Thursday, December 22, 2011
Refraction
Light waves are part of the EM wave spectrum. When moving through an optical medium (i.e. air, glass, etc. …), the E field of the wave excites the electrons within the medium, causing them to oscillate, as a result, the light wave slows down slightly due to the loss of some of its kinetic energy. Its new speed is always less than that of the speed of light in a vacuum (v<c). Materials are characterized by their ability to bend as well as slow down light, which is known as optical refractive index (n).
         c
     n = -
         v
          speed of light in a vacuum
       = ----------------------------
         speed of light in the medium
n = 1 in a vacuum
n = more than 1 in all other media
Refraction itself occurs when light passes across an interface between two media with different indices of refraction. As a general rule (which can be derived by Snell’s law below), light refracts towards the normal when passing to a medium with a higher refractive index, and away from the normal when moving to a medium of lower refractive index.
Snell’s Law:
n₁sinα = n₂sinβ
where n₁ is the refractive index of the first medium
Reflection
One of the properties of a boundary between optical media is that some of the light that’s approaching the interface at the angle of incidence (α) is reflected back into the first medium, while the rest continues on into the second medium at the angle of refraction (β).
Angle of incidence = Angle of Reflection

Refraction

Light waves are part of the EM wave spectrum. When moving through an optical medium (i.e. air, glass, etc. …), the E field of the wave excites the electrons within the medium, causing them to oscillate, as a result, the light wave slows down slightly due to the loss of some of its kinetic energy. Its new speed is always less than that of the speed of light in a vacuum (v<c). Materials are characterized by their ability to bend as well as slow down light, which is known as optical refractive index (n).

         c
     n = -
         v
          speed of light in a vacuum
       = ----------------------------
         speed of light in the medium
n = 1 in a vacuum
n = more than 1 in all other media

Refraction itself occurs when light passes across an interface between two media with different indices of refraction. As a general rule (which can be derived by Snell’s law below), light refracts towards the normal when passing to a medium with a higher refractive index, and away from the normal when moving to a medium of lower refractive index.

Snell’s Law:

n₁sinα = n₂sinβ

where n₁ is the refractive index of the first medium

Reflection

One of the properties of a boundary between optical media is that some of the light that’s approaching the interface at the angle of incidence (α) is reflected back into the first medium, while the rest continues on into the second medium at the angle of refraction (β).

Angle of incidence = Angle of Reflection

Saturday, December 17, 2011
The Hamilton-Jacobi Equation

This blog has posted more than a few times in the past about classical mechanics. Luckily, classical mechanics can be approached in several ways. This approach, which uses the Hamilton-Jacobi equation (HJE), is one of the most elegant and powerful methods.

Why is the HJE so powerful? Consider a dynamical system with a Hamiltonian H=H(q,p,t). Suppose we knew of a canonical transformation (CT) that generated a new Hamiltonian K=K(Q,P,t) which (for a local chart on phase space) vanishes identically. Then the canonical equations would give that the transformed coordinates (Q,P) are constant in this region. How easy it would be to solve a system where you know that most of the important quantities are constant!

The rub is in finding such a canonical transformation. Sometimes it can&#8217;t even be done analytically, but nevertheless this is the goal of the Hamilton-Jacobi method of solving mechanical systems. In the equation given above, S is the generating function of the CT. Coincidentally, it often comes out to just equal the classical action up to an additive constant! This is due to the connection between canonical transformations and mechanical gauge transformations; it turns out that the additive function used to define the latter is the generating function of the former. In general the HJE is a partial differential equation that might be solvable by additive separation of variables&#8230; but don&#8217;t get too hopeful! Oftentimes the value of the HJE comes not in finding the actual equations of motion but in revealing symmetry and conservation properties of the system.

The Hamilton-Jacobi Equation

This blog has posted more than a few times in the past about classical mechanics. Luckily, classical mechanics can be approached in several ways. This approach, which uses the Hamilton-Jacobi equation (HJE), is one of the most elegant and powerful methods.

