ and $\vec{p}{\text{there}}$. The numerator of the force relationship is the difference:

which can be constructed here from our rules about subtracting vectors described in Lesson 3, Mathematics. The here and there vectors moved around in order to create the difference. (a) shows the two vectors translated to be together. Remember that subtraction of vectors is the same as the first vector plus the negative of the second, which is shown in (b). The resulting sum is in (c).

That difference is labeled $\Delta\vec{p}$ points very close to the center of the circle! This was his brilliance. If the “here” and “there” points were closer and closer to one another, then the difference would point closer and closer to the center.

So like we knew all along, the rope is what causes the ball to go in a circle, the combination of Newton’s First law with his Second law, and the crucial recognition that momentum is a vector, leads to the demonstration of the force towards the center is responsible for the change of momentum.

This force is that special “centripetal force” that Hazel and Ira encountered going around the curve in their car.

All non-straight motions are caused by a centripetal force.

If the trajectory is circular, it’s easy to see that the centripetal force points to the center of the imaginary circle. If the trajectory is uniformly curvy, at each point there can be many instantaneous “circles” and the forces would point towards their centers.

## Centripetal Force and Centripetal Acceleration

If this understanding of circular motion weren’t enough, Newton went a step further in his paranoid sort of way. He actually found a relationship for what the centripetal force would be and did it both using his new calculus and in a strictly geometrical fashion. The latter he published in Principia, and like other such derivations, kept the calculus version to himself. Why? He feared being scooped. Calculus was his (“my precious”) private tool for a long time.

I’ll just enunciate the result without his tedious geometrical explanation or the more complicated calculus explanation.

The essence of this is on the left and right figures here where we see a representation of the momentum vector and the centripetal force, $\vec{F}_C,$ on the left, and following Hazel and Ira’s experiences, pointing to the center. But where there’s smoke, there’s fire…or rather, where there’s a force, there’s an acceleration, because $F=ma,$ right? The right hand side of the figure shows this “centripetal acceleration,” a different vector but pointing in the same direction as the centripetal force vector. So we have:

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Here’s the punchline for centripetal acceleration:

Centripetal acceleration is special and has a very particular form (found by Newton and his Dutch competitor, Christiaan Huygens) for all curvy motions. It’s

Here, $R$ is the radius of that circle and $v$ is the magnitude of the velocity. So, we have for circular motion only:

The speed of an object moving in a circle is of interest and it’s easy to model that by multiplying the $F_C$ by $R$ and dividing by $m$:

There are two ways to use this concept:

🖋

• If you want something to move in a curved path at a particular speed, you can calculate and apply a precise (centripetal) force—tug it—to make that happen. Depending on the curve ($R$) and the speed you want, you can dial up the $F.$ This is instinctive…your muscle-brain connection does it all the time.
• If you see that an object is not moving in a straight line, then there has to be a centripetal force being applied somewhere! You should be able to find the cause, or infer a cause.

Sometimes identification of such a force is tricky. Remember that in Hazel’s car, what actually causes the car itself to go around a circular curve is the force of friction between the road surface and the four tires of the car—four little force vectors all pointing along the road-tire interface towards the center. Each little $F_C$ adds together to equal the $F_C=m\frac{v^2}{R}$ involving the speed and the turning radius. Here $m$ is the mass of the car plus that of its occupants (less that apple that went out the window).

• Reduce the stickiness of the road (ice, snow, rain?) and that friction force is reduced on the left — we’ve all been there — and that available force is reduced, sometimes considerably. You instinctively know this, so you drive slower (reducing $v$ in the numerator to match the $F_C$ that can be produced by the tires and road given the conditions.
• Make the turn tighter and so the $m\frac{v^2}{R}$ goes up and it better be balanced by the tires’ stickiness on the road.

If any of these limits are reached, you cease doing circular motion and Newton’s 1st law wins and you start going straight. Not good in most circumstances where the road curved.

Everyone’s favorite playground device not requiring an apple:

An Example: Playground merry-go-round

The Question: The figure shows an upper view of a merry-go-round with two children, Arnie and Bertha at two different distances from the center. (a) What is the force of friction required to hold Arnie on board? (b) Is the force of friction required less or more to hold Bertha? ;<br Here are the particulars: $R_A = 3$ m and $R_B = 5$ m. The merry-go-round makes one complete revolution in 10 seconds and each child weighs 50 pounds, so 22.7 kg.

In order to know the force of friction required, we need to know the speed.

