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## Partial Derivatives

A partial derivative uses this nice formula. (f)/(x), where f:R^2->R is lim h->0 (f(x+h,y)-f(x,y))/h. Physics is everywhere, waiting, watching.

## Mass CAN Change?

In high school physics we've always been told that test will try to trick you. They'll ask if a 10kg person goes from the earth to the moon how will their mass change. And the answer is always it doesn't. Mass doesn't change, mass doesn't change, mass doesn't change. It's been hammered into our brains. But it's a lie. So the speed of light in a vacuum is 300,000 km/s. This is the fastest speed any object in the universe can travel at. So what happens if you try to accelerate an object going the speed of light? Well picture this: a rocket accelerate to the speed of light, but the thrusters are still pushing on the rocket. You might be tempted to say that the frictional force balances with the thrust of the rocket, so there's no net force. But then how would the rocket have accelerated to the speed of light? There must be a net force. Given that there is a net force, work is being done on the rocket. Therefore, there is a change in kinetic energy, but velocity isn't increasing. That means the other component of kinetic energy must be increasing: mass. In most cases mass is a constant, but when energy cannot be transferred into speed any longer, it has to be transferred into mass instead.

Black holes are often thought of as dark holes sucking matter in towards them by there massive amount of gravitational force. Interestingly enough, however, black holes are anything but black. Black holes might be dark, but they glow. It is well known that black holes decay until they don't have enough energy to sustain their mass, thereby not allowing them to exist any longer. But what does this loss of energy turn into? The slight glow in black holes. This slight glow is due to "Hawking Radiation". It is the slight decay of energy into radiation from black holes over the time of their existence. It is intriguing all the unknown facts about the universe and how much more is left to be discovered!

The speed of light is known as 300,000 km/s and we leave it at that. But this speed is only the speed of light through a vacuum and light doesn't always travel in a vacuum. The slowest recorded speed of light is actually 17 m/s, a speed easily attainable by a car. So what happens then if particles can travel faster than light? Well in many nuclear reactors, this is what happens. Particles travel at a speed greater than the speed of light in that specific atmosphere. When this happens an emission of blue light emerges. This is called Cherenkov Radiation and it can be compared to a sonic boom, which happens when an object is travelling faster than the speed of sound, but with light. It is interesting the concrete ideas we have about physics and specifically light, but all of these concrete "facts" can be manipulated and produce unforeseen outcomes.

## An Observer Can Change Everything

Recently in our physics class we were discussing the theory of relativity and how it works in nature. Without learning the math behind the theory yet, the theory is incredibly confusing, but it reminded me of a video we watched last year in my physics class that discussed how observers can change the way particles act. In a certain experiment, physicists shot electrons through a small slit to see the nature of an electron, whether it would act as a wave or as a particle. Incredibly, even though an electron is a particle, when the experiment was first run, it acted as a wave and diffraction occured from its passing through the small slit. The physicists desired to know more about this remarkable discovery so they ran the experiment again, except this time with an extremely accurate slow motion camera recording the electrons movement. In this trial the electrons acted as particles. The physicists were astounded, but checked again and again and realized it was the camera that changed the electrons behavior. A particle, which has no ability to think, changes its behavior based on whether or not it is being observed. I believe this is one of the most fascinating things about physics, how particles, and our planet, changes its actions based on whether or not its certain actions are being observed. Here's a short video explaining the experiment.

## Slipping in the Rain

I was recently driving on a day when it was raining fairly aggressively. I was driving fine when all of a sudden a car headed the opposite direction from me slid right in front of me almost hitting my car. After assessing the accident and making sure everyone was okay I began to think about what made the car slide all the way to the opposite side of the road. As the pavement was wet, the coefficient of friction between the car and the road was decreased. This made it so the traction in his tires didn't help him with turning. He was beginning to slide to his right side, and tried to compensate for the sliding by turning to the left. He turned to far to the left, however, and when the tires hit a dryer spot on the pavement the traction between the tires and the road suddenly increased. This pulled his car hard to the left and then allowed his car to slide all the way over to the other side of the road. It is intriguing how simple physics can become complex when situations, or in this case a changing coefficient of friction, change. So if you're driving in the rain or the snow and begin to slide, remember to keep your wheel straight, don't try to turn to far.

## Breaking Your Phone with Physics

Most people today have iPhone's that have an immensely complex system of wires in them to allow them to function properly. They are filled with wires, small batteries and capacitors to allow for the story of data and basic functions on your phone. But this complex system presents a problem when faced with a magnet. If a magnet is brought closer to a phone it will cause a changing  magnetic field around the phone's wires. The change in the magnetic field will cause current to move in the direction opposing the change in the magnetic field. But doesn't the complexity of iPhone's help prevent this problem? Actually it makes it easier to destroy an iPhone with a magnet. Since magnetic fields can only affect current perpendicular to their direction, the complexity of an iPhone's circuits provide ample opportunity for the changing magnetic field to align properly with a coil of wire thereby inducing a current in your iPhone and destroying it. So next time you're near a magnet don't rub your phone up against it!