Why is the HJE so powerful? Consider a dynamical system with a Hamiltonian H=H(q,p,t). Suppose we knew of a canonical transformation (CT) that generated a new Hamiltonian K=K(Q,P,t) which (for a local chart on phase space) vanishes identically. Then the canonical equations would give that the transformed coordinates (Q,P) are constant in this region. How easy it would be to solve a system where you know that most of the important quantities are constant!

The rub is in finding such a canonical transformation. Sometimes it can’t even be done analytically, but nevertheless this is the goal of the Hamilton-Jacobi method of solving mechanical systems. In the equation given above, S is the generating function of the CT. Coincidentally, it often comes out to just equal the classical action up to an additive constant! This is due to the connection between canonical transformations and mechanical gauge transformations; it turns out that the additive function used to define the latter is the generating function of the former. In general the HJE is a partial differential equation that might be solvable by additive separation of variables… but don’t get too hopeful! Oftentimes the value of the HJE comes not in finding the actual equations of motion but in revealing symmetry and conservation properties of the system.

Monday, November 14, 2011
Variable Star Astronomy
Variable stars are stars whose brightness changes because of physical changes within the star. There exist more than 30,000 variable stars in just the Milky Way. Variable star astronomy is a popular part of astronomy because amateur astronomers play a key role. They have submitted thousands of observed data and these data are logged onto a database. American readers can find information on it on the American Association of Variable Star Observers page. 
One of such variable stars are called Cepheids. Cepheids are pulsating variable stars because they undergo  a &#8220;repetitive expansion and contraction of their outer layers&#8221; [1]. In Cepheids, the star&#8217;s period of variation (about 1-70 days) is related to its luminosity; the longer the period, the higher the luminosity. In fact, when graphed, the relationship is shown by a straight line (as can be seen on the title image). Henrietta Swan Leavitt, an American astronomer, first discovered this and understood the significance of this knowledge.  Combined with understanding of the star&#8217;s apparent magnitude (a previously written post on this subject can be found here), astronomers can use this information to find a star&#8217;s distance from Earth. Cepheids are famously known for their usefulness in finding distances to far-away galaxies and other deep sky objects. Leavitt died early from cancer but was to be nominated for the Nobel Prize in Physics by Professor Mittag-Leffler (Swedish Academy of Sciences). 
Edwin Hubble used Leavitt&#8217;s discovery to prove that the Andromeda Galaxy (M31) is not part of the Milky Way, but was able to find the distance to the Andromeda Galaxy (between 2-9 million light years away). At first his calculation was incorrect (900,000 light years) because he observed Type I (classical) Cepheid Stars. Type I Cepheid stars are brighter, newer Population I stars. Hubble later used type II Cepheids (also called W Virginis stars), which are smaller, dimmer, Population II stars, and he was able to make more accurate calculations.

To determine the star&#8217;s distance, use the inverse square law of light brightness. 


A similar type of star are RR Lyrae Variable Stars. They are smaller than Cepheids and have a much shorter period (from a few hours to a day). On the other hand, they are far more common. Likewise, they can be used to solve for distances as well. Low mass stars live longer, and thus Cepheid stars are generally younger because they are more massive. 
Both Cepheids and RR Lyrae Variable stars are referred to as standard candles: objects with known luminosity. If you&#8217;ve ever wondered how astronomers came to those enormous figures when describing how far away galaxies and stars are from us, you can now better understand why and how. 

Variable Star Astronomy

Variable stars are stars whose brightness changes because of physical changes within the star. There exist more than 30,000 variable stars in just the Milky Way. Variable star astronomy is a popular part of astronomy because amateur astronomers play a key role. They have submitted thousands of observed data and these data are logged onto a database. American readers can find information on it on the American Association of Variable Star Observers page. 