We need the speed and then we can get the centripetal force since we know the mass and the radius, $R_A$:

The force is then

which is about 6 pounds. Maybe sticky tennis shoes?

You’ve all done it and you know that it can be very hard to live on the edge. So to speak.

An Example: Edginess

The Question:
(a) If Bertha stands up all the way at the rim at $R_B$, is it harder or easier for her to stay on as compared with Arnie? (b) A different, but related question: at the edge is Bertha moving faster or slower than shy Arnie who’s closer to the center?

Some symbolic manipulation will address (a):

so the further out she squinches towards the edge ($R_B$ getting larger), the higher is the sticky force of her shoes and the surface that she needs in order to not slide off—to stay on the merry-go-round. Bertha had better be wearing baseball spikes.

The answer to the speed question, (b) comes directly from the manipulation of the centripetal force equation, \eqref{vcentrip}: %

Since Bertha’s radius is larger than Arnie’s radius and since for friction, the force is independent of the radius for each…then Bertha is moving much faster and so requires a larger centripetal force.

An Example: Hammer throw

The Question: The Hammer Throw is an ancient track and field event. For men, a 16 lb ball (7.3 kg) is attached to a chain that’s approximately 4 ft long (1.22 m) and whirled around a circle and let go. Olympic-class hammer throwers spin their bodies incredibly fast—in their last “wind” before release they are spinning less than a second per revolution. Let’s call it 0.3 seconds. The figure shows a collegiate hammer champion at work. Cynthia Watt who won the 2015 Big Ten Outdoor Track & Field Championships in East Lansing, Mich. (Photo by Matt Mitchell)

Calculate how fast the ball is moving at that rotational rate.

Let’s attack this strategically by collecting what we know and what we need to know. Here’s what we know:

• We know how long it takes to make a revolution.
• We know how long the chain is, the radius.

In order to calculate how fast the hammer is traveling around its arc, we know how long it takes to make a complete revolution and we know how far it goes in one revolution is the circumference, C, of that circular path. Figure [hammer] shows the forces and the distances for our situation.

The circumference is: $C = 2\pi R = (2)(\pi)(1.22\text{ m})=7.67 \text{ m} \nonumber$

So the speed is: $v= \frac{C}{t} = \frac{7.67}{0.3} = 25.6 \text{ m/s} \nonumber$

(This is about what the measured “escape velocity” is for world-class throwers, who can toss the hammer more than 80 m. Mr Google will quickly tell you that this is about 60 mph.)

Now at that speed, the centripetal force required must be pretty impressive. Calculate that:

 Interactive: 5.6; 2019.09.11; 15:02 Hammer throw centripetal force 

 Interactive: 5.7; 2019.09.11; 15:03 what’s not accelerated? 

 Interactive: 5.8; 2019.09.07; 14:17 string break 

# What to Remember from Lesson 5?

This has been a big lesson with lots of ideas. Here’s what we’ll refer to in lessons to come.

### Force and Acceleration

The T-shirt equation, Newton’s 2nd little-l law is simple and we’ll not require any sophisticated or tricky use of it:

If there’s a force on an object, then that object will accelerate. The larger is the mass of an object, the more force is required to accelerate it, or the smaller will its acceleration be. All common-sense ideas nicely packaged in this simple model for acceleration.

### Mass and Momentum

Mass is the resistance to being accelerated—its nickname is inertia. You knew that, but now Newton’s 2nd law embodies it:

big $m$, small $a$.

Momentum is the oomph that an object moving at a speed has. It’s a little bit of mass and a little bit of velocity and together they make momentum:

### Weight

Still going with $F=ma$, but now for a particular acceleration, that of gravity near the surface of the Earth. That force is nearly constant and so the acceleration is nearly constant and we give it a special name, $g$…”little gee.” It’s actual value is $g=9.8$ m/s$^2$ but we can often get away with using $g=10$ m/s$^2$ and I’ll do that whenever I can.

### Circular Motion

The model for circular motion will recur a number of times. Just remember that

or its alter-ego,

which has the merry-go-round experiences encapsulated in this one-line formula. If the force is constant, for any object “making a turn” (or in an orbit!), the further it is from the center of its circular path, the faster it will go. For everyday life, where the centripetal force responsible for keeping an object moving in a circle (tires, seat belts, jeans-on-the merry-go-round, the rope overhead). These frictional or tension forces have a limit and when that’s reached, straight line motion is the result! We’ll encounter circular motion with a force that’s not constant, you just wait.

But first, let’s bang things together.