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After all this time you’d think that we know how an airplane flies. There must be some simple, straightforward explanation. Air hits a plane, air and plane then follow appropriate laws of physics, and voilà, the plane flies, right?
But although the flight of airplanes indeed is in accord with the laws of physics, all of the traditional, straightforward ideas about how this happens are notoriously incomplete, if not incorrect.
What’s the controversy about, and what new ideas today are being proposed?
Bernoulli theorem idea
Newton’s laws of motion
New Theories of lift
No One Can Explain Why Planes Stay in the Air: Do recent explanations solve the mysteries of aerodynamic lift? By Ed Regis
Scientific American, February 2020, Volume 322, Issue 2
Aerodynamic Lift, Part 1: The Science, Doug McLean, The Physics Teacher Vol. 56, issue 8, 516 (2018)
Aerodynamic Lift, Part 2: A Comprehensive Physical Explanation, Doug McLean, The Physics Teacher Vol. 56, 521 (2018)
Understanding Aerodynamics: Arguing from the Real Physics, Doug McLean. Wiley, 2012
You Will Never Understand Lift. Peter Garrison, Flying; June 4, 2012.
Flight Vehicle Aerodynamics. Mark Drela, MIT Press, 2014.
#Flight #aerodynamics #Bernoulli #Lift
A catapult is any one of a number of non-handheld mechanical devices used to throw a projectile a great distance without the aid of an explosive substance—particularly various types of ancient and medieval siege engines.
The name is the Latinized form of the Ancient Greek καταπέλτης – katapeltes, from κατά – kata (downwards, into, against) and πάλλω – pallo (to poise or sway a missile before it is thrown.) [from Wikipedia]
Ideas on how to build them at home
Today’s Latin lesson:
“Cum catapultae proscriptae erunt tum soli proscripti catapultas habebunt.”
( “When catapults are outlawed, only outlaws will have catapults.” )
“Catapultam habeo. Nisi pecuniam omnem mihi dabis, ad caput tuum saxum immane mittam”
( “I have a catapult. Give me all your money, or I will fling an enormous rock at your head.” )
If you lived in the Dark Ages, and you were a catapult operator, I bet the most common question people would ask is, ‘Can’t you make it shoot farther?’ No. I’m sorry. That’s as far as it shoots.”
– Jack Handy, Deep Thoughts, Saturday Night Live
Build an onager, ballista or trebuchet.
Grading rubric. The project is worth 100 points.
Timeliness: Late projects lose 5 points per day.
A. Catapults use torsion (energy stored in a twisted rope or other material.) Do not merely use a stretched elastic (e.g. rubber band.)
If you build a trebuchet then you will need to use a pivoting beam and a counterweight.
B. It will have some kind of trigger or switch. (Without such a trigger, you would merely have a large slingshot.)
C. The payload range will be nearly constant (each payload lands within 15% of the other payloads.)
D. It will have adjustable firing: One setting will yield a shorter range (at least 4 feet.), while another setting yields a longer range (at least 8 feet.)
E. The weight limit is 10 pounds.
F. The longest allowable dimensions of height, length and width are 50 centimeters for each.
100 points Machine built according to the above characteristics
– 20 points Minimum range is not met.
– 20 points Too large or too heavy.
– 10 points Firing range is not adjustable.
– 10 points Uses a stretched elastic material (e.g. rubber band) as the only source of power. (Not applicable for trebuchets, of course.)
– 10 points No trigger.
– 5 points Payload range is not constant
Practical problem solving: When we do use conservation of momentum to solve a problem? When do we use Newton’s laws of motions?
Sometimes we need to use only one or the other; other times both are equally useful. And on other occasions some problems may require the use of both approaches. Rhett Allain on Wired.com discusses this in “Physics Face Off: The Momentum Principle Vs Newton’s 2nd Law”
CONSIDER THE FOLLOWING physics problem.
An object with a mass of 1 kg and a velocity of 1 m/s in the x-direction has a net force of 1 Newton pushing on it (also in the x-direction). What will the velocity of the object be after 1 second? (Yes, I am using simple numbers—because the numbers aren’t the point.)