One of such variable stars are called Cepheids. Cepheids are pulsating variable stars because they undergo  a “repetitive expansion and contraction of their outer layers” [1]. In Cepheids, the star’s period of variation (about 1-70 days) is related to its luminosity; the longer the period, the higher the luminosity. In fact, when graphed, the relationship is shown by a straight line (as can be seen on the title image). Henrietta Swan Leavitt, an American astronomer, first discovered this and understood the significance of this knowledge.  Combined with understanding of the star’s apparent magnitude (a previously written post on this subject can be found here), astronomers can use this information to find a star’s distance from Earth. Cepheids are famously known for their usefulness in finding distances to far-away galaxies and other deep sky objects. Leavitt died early from cancer but was to be nominated for the Nobel Prize in Physics by Professor Mittag-Leffler (Swedish Academy of Sciences). 

Edwin Hubble used Leavitt’s discovery to prove that the Andromeda Galaxy (M31) is not part of the Milky Way, but was able to find the distance to the Andromeda Galaxy (between 2-9 million light years away). At first his calculation was incorrect (900,000 light years) because he observed Type I (classical) Cepheid Stars. Type I Cepheid stars are brighter, newer Population I stars. Hubble later used type II Cepheids (also called W Virginis stars), which are smaller, dimmer, Population II stars, and he was able to make more accurate calculations.

To determine the star’s distance, use the inverse square law of light brightness. 

A similar type of star are RR Lyrae Variable Stars. They are smaller than Cepheids and have a much shorter period (from a few hours to a day). On the other hand, they are far more common. Likewise, they can be used to solve for distances as well. Low mass stars live longer, and thus Cepheid stars are generally younger because they are more massive. 

Both Cepheids and RR Lyrae Variable stars are referred to as standard candles: objects with known luminosity. If you’ve ever wondered how astronomers came to those enormous figures when describing how far away galaxies and stars are from us, you can now better understand why and how. 

Saturday, October 29, 2011

Anonymous said: How can I be a writer on sayitwithscience?

Hey, there!

At this moment, we’re not sure if we can immediately recruit someone new to the team. There are a few things, however, that you can do to show us your interest:

  • Email us! Message us (non anonymously)! Let us know who you are! Tell us about your science interests!
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  • Provide general feedback and comments.


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-Say It With Science

Friday, October 28, 2011
The Virial Theorem



In the transition from classical to statistical mechanics, are there familiar quantities that remain constant? The Virial theorem defines a law for how the total kinetic energy of a system behaves under the right conditions, and is equally valid for a one particle system or a mole of particles.



Rudolf Clausius, the man responsible for the first mathematical treatment of entropy and for one of the classic statements of the second law of thermodynamics, defined a quantity G (now called the Virial of Clausius):



G ≡ Σi(pi · ri)



Where the sum is taken over all the particles in a system. You may want to satisfy yourself (it’s a short derivation) that taking the time derivative gives:



dG/dt = 2T + Σi(Fi · ri)



Where T is the total kinetic energy of the system (Σ  ½mv2) and dp/dt = F. Now for the theorem: the Virial Theorem states that if the time average of dG/dt is zero, then the following holds (we use angle brackets ⟨·⟩ to denote time averages):



2⟨T⟩ = - Σi(Fi · ri)



Which may not be surprising. If, however, all the forces can be written as power laws so that the potential is V=arn (with r the inter-particle separation), then



2⟨T⟩ = n⟨V⟩



Which is pretty good to know! (Here, V is the total kinetic energy of the particles in the system, not the potential function V=arn.) For an inverse square law (like the gravitational or Coulomb forces), F∝1/r2 ⇒ V∝1/r, so 2⟨T⟩ = -⟨V⟩.