## Is Gravity an Illusion?

From the earliest discoveries of gravity and when students first learn about gravity, they are told it's a force. The force of gravity is equal to mass times the acceleration due to gravity or Fg=(m1m2/r^2. That is just a fact. Or is it? Gravitational forces are actually much more interesting than just the relationship between one mass and another. Gravity is the act of changing space-time. Gravity causes space-time to curve into a bowl like shape pulling masses into the center of it. As a planet is a massive object, its core pulls the space-time surrounding it toward itself, pulling the atoms of the planet toward its core as well as holding surrounding objects on the surface. As one goes onto the outer edges of the bowl, however, the force pulling them in decreases, because the slope of space-time is less. For this reason, Einstein's theory of relativity makes sense. As masses change the shape of space and time according to their size and mass, time and distance on one planet would differ from that of another.

## High Notes on the Tuba

As an experienced tubist, I have been practicing bettering the quality of my higher range for years now, but it is still a challenge. The challenging aspect of playing clear high notes through a tuba can be attributed to physics. The higher the pitch, the higher the frequency of the sound waves. To increase the frequency of the sound waves, one must increase the speed of the air through the tuba. To do so, you increase the pressure of the air in the mouthpiece by pressing your lips close together to generate a fast stream of air. This seems simple enough, so what's the challenge. As frequency is proportional to velocity, it is inversely proportional to length. Therefore, even though the speed of the air is increasing, the tuba is 16 feet of brass tubing, one of the longest instruments ever made. This makes generating a higher pitch by increasing the frequency extremely hard. The pressure in the mouthpiece need to increase an incredible amount due to the length of the instrument. So I suppose I'll just have to give up on the tuba because I can never defy physics.

## Why is the sun hot?

I was recently reading a on a physics website (http://www.telegraph.co.uk/news/science/) an article revealing the actual reason why the sun is hot. Most people, like I, would think it is because of the energy dissipated from the collision of billions and billions of hydrogen and helium atoms. It seems like straight forward mechanics, the particles collide and dissipate a certain amount of heat energy due to the collision and the addition of the billions of collisions that happen every second makes the sun hot. In actuality, however, the sun is hot because of its immense weight. Its weight creates a gravitational force toward the core of the sun increasing the pressure. The increased pressure is what increases the temperature of the sun. It is, however, still important that the sun is made up of hydrogen and helium because without fusion occurring, the sun would have been initially hot a very long time ago, but then would have rapidly cooled. Instead the sun keeps burning for billions and billions of years due to fusion.

## Soun Board

Recently I participated in my high school's musical The Addams Family. Many times during rehearsal I would go to the soundboard to get mic'd and then have my mic checked. I never really knew what a soundcheck composed of, so I asked my friend Jack who worked the soundboard. In the simplest terms possible, one listens to the speaker/singer and they determine which frequencies sound good in the room and which sound awful. The ones that sound awful are cut out. I thought it was pretty interesting, but had no idea how certain frequencies could simply be cut out or stopped from being amplified by the mic. Then today in my Physics C class, we learned that inductors can be used to do this. Inductors store a magnetic field like capacitors store an electric field. By using LC circuits, the soundboard is able to activate certain LC circuits that cut out different frequencies from exiting the microphone. I still don't know totally how LC circuits are able to work, but it was interesting realizes a real life use of them.

## Time Travel

Many people think time travel is absolutely ludicrous, but one has to consider what kind of time travel they are referring to. To travel back in time is ludicrous, because if this were ever to become possible, there would have been discovered evidence of time travelers from the future that came to our time. Time travel according to Einstein's theory of Relativity, however, is not only plausible, but true. According to Einstein, as one increases the speed at which they travel, the rate of change of time is less for them than it is for an outside observer. Based on this idea, one can travel in time by going at incredibly high speeds. By traveling at high speeds, a person will age slower than an outside observer, showing the person traveling so quickly will have, in essence, time traveled forward. So time travel backward will, most likely, never exist, but time travel forward, if great enough speeds are attainable, is fairly simple to accomplish.

## The Piano

Many people understand that the third pedal on a piano allows the notes to be held out for longer by not allowing the strings to be muffled inside, but the first and second pedals are a mystery. The first pedal is also a mystery to me so I won't discuss that one, but the second pedal makes the notes played softer. There is a fair amount of physics that goes into making this happen in a piano. To reduce the sound, the strings are lightly touched so that they cannot vibrate as vigorously, but not too much so that they are cut out. But why does reducing the amount a string vibrates reduce its volume? It is not because the speed at which the string rotates is reduced, but rather because its amplitude is reduced. The amplitude of a strings vibration is directly proportional to its volume. So as the amplitude is decreased by the mechanisms in the piano, the volume that the piano plays at is decreased. If anyone does know what the first pedal does, I am interested so leave a comment.