Let’s solve this simple problem two different ways. For the first method, I will use Newton’s Second Law. In one dimension, I can write this as:
F (net – x) = m x ax
Using this equation, I can get the acceleration of the object (in the x-direction). I’ll skip the details, but it should be fairly easy to see that it would have an acceleration of 1 m/s2. Next, I need the definition of acceleration (in the x-direction). Oh, and just to be clear—I’m trying to be careful about these equations since they are inherently vector equations.
a = delta Vx / time
The article continues here:
Let’s assume that the memory fiber used in “The Dark Knight” is real, and that it can be used to change the shape of a cape into gliding wings with the application of an electrical current. (No such material yet exists, but materials scientists may be getting close.)
Why don’t people use some form of bat wings? Let’s analyze the forces your arms would have to exert in order to successfully use bat wings.
Adapted from “The Physics of Batman: The Dark Knight – High Dive”, Adam Weiner, 08.15.2008
Batman spreads his wings & moves into a circular path.
His motion goes from vertical to horizontal.
The force of air resistance increases dramatically when he opens his wings.
This force turns his linear path into a circular path.
This inward pointing force is a centripetal force.
Law of physics: No object travels in a circular path (Newton’s 1st law), unless some force continually pulls it radially inward.
The balance of inertia and a radially inward force can create circular motion.
Centripetal force depends on the radius of the curve (r) and the radial velocity (v)
F = mv2/r
When a glider – or a Batwing – is bent into the wind, one can use the force to deflect the glider, plane or Batman.
Red arrow to upper right = “lift” (due to the wind hitting the wings)
Red arrow down = weight
Horizontal green arrow is the horizontal component of lift (aka centripetal force)
Vertical green arrow is the vertical component of lift. (If itis big enough, then one can glide for long periods of time)
What about Newton’s 3rd law of motion?
To hold his arms out, Batman has to exert the same force back on the air. So while he moves in a circle, we can calculate the force that will be exerted on Batman’s arms.
circle radius = 20 meters
man + equipment mass = 80 kg
speed remains constant during this turn
Let’s estimate the force on Batman’s arms as he sweeps through the bottom of the arc.
F = weight + centripetal force
F = m g + m v2/r = m ( g + v2/r )
= 80 kg (9.8 m/s2 + [40 m/s]2 /20 m) = 7200 N
= about 1600 pounds
This means that Batman has to hold 800 pounds on each arm! Imagine lying on your back, on a workout bench, holding your arms out and having 800 pounds of weights placed on each one! This is probably impossible for someone to do without super-strength.
Perhaps there is a way out of this. Maybe there are some hinges that connect the wings to the Bat suit. If so, then these hinges could be doing some of the supporting, rather than Batman’s arms.
Cartoon Law I
Any body suspended in space will remain in space until made aware of its situation.
Daffy Duck steps off a cliff, expecting further pastureland. He loiters in midair, soliloquizing flippantly, until he chances to look down. At this point, the familiar principle of 32 feet per second per second takes over.
Cartoon Law II
Any body in motion will tend to remain in motion until solid matter intervenes suddenly.
Whether shot from a cannon or in hot pursuit on foot, cartoon characters are so absolute in their momentum that only a telephone pole or an outsize boulder retards their forward motion absolutely. Sir Isaac Newton called this sudden termination of motion the stooge’s surcease.
Cartoon Law III
Any body passing through solid matter will leave a perforation conforming to its perimeter.
Also called the silhouette of passage, this phenomenon is the speciality of victims of directed-pressure explosions and of reckless cowards who are so eager to escape that they exit directly through the wall of a house, leaving a cookie-cutout-perfect hole. The threat of skunks or matrimony often catalyzes this reaction.
Cartoon Law IV
The time required for an object to fall twenty stories is greater than or equal to the time it takes for whoever knocked it off the ledge to spiral down twenty flights to attempt to capture it unbroken.