Try it out on a simple harmonic oscillator (like a mass on a spring with no gravity) to see for yourself. The potential V ∝ kx², so it should be the case that the time average of the potential energy is equal to the time average of the kinetic energy (n=2 matches the coefficient in 2⟨T⟩). Indeed, if x = A sin( √[k/m] · t ), then v = A√[k/m] cos( √[k/m] · t ); then x2 ∝ sin² and v² ∝ cos², and the time averages (over an integral number of periods) of sine squared and cosine squared are both ½. Thus the Virial theorem reduces to



2 · ½m·(A²k/2m) = 2 · ½k(A²/2)



Which is easily verified. This doesn’t tell us much about the simple harmonic oscillator; in fact, we had to find the equations of motion before we could even use the theorem! (Try plugging in the force term F=-kx in the first form of the Virial theorem, without assuming that the potential is polynomial, and verify that the result is the same). But the theorem scales to much larger systems where finding the equations of motion is impossible (unless you want to solve an Avogadro’s number of differential equations!), and just knowing the potential energy of particle interactions in such systems can tell us a lot about the total energy or temperature of the ensemble.

The Virial Theorem

In the transition from classical to statistical mechanics, are there familiar quantities that remain constant? The Virial theorem defines a law for how the total kinetic energy of a system behaves under the right conditions, and is equally valid for a one particle system or a mole of particles.

Rudolf Clausius, the man responsible for the first mathematical treatment of entropy and for one of the classic statements of the second law of thermodynamics, defined a quantity G (now called the Virial of Clausius):

G ≡ Σi(pi · ri)

Where the sum is taken over all the particles in a system. You may want to satisfy yourself (it’s a short derivation) that taking the time derivative gives:

dG/dt = 2T + Σi(Fi · ri)

Where T is the total kinetic energy of the system (Σ  ½mv2) and dp/dt = F. Now for the theorem: the Virial Theorem states that if the time average of dG/dt is zero, then the following holds (we use angle brackets ⟨·⟩ to denote time averages):

2⟨T⟩ = - Σi(Fi · ri)

Which may not be surprising. If, however, all the forces can be written as power laws so that the potential is V=arn (with r the inter-particle separation), then

2⟨T⟩ = n⟨V⟩

Which is pretty good to know! (Here, V is the total kinetic energy of the particles in the system, not the potential function V=arn.) For an inverse square law (like the gravitational or Coulomb forces), F∝1/r2 ⇒ V∝1/r, so 2⟨T⟩ = -⟨V⟩.

Try it out on a simple harmonic oscillator (like a mass on a spring with no gravity) to see for yourself. The potential Vkx², so it should be the case that the time average of the potential energy is equal to the time average of the kinetic energy (n=2 matches the coefficient in 2⟨T⟩). Indeed, if x = A sin( √[k/m] · t ), then v = A√[k/m] cos( √[k/m] · t ); then x2 ∝ sin² and v² ∝ cos², and the time averages (over an integral number of periods) of sine squared and cosine squared are both ½. Thus the Virial theorem reduces to

2 · ½m·(A²k/2m) = 2 · ½k(A²/2)

Which is easily verified. This doesn’t tell us much about the simple harmonic oscillator; in fact, we had to find the equations of motion before we could even use the theorem! (Try plugging in the force term F=-kx in the first form of the Virial theorem, without assuming that the potential is polynomial, and verify that the result is the same). But the theorem scales to much larger systems where finding the equations of motion is impossible (unless you want to solve an Avogadro’s number of differential equations!), and just knowing the potential energy of particle interactions in such systems can tell us a lot about the total energy or temperature of the ensemble.

Tuesday, October 18, 2011

∑ F = ma

… is a differential equation:

where acceleration a(t), velocity v(t), and displacement s(t) are all vectors and functions of time. This equation is second-order in position because the highest derivative is the second time derivative of position. Combined with the right boundary conditions, s(t) (also called the trajectory: path through space and time) can be determined.

This differential equation can be solved one component, or dimension, at a time. Let us focus on one of these, and call it the x component. The equations for y and z can be found exactly the same way.

Constant acceleration

If the graph of a(t) signifying acceleration in the x direction is constant

then the graph of v(t), the velocity in the x direction, is a straight line with slope a0

and the graph of x(t), the position along the x axis, is a parabola

It is also possible for the acceleration, or either of the initial velocity or initial position, to be negative. Thus the displacement/projectile motion formula is derived.

Friday, October 14, 2011
Going superfluid!
A liquid goes superfluid when it suddenly loses all internal friction and gains near infinite thermal conductivity. The combination of zero viscosity but nonzero surface tension allows a superfluid to creep up walls and back down the outside to drip from the bottom of open containers, or to completely cover the inner surface of sealed containers. Lack of viscosity also allows a superfluid to leak through a surface that is porous to any degree, because the molecules can slip through even microscopic holes. Superfluids furthermore exhibit a thermo-mechanical effect where they flow from colder to warmer temperatures, exactly the opposite of heat flow as stated by the laws of thermodynamics! That implies the remarkable property of superfluids of carrying zero entropy. Because of this, a perpetual fountain can be set up by shining light on a superfluid bath just below a vertical open capillary tube, causing the fluid to shoot up through and beyond the tube until its  contact with the air causes it to cease being a superfluid and fall back  down into the bath, whereby it will cool back into the superfluid state  and repeat the process.
So how does superfluidity work, exactly?
Makings of a superfluid
Physicists first got the inkling of something stranger than the norm when, around 1940, they cooled liquid helium (specifically, the 4He isotope) down to 2.17&#160;K and it started exhibiting the above-mentioned properties. Since the chemical makeup of the helium didn&#8217;t change (it was still helium), the transformation to a superfluid state is a physical change, a phase transition, just like ice melting into liquid water. Perhaps for cold matter researchers, this transition to a new phase of matter makes up for the fact that helium doesn&#8217;t solidify even at 0&#160;K except under large pressure - whereas ALL other substances solidify above 10&#160;K.
[Phase diagram of 4He, source]
Helium is truly the only substance that never solidifies under its own vapor pressure.
Instead, when the temperature reaches the transition or lambda point, quantum physics takes hold and a fraction of the liquid particles drop into the same ground-energy quantum state. They move in lock-step, behaving identically and never getting in each others&#8217; way. Thus we come to see that superfluidity is a kind of Bose-Einstein condensation, the general phenomenon of a substance&#8217;s particles simultaneously occupying the lowest-energy quantum state.
Read more:&#8221;This Month in Physics History: Discovery of Superfluidity, January 1938&#8221;. APS News: January 2006
Based on a project by Barbara Bai, Frankie Chan, and Michele Silverstein at Cornell University.

Going superfluid!

A liquid goes superfluid when it suddenly loses all internal friction and gains near infinite thermal conductivity. The combination of zero viscosity but nonzero surface tension allows a superfluid to creep up walls and back down the outside to drip from the bottom of open containers, or to completely cover the inner surface of sealed containers. Lack of viscosity also allows a superfluid to leak through a surface that is porous to any degree, because the molecules can slip through even microscopic holes. Superfluids furthermore exhibit a thermo-mechanical effect where they flow from colder to warmer temperatures, exactly the opposite of heat flow as stated by the laws of thermodynamics! That implies the remarkable property of superfluids of carrying zero entropy. Because of this, a perpetual fountain can be set up by shining light on a superfluid bath just below a vertical open capillary tube, causing the fluid to shoot up through and beyond the tube until its contact with the air causes it to cease being a superfluid and fall back down into the bath, whereby it will cool back into the superfluid state and repeat the process.

So how does superfluidity work, exactly?

Makings of a superfluid

Physicists first got the inkling of something stranger than the norm when, around 1940, they cooled liquid helium (specifically, the 4He isotope) down to 2.17 K and it started exhibiting the above-mentioned properties. Since the chemical makeup of the helium didn’t change (it was still helium), the transformation to a superfluid state is a physical change, a phase transition, just like ice melting into liquid water. Perhaps for cold matter researchers, this transition to a new phase of matter makes up for the fact that helium doesn’t solidify even at 0 K except under large pressure - whereas ALL other substances solidify above 10 K.