Such an object is inevitably priceless, the attempt to capture it inevitably unsuccessful.
Cartoon Law V
All principles of gravity are negated by fear.
Psychic forces are sufficient in most bodies for a shock to propel them directly away from the earth’s surface. A spooky noise or an adversary’s signature sound will induce motion upward, usually to the cradle of a chandelier, a treetop, or the crest of a flagpole. The feet of a character who is running or the wheels of a speeding auto need never touch the ground, especially when in flight.
Cartoon Law VI
As speed increases, objects can be in several places at once.
This is particularly true of tooth-and-claw fights, in which a character’s head may be glimpsed emerging from the cloud of altercation at several places simultaneously. This effect is common as well among bodies that are spinning or being throttled. A ‘wacky’ character has the option of self- replication only at manic high speeds and may ricochet off walls to achieve the velocity required.
Cartoon Law VII
Certain bodies can pass through solid walls painted to resemble tunnel entrances; others cannot.
This trompe l’oeil inconsistency has baffled generations, but at least it is known that whoever paints an entrance on a wall’s surface to trick an opponent will be unable to pursue him into this theoretical space. The painter is flattened against the wall when he attempts to follow into the painting. This is ultimately a problem of art, not of science.
Cartoon Law VIII
Any violent rearrangement of feline matter is impermanent.
Cartoon cats possess even more deaths than the traditional nine lives might comfortably afford. They can be decimated, spliced, splayed, accordion-pleated, spindled, or disassembled, but they cannot be destroyed. After a few moments of blinking self pity, they reinflate, elongate, snap back, or solidify.
Corollary: A cat will assume the shape of its container.
Cartoon Law IX
Everything falls faster than an anvil.
Cartoon Law X
For every vengea nce there is an equal and opposite revengeance.
This is the one law of animated cartoon motion that also applies to the physical world at large. For that reason, we need the relief of watching it happen to a duck instead.
Cartoon Law Amendment A
A sharp object will always propel a character upward.
When poked (usually in the buttocks) with a sharp object (usually a pin), a character will defy gravity by shooting straight up, with great velocity.
Cartoon Law Amendment B
The laws of object permanence are nullified for “cool” characters.
Characters who are intended to be “cool” can make previously nonexistent objects appear from behind their backs at will. For instance, the Road Runner can materialize signs to express himself without speaking.
Cartoon Law Amendment C
Explosive weapons cannot cause fatal injuries.
They merely turn characters temporarily black and smoky.
Cartoon Law Amendment D
Gravity is transmitted by slow-moving waves of large wavelengths.
Their operation can be wittnessed by observing the behavior of a canine suspended over a large vertical drop. Its feet will begin to fall first, causing its legs to stretch. As the wave reaches its torso, that part will begin to fall, causing the neck to stretch. As the head begins to fall, tension is released and the canine will resume its regular proportions until such time as it strikes the ground.
Cartoon Law Amendment E
Dynamite is spontaneously generated in “C-spaces” (spaces in which cartoon laws hold).
The process is analogous to steady-state theories of the universe which postulated that the tensions involved in maintaining a space would cause the creation of hydrogen from nothing. Dynamite quanta are quite large (stick sized) and unstable (lit). Such quanta are attracted to psychic forces generated by feelings of distress in “cool” characters (see Amendment B, which may be a special case of this law), who are able to use said quanta to their advantage. One may imagine C-spaces where all matter and energy result from primal masses of dynamite exploding. A big bang indeed.
© 1997 William Geoffrey Shotts. Last update: Thursday, December 4, 1997
Topic goal: Write a paper on the science of one of the Olympic sports.
ELA goals: Develop your ability to “Gather relevant information from multiple print and digital sources, assess the credibility and accuracy of each source, and integrate the information while avoiding plagiarism.
Physics goals: Given real-world situations, identify the objects involved in the interaction, identify the pattern of motion; and explain & represent the forces with a free-body diagram.
You may choose any Olympic sport. Suggested topics are offered below. Use the following template.
Here a student created a free-body diagram, showing the forces on people in Karate.
Diving and swimming
Can runners benefit from drafting
Does the density of air, and altitude affect the ability to do a long jump
Gymnastics and stunts
Water drag and swimming
How the hammer throw is like a particle accelerator
Why is the iron cross so difficult?
PBS: The Olympics Mind and Body
The discus throw is a track and field event in which an athlete throws a heavy frisbee—called a discus—in an attempt to mark a farther distance than their competitors.
Discus throwing is an ancient sport, as demonstrated by the fifth-century-BC Myron statue, Discobolus. Although not part of the modern pentathlon, it was one of the events of the ancient Greek pentathlon, which can be dated back to at least to 708 BC.
There is a great scene of this in the classic film Jason and the Argonauts, 1963, directed by Don Chaffey with animation by Ray Harryhausen. In one scene Greek athletes compete to win spots on the ship Argo. This culminates in a challenge between Hercules (Nigel Green) and Hylas (John Cairney.) We can examine it here: Discus scene: Jason And The Argonauts
The physics of discus
Video: Physics behind discus throwing
Sports Science discus throw
Re: what are the physics behind discus throwing?
The Physics behind discus
Rotational speed of a discus
Massachusetts 2016 Science and Engineering Practices
8. Obtaining, Evaluating, and Communicating Information
Compare, integrate, and evaluate sources of information presented in different media or formats, as well as in words in order to address a scientific question or solve a problem.
Communicate scientific and/or technical information or ideas (e.g., about phenomena and/or the process of development and the design and performance of a proposed process or system) in multiple formats.
Next Generation Science Standards: Science and Engineering Practice: “Ask questions that arise from examining models or a theory to clarify relationships.” (HS-LS3-1)
CCRA.R.1 – Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual evidence when writing or speaking to support conclusions drawn from the text.
CCRA.R.9 – Analyze how two or more texts address similar themes or topics in order to build knowledge or to compare the approaches the authors take.
CCRA.W.7 – Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the subject under investigation.
CCRA.W.8 – Gather relevant information from multiple print and digital sources, assess the credibility and accuracy of each source, and integrate the information while avoiding plagiarism.
CCRA.W.9 – Draw evidence from literary or informational texts to support analysis, reflection, and research.
Acquire and use accurately a range of general academic and domain-specific words and phrases sufficient for reading, writing, speaking, and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when encountering an unknown term important to comprehension or expression.
Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing.
Provide a concluding statement or section that follows from or supports the argument presented.
This article is archived for use with my students from Ask Ethan: If Matter Is Made Of Point Particles, Why Does Everything Have A Size?
Forbes, Stars With a Bang, by Ethan Siegel 9/16/17
The big idea of atomic theory is that, at some smallest, fundamental level, the matter that makes up everything can be divided no further. Those ultimate building blocks would be literally ἄ-τομος, or un-cuttable.
As we’ve gone down to progressively smaller scales, we’ve found that molecules are made of atoms, which are made of protons, neutrons, and electrons, and that protons and neutrons can be further split into quark and gluons. Yet even though quarks, gluons, electrons, and more appear to be truly point-like, all the matter made out of them has a real, finite size. Why is that? That’s what Brian Cobb wants to know:
Many sources state that quarks are point particles… so one would think that objects composed of them — in this instance, neutrons — would also be points. Is my logic flawed? Or would they be bound to each other in such a way that they would cause the resulting neutron to have angular size?
Let’s take a journey down to the smallest scales, and find out what’s truly going on.
If we take a look at matter, things behave similar to how we expect they should, in the macroscopic world, down to about the size of molecules: nanometer (10-9meter) scales. On smaller scales than that, the quantum rules that govern individual particles start to become important.
Single atoms, with electrons orbiting a nucleus, come in at about the size of an Angstrom: 10-10 meters. The atomic nucleus itself, made up of protons and neutrons, is 100,000 times smaller than the atoms in which they are found: a scale of 10-15 meters. Within each individual proton or neutron, quarks and gluons reside.