[Phase diagram of 4He, source]

Helium is truly the only substance that never solidifies under its own vapor pressure.

Instead, when the temperature reaches the transition or lambda point, quantum physics takes hold and a fraction of the liquid particles drop into the same ground-energy quantum state. They move in lock-step, behaving identically and never getting in each others’ way. Thus we come to see that superfluidity is a kind of Bose-Einstein condensation, the general phenomenon of a substance’s particles simultaneously occupying the lowest-energy quantum state.

Read more:
This Month in Physics History: Discovery of Superfluidity, January 1938”. APS News: January 2006

Based on a project by Barbara Bai, Frankie Chan, and Michele Silverstein at Cornell University.

Wednesday, October 12, 2011
Hypercubes
What is a hypercube (also referred to as a tesseract) you say! Well, let&#8217;s start with what you know already. We know what a cube is, it&#8217;s a box! But how else could you describe a cube? A cube is 3 dimensional. Its 2 dimensional cousin is a square. 
A hypercube is just to a cube what a cube is to a square. A hypercube is 4 dimensional! (Actually&#8212; to clarify, hypercubes can refer to cubes of all dimensions. &#8220;Normal&#8221; cubes are 3 dimensional, squares are 2 dimensional &#8220;cubes, etc. This is because a hypercube is an n-dimensional figure whose edges are aligned in each of the space&#8217;s dimensions, perpendicular to each other and of the same length. A tesseract is specifically a 4-d cube). 

[source]
Another way to think about this can be found here:

Start with a point. Make a copy of the point, and move it some distance away. Connect these points. We now have a segment. Make a copy of the segment, and move it away from the first segment in a new (orthogonal) direction. Connect corresponding points. We now have an ordinary square. Make a copy of the square, and move it in a new (orthogonal) direction. Connect corresponding points. We now have a cube. Make a copy and move it in a new (orthogonal, fourth) direction. Connect corresponding points. This is the tesseract.

If a tesseract were to enter our world, we would only see it in our three dimensions, meaning we would see forms of a cube doing funny things and spinning on its axes. This would be referred to as a cross-section of the tesseract. Similarly, if we as 3-dimensional bodies were to enter a 2-dimensional world, its 2-dimension citizens would &#8220;observe&#8221; us as 2-dimensional cross objects as well! It would only be possible for them to see cross-sections of us.
Why would this be significant? Generally, in math, we work with multiple dimensions very often. While it may seem as though a mathematican must then work with 3 dimensions often, it is not necessarily true. The mathematician deals with these dimensions only mathematically. These dimensions do not have a value because they do not correspond to anything in reality; 3 dimensions are nothing ordinary nor special. 
Yet, through modern mathematics and physics, researchers consider the existence of other (spatial) dimensions.  What might be an example of such a theory? String theory is a model of the universe which supposes there may be many more than the usual 4 spacetime dimensions (3 for space, 1 for time). Perhaps understanding these dimensions, though seemingly impossible to visualize, will come in hand. 
Carl Sagan also explains what a tesseract is. 
Image: Peter Forakis, Hyper-Cube, 1967, Walker Art Center, Minneapolis

Hypercubes

What is a hypercube (also referred to as a tesseract) you say! Well, let’s start with what you know already. We know what a cube is, it’s a box! But how else could you describe a cube? A cube is 3 dimensional. Its 2 dimensional cousin is a square. 

A hypercube is just to a cube what a cube is to a square. A hypercube is 4 dimensional! (Actually— to clarify, hypercubes can refer to cubes of all dimensions. “Normal” cubes are 3 dimensional, squares are 2 dimensional “cubes, etc. This is because a hypercube is an n-dimensional figure whose edges are aligned in each of the space’s dimensions, perpendicular to each other and of the same length. A tesseract is specifically a 4-d cube). 