While molecules, atoms, and nuclei all have sizes associated with them, the fundamental particles they’re made out of — quarks, gluons, and electrons — are truly point-like.
The way we determine whether something is point-like or not is simply to collide whatever we can with it at the highest possible energies, and to look for evidence that there’s a composite structure inside.
In the quantum world, particles don’t just have a physical size, they also have a wavelength associated with them, determined by their energy. Higher energy means smaller wavelength, which means we can probe smaller and more intricate structures. X-rays are high-enough in energy to probe the structure of atoms, with images from X-ray diffraction and crystallography shedding light on what molecules look like and how individual bonds look.
At even higher energies, we can get even better resolution. Particle accelerators could not only blast atomic nuclei apart, but deep inelastic scattering revealed the internal structure of the proton and neutron: the quarks and gluons lying within.
It’s possible that, at some point down the road, we’ll find that some of the particles we presently think are fundamental are actually made of smaller entities themselves. At the present point, however, thanks to the energies reached by the LHC, we know that if quarks, gluons, or electrons aren’t fundamental, their structures must be smaller than 10-18 to 10-19 meters. To the best of our knowledge, they’re truly points.
So how, then, are the things made out of them larger than points? It’s the interplay of (up to) three things: Forces, Particle properties, and Energy.
The quarks that we know don’t just have an electric charge, but also (like the gluons) have a color charge. While the electric charge can be positive or negative, and while like charges repel while opposites attract, the force arising from the color charges — the strong nuclear force — is always attractive. And it works, believe it or not, much like a spring does.
Warning: Analogy ahead!
Here we go:
Above: The internal structure of a proton, with quarks, gluons, and quark spin shown. The nuclear force acts like a spring, with negligible force when unstretched but large, attractive forces when stretched to large distances
When two color-charged objects are close together, the force between them drops away to zero, like a coiled spring that isn’t stretched at all.
When quarks are close together, the electrical force takes over, which often leads to a mutual repulsion.
But when the color-charged objects are far apart, the strong force gets stronger. Like a stretched spring, it works to pull the quarks back together.
Based on the magnitude of the color charges and the strength of the strong force, along with the electric charges of each of the quarks, that’s how we arrive at the size of the proton and the neutron: where the strong and electromagnetic forces roughly balance.
The three valence quarks of a proton contribute to its spin, but so do the gluons, sea quarks and antiquarks, and orbital angular momentum as well. The electrostatic repulsion and the attractive strong nuclear force, in tandem, are what give the proton its size.
On slightly larger scales, the strong force holds protons and neutrons together in an atomic nucleus, overcoming the electrostatic repulsion between the individual protons. This nuclear force is a residual effect of the strong nuclear force, which only works over very short distances.
Because individual protons and neutrons themselves are color-neutral, the exchange is mediated by virtual, unstable particles known as pions, which explains why nuclei beyond a certain size become unstable; it’s too difficult for pions to be exchanged across larger distances. Only in the case of neutron stars does the addition of gravitational binding energy suppress the nucleus’ tendency to rearrange itself into a more stable configuration.
And on the scale of the atom itself, the key is that the lowest-energy configuration of any electron bound to a nucleus isn’t a zero-energy state, but is actually a relatively high-energy one compared to the electron’s rest mass.
This quantum configuration means that the electron itself needs to zip around at very high speeds inside the atom; even though the nucleus and the electron are oppositely charged, the electron won’t simply hit the nucleus and remain at the center.
Instead, the electron exists in a cloud-like configuration, zipping and swirling around the nucleus (and passing through it) at a distance that’s almost a million times as great as the size of the nucleus itself.
The energy levels and electron wavefunctions that correspond to different states within a hydrogen atom, although the configurations are extremely similar for all atoms. The energy levels are quantized in multiples of Planck’s constant, but the sizes of the orbitals and atoms are determined by the ground-state energy and the electron’s mass.
There are some fun caveats that allow us to explore how these sizes change in extreme conditions. In extremely massive planets, the atoms themselves begin to get compressed due to large gravitational forces, meaning you can pack more of them into a small space.