[source]

Another way to think about this can be found here:

Start with a point. Make a copy of the point, and move it some distance away. Connect these points. We now have a segment. Make a copy of the segment, and move it away from the first segment in a new (orthogonal) direction. Connect corresponding points. We now have an ordinary square. Make a copy of the square, and move it in a new (orthogonal) direction. Connect corresponding points. We now have a cube. Make a copy and move it in a new (orthogonal, fourth) direction. Connect corresponding points. This is the tesseract.

If a tesseract were to enter our world, we would only see it in our three dimensions, meaning we would see forms of a cube doing funny things and spinning on its axes. This would be referred to as a cross-section of the tesseract. Similarly, if we as 3-dimensional bodies were to enter a 2-dimensional world, its 2-dimension citizens would “observe” us as 2-dimensional cross objects as well! It would only be possible for them to see cross-sections of us.

Why would this be significant? Generally, in math, we work with multiple dimensions very often. While it may seem as though a mathematican must then work with 3 dimensions often, it is not necessarily true. The mathematician deals with these dimensions only mathematically. These dimensions do not have a value because they do not correspond to anything in reality; 3 dimensions are nothing ordinary nor special. 

Yet, through modern mathematics and physics, researchers consider the existence of other (spatial) dimensions.  What might be an example of such a theory? String theory is a model of the universe which supposes there may be many more than the usual 4 spacetime dimensions (3 for space, 1 for time). Perhaps understanding these dimensions, though seemingly impossible to visualize, will come in hand. 

Carl Sagan also explains what a tesseract is

Image: Peter Forakis, Hyper-Cube, 1967, Walker Art Center, Minneapolis

Tuesday, October 11, 2011

Hello followers!

The contributors have been on a bit of a break, but we’ll be back soon with a new batch of posts. In the meantime, we’d like to get to know our readers!

What subject areas do you study/where are you in your education/what classes are you taking?

Any requests for future Say it with Science posts?

Friday, September 23, 2011
Uniform circular motion describes an object that is travelling at constant speed in a circular pathway. Though the speed which is the magnitude of the velocity remains the same, the direction of velocity is constantly changing as the object curves along the edge of the circle. A changing velocity implies a nonzero acceleration, and thus a net force acting on the object. What might such an acceleration or force look like?
The equation which characterizes UCM is
||Fnet|| = mar = mv2/r
where m is the mass of the object, v the speed, and r the radius of the pathway.  The acceleration ar = v2/r and net force point at all times toward the center of the circle. You might realize this is exactly Newton&#8217;s law with the added information that the acceleration is entirely radial.
Why would that be? If this were true, it means the external forces on the object in sum continually change the direction of an object&#8217;s motion but preserve its speed. This means the object never gains nor loses kinetic energy! It also means that acceleration, pointing radially toward the center of the circle, is always at odds to the velocity, which points along a tangent to the circle. By geometry the acceleration and velocity vectors for a particle in UCM are always perpendicular, because in a circle a radial line and a tangent must form a right angle at the circumference.
We gain more insight by looking to Newton&#8217;s laws. The first defines inertia, which is the ability of an object with mass to remain at the same velocity unless acted upon by a force. So we know that, absent this curious radial net force, the object would speed through space not only at a constant speed but in a straight line. With the forces in place, a net acceleration arises that snaps the trajectory from line to circle.
The second law states ∑ F = ma, that the acceleration of the object is result of all the forces acting on it. UCM is usually defined using this equation, so there&#8217;s not much to glean here except to remember that the characteristic acceleration, v2/r, is constant in time because v and r are constant, and that we must sum all the forces before equating to mv2/r. But this is a sticking point: the net force in UCM is often called centripetal force, but it is not actually present in the setup. When the initial conditions of the system are just right that they effect a constant radial acceleration, only then do we call the net force centripetal in nature. The UCM version of Newton&#8217;s second law conveniently allows us to solve for forces or other unknowns precisely because we know what the resultant net force must be.
Finally, Newton&#8217;s third law states for any force from a first object on a second, the force from the second on the first is equal in magnitude and opposite in direction. This is useful for looking at specific situations: celebrated (or not) problems include cars travelling on a circular track, cars travelling on a circular banked track, the carnival ride where a person is pressed against the inside of a spinning cylinder, a dish being carried on a lazy susan, a puck or other object being spun around on a string, or an electron shot through some kinds of magnetic or electric fields. Forces between objects can be clues to weigh in the equation that equates to the centripetal (net) force.
If you have some calculus under your belt, here is a fun exercise: imagine a bicycle chain, which is a closed loop made of metal links. You set the chain spinning so that it continues spinning in a perfect circle. If the chain has n links, what is the tension in each link? Then what is the tension in each link as n approaches infinity?
If you&#8217;ve been following the blog, try formulating UCM using the Frenet-Serret equations!