Jupiter, for example, has three times the mass of Saturn, but is only about 20% larger in size. If you replace an electron in a hydrogen atom with a muon, an unstable electron-like particle that has the same charge but 206 times the mass, the muonic hydrogen atom will be only 1/206th the size of normal hydrogen.
And a Uranium atom is actually larger in size than the individual protons-and-neutrons would be if you packed them together, due to the long-range nature of the electrostatic repulsion of the protons, compared to the short-range nature of the strong force.
The planets of the Solar System, shown to the scale of their physical sizes, show a Saturn that’s almost as large as Jupiter. However, Jupiter is 3 times as massive, indicating that its atoms are substantially compressed due to gravitational pressure.
By having different forces at play of different strengths, you can build a proton, neutron, or other hadron of finite size out of point-like quarks. By combining protons and neutrons, you can build nuclei of larger sizes than their individual components, bound together, would give you. And by binding electrons to the nucleus, you can build a much larger structure, all owing to the fact that the zero-point energy of an electron bound to an atom is much greater than zero.
In order to get a Universe filled with structures that take up a finite amount of space and have a non-zero size, you don’t need anything more than zero-dimensional, point-like building blocks. Forces, energy, and the quantum properties inherent to particles themselves are more than enough to do the job.
Ethan Siegel is the founder and primary writer of Starts With A Bang!
This website is educational. Materials within it are being used in accord with the Fair Use doctrine, as defined by United States law.
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the effect of the use upon the potential market for or value of the copyrighted work. (added pub. l 94-553, Title I, 101, Oct 19, 1976, 90 Stat 2546)
Torque in Everyday Life
When we hear the term ‘torque’ brought up, it’s most often in the context of automobiles. Torque is one of the terms commonly thrown around to describe how powerful a car is, but what exactly does it mean?
In a car, torque is the force that pistons put on the crankshaft, causing it and the wheels to turn.
– Damien Howard
Torque is a force that causes an object to rotate about an axis.
Here we see the piston and crankshaft motion in an internal combustion engine (like in an automobile.) Image from Wikipedia.
While often considered an automotive term, torque is actually a general physics term that has many applications.
Torque is defined as a twisting force that tends to cause rotation.
We call the point where the object rotates the axis of rotation.
You use torque every day without realizing it. You apply torque three times when you simply open a locked door: (1) Turning the key, (2) turning the doorknob, and (3) pushing the door open so it swings on its hinges are all methods of applying a torque.
– Damien Howard, Holt McDougal Physics Chapter 4: Forces and the Laws of Motion
Torque is important when studying physiology (the study of how cells and organs interact to form a whole being, including skeletons, muscles, organs, etc.)
In this lesson we learn about Forces and Torques in Muscles and Joints.
Many body motions require or take advantage of torque, for instance:
A cheetah’s tail provides an excellent example of torque and angular momentum in action. With a simple clockwise flick of the tail, the cheetah’s body (in a response which conserves angular momentum) rolls in the anti-clockwise direction (and conversely).
This enables the cheetah to position its body mid-flight so that it is ready to turn the instant its feet make contact with the ground.
Torque: Introductory physics
Chapter 8, Rotational Motion, Section 2, Rotational dynamics
A lever arm is… (p.209)
Torque is defined as a force that tends to cause rotation. It changes the rotational motion of an object.
We call the point where the object rotates the axis of rotation.
Torque = τ (Greek letter tau)
τ = [Force applied] x [lever arm]
τ = F·r
The amount of torque applied depends on the angle, θ, so then:
τ = F·r·sin(θ)
lever arm = perpendicular distance from the axis of rotation to the line of action of the force.
Finding the net torque
HS-ETS1-2. Break a complex real-world problem into smaller, more manageable problems that each can be solved using scientific and engineering principles.
HS-ETS4-1(MA). Research and describe various ways that humans use energy and power
systems to harness resources to accomplish tasks effectively and efficiently.
SAT Physics test
Torque and equilibrium
“When SAT II Physics tests you on equilibrium, it will usually present you with a system where more than one torque is acting upon an object, and will tell you that the object is not rotating. That means that the net torque acting on the object is zero, so that the sum of all torques acting in the clockwise direction is equal to the sum of all torques acting in the counterclockwise direction.” – SparkNotes Rotational dynamics
Enduring Understanding 3F: A force exerted on an object can cause a torque on that object.
Enduring Understanding 4D: A net torque exerted on a system by other objects or systems will change the angular momentum of the system.
PS2.A: FORCES AND MOTION: How can one predict an object’s continued motion, changes in motion, or stability?
Interactions of an object with another object can be explained and predicted using the concept of forces, which can cause a change in motion of one or both of the interacting objects… At the macroscale, the motion of an object subject to forces is governed by Newton’s second law of motion… An understanding of the forces between objects is important for describing how their motions change, as well as for predicting stability or instability in systems at any scale.
Play hockey with electric charges. Place charges on the ice, try to get the puck in the goal. View the electric field. Trace the puck’s motion.
Determine the variables that affect how charged bodies interact.
Predict how charged bodies will interact.
Describe the strength and direction of the electric field around a charged body.
By Rhett Allain , 07.09.15
This is a classic introductory physics problem. Basically, you have a cart on a frictionless track (call this m1) with a string that runs over a pulley to another mass hanging below (call this m2). Here’s a diagram.
Now suppose I want to find the acceleration of the cart, after it is let go.
The string that attaches the two carts does two things.
First, the string makes the magnitude of the acceleration for both carts is the same.
Second, the magnitude of the tension on cart 1 and cart 2 has the same value (since it’s the same string).
This means I can draw the following two force diagrams for the two masses.
So, how do you find the acceleration of cart 1? It seems clear, right?
You just need to find the tension in the string since that’s the only force in the horizontal direction. You could write:
If I know the tension, then I can calculate the acceleration. Simple, right?
Even simpler, the tension would just be equal to the gravitational force on the hanging mass (m2).
WRONG! This is not the correct way to solve this problem — I actually remember making this exact mistake when I was an undergraduate student. But why is it wrong?
Here’s the link to the full article:
Why is the tension not the same as the weight of mass 2? The answer is simple — mass 2 is not in equilibrium but instead it is accelerating downward.
Since it’s accelerating, the net force is not equal to zero (vector). This means that the tension should be smaller than the weight of mass 2 — which it is.
Solution to the Half-Atwood Machine
The tension in the string depends on the weight of mass 2 as well as the acceleration of mass 2. However, the acceleration of mass 2 is the same as mass 1 — but the acceleration of mass 1 depends on the tension. Does this mean you can’t solve the problem? Of course not, it just means that it’s slightly more complicated.
Let’s say mass 2 is accelerating in the negative y-direction. This means that I can write the following force equation (in the y-direction).
Now I can do a similar thing for mass 1 with its acceleration in the x-direction. Since the magnitudes of these two accelerations are the same, I will use the same variable.
With two equations and two variables (a and T), I can solve for both variables. If I substitute the expression for T for mass 1 into the equation for mass 2, I get:
Instead of completely solving for the acceleration, let me leave it in the form above. Think of the problem like this: suppose you consider the system that consists of both mass 1 and mass 2 and it’s accelerating.
What force causes this whole system to accelerate? It’s just the weight of mass 2. So, that is exactly what this equation shows — there is only one force (m2g) and it accelerates the total mass (m1 + m2).
From this I can solve for the acceleration.
Using the values of mass 1 = 1.207 kg and mass 2 = 0.145 kg, I get an acceleration of 1.05 m/s2. This is pretty close to the experimental value (seen above) at 1.109 m/s2. I’m happy.
With the value of the acceleration, I can plug back into the original equation to solve for the tension. With this, I get a tension of 1.267 N. This is fairly close to the experimental value of 1.285 N. Again, I’m happy. It seems physics still works.