Uniform circular motion describes an object that is travelling at constant speed in a circular pathway. Though the speed which is the magnitude of the velocity remains the same, the direction of velocity is constantly changing as the object curves along the edge of the circle. A changing velocity implies a nonzero acceleration, and thus a net force acting on the object. What might such an acceleration or force look like?

The equation which characterizes UCM is

||Fnet|| = mar = mv2/r

where m is the mass of the object, v the speed, and r the radius of the pathway.  The acceleration ar = v2/r and net force point at all times toward the center of the circle. You might realize this is exactly Newton’s law with the added information that the acceleration is entirely radial.

Why would that be? If this were true, it means the external forces on the object in sum continually change the direction of an object’s motion but preserve its speed. This means the object never gains nor loses kinetic energy! It also means that acceleration, pointing radially toward the center of the circle, is always at odds to the velocity, which points along a tangent to the circle. By geometry the acceleration and velocity vectors for a particle in UCM are always perpendicular, because in a circle a radial line and a tangent must form a right angle at the circumference.

We gain more insight by looking to Newton’s laws. The first defines inertia, which is the ability of an object with mass to remain at the same velocity unless acted upon by a force. So we know that, absent this curious radial net force, the object would speed through space not only at a constant speed but in a straight line. With the forces in place, a net acceleration arises that snaps the trajectory from line to circle.

The second law states ∑ F = ma, that the acceleration of the object is result of all the forces acting on it. UCM is usually defined using this equation, so there’s not much to glean here except to remember that the characteristic acceleration, v2/r, is constant in time because v and r are constant, and that we must sum all the forces before equating to mv2/r. But this is a sticking point: the net force in UCM is often called centripetal force, but it is not actually present in the setup. When the initial conditions of the system are just right that they effect a constant radial acceleration, only then do we call the net force centripetal in nature. The UCM version of Newton’s second law conveniently allows us to solve for forces or other unknowns precisely because we know what the resultant net force must be.

Finally, Newton’s third law states for any force from a first object on a second, the force from the second on the first is equal in magnitude and opposite in direction. This is useful for looking at specific situations: celebrated (or not) problems include cars travelling on a circular track, cars travelling on a circular banked track, the carnival ride where a person is pressed against the inside of a spinning cylinder, a dish being carried on a lazy susan, a puck or other object being spun around on a string, or an electron shot through some kinds of magnetic or electric fields. Forces between objects can be clues to weigh in the equation that equates to the centripetal (net) force.

If you have some calculus under your belt, here is a fun exercise: imagine a bicycle chain, which is a closed loop made of metal links. You set the chain spinning so that it continues spinning in a perfect circle. If the chain has n links, what is the tension in each link? Then what is the tension in each link as n approaches infinity?

If you’ve been following the blog, try formulating UCM using the Frenet-Serret equations!