## Walking the plank

In a show I recently stumbled upon, a man was told to walk the plank. This plank was nailed down, but considering a plank that wasn't nailed down, one could find the length at which to extend the plank off the ship so that it wouldn't tip over when a person with a known mass walked across it. To calculate this, one has to think about the torques applied to the plank. The torques applied, assuming the person is at the end of the plank and the plank has a uniform mass, is only the torque applied by the person and the plank. The torque provded by the person is calculated by the person's mass multiplied by the acceleration due to gravity multiplied by the person's distance from the position at which the plank is being pivoted. The torque provided by the plank is the plank's mass multiplied by the acceleration due to gravity and the plank's distance from the pivot to its center of mass. An equation can then be solved by knowing that if the plank doesn't rotate, its net torque is zero, and therefore the torque provided by the person is equal to the torque provided by the plank. By setting up this equation the position at which the plank should be placed can be determined.

## Bowling

When looking at the sport of bowling, one can easily say the velocity at which the ball is thrown and its mass are the factors in whether or not the pins fall down, but which one matters more, or do they have the same amount of importance? When looking at this question, momentum has to be focused on. The momentum of the ball as it is thrown is what causes the pins to fall down. As momentum is conserved as each pin hits another, the initial momentum of the ball is what matters most. But what is momentum? Momentum is defined by an objects mass multiplied by its velocity. Therefore, the balls mass and the velocity at which the ball is thrown have equal amounts of importance in knocking down the pins. Therefore it's best for a bowler to pick a ball that is heavy, but not so heavy that the bowler cannot throw it with a sufficient velocity.

## Ice Skating

When a skater goes into a spin, they usually start it with their arms out wide, spinning at a slow pace. Then the skater pulls their arms in and the speed at which their rotating increases and finally as the spin comes to an end, their arms extend again and they slow down. Many people understand that physics is incorporated in skating, but they don't understand how much goes into a simple spin in terms of physics. Rotational momentum is defined by the objects moment of inertia multiplied by their angular velocity. An object's moment of inertia is defined by their mass multiplied by their radius squared, multiplied by a constant determined by the shape of the object. Therefore, as a skater pulls in their arms, their radius decreases, decreasing their moment of inertia. Since rotational momentum is conserved during the skaters spin, their rotational velocity increases as their moment of inertia decreases. It is astonishing how simple something so mesmerizing can be after the physics behind it is understood.

## Sleding

Many people spend the winter practicing thrilling winter sports such as skiing or snowboarding, but I like to stick with simplicity. Sleding requires very limited skill to still have the thrill of gliding down a hill. There is also a lot of physics behind sleding, specifically how to turn on a sled. People seem to automatically know that they should lean to a side to turn to that side on a sled, but why? It's all about the normal force. The sled glides down the hill because of the force of gravity on the sled and the person in the sled but turning is a different story. Once a person leans to the side they are push by the snow because they have rotated the snows normal force on the sled. Initially the normal force is perpendicular to the sled but once the sled is turned, the normal force is at an angle, causing the sled and the person to be pushed to the side. This is why simply leaning to the slide one wants to turn works in sleding, and the basic concept even holds true in skiing and snowboarding.

## Returning a serve

In my limited time playing tennis for school and ping pong in my free time, I've learned how to properly return a fast serve. I would always see a quick serve coming at me and be tempted to swing hard back at it, but that would always end in the ball soaring off to either side. My coach instead told me to just hold my racket still and steady and let the ball bounce off of it. This technique has a lot of physics behind it that makes sense. Think of a ball being bounced on the floor. The floor does not swing at the ball to propel it back to your hand, rather the ball merely hits the still floor and goes back up. This can be thought of an elastic collision where all of the potential and kinetic energy of the ball is conserved causing the ball to bounce back up to one's hand. Similarly in tennis and ping pong, a fast serve met with a still racket causes the ball to go across the net with the same speed as it was served with but in a controlled manner. Therefore, even though swinging at the ball will cause its speed to increase, to get a fast AND controlled return, one should cause and elastic collision with the ball and the racket by holding the racket steady and still.

I often play pickup basketball with my brothers, the teams usually split up as me and Paul vs Nathan and Dave. Paul is garbage, however his terrible form and his "signature move" has a lot of physics involved with it. Paul believes the greatest shot is one where he dribbles along the three point arc and chucks up a shot one handed while falling backward. He believes the best way to make this shot is by aiming for the white square on the backboard. This is surprisingly not the best tactic however. Even though every coach tells their 5 year old players to aim for the magic white box, in Paul case, they shouldn't. Since Paul is moving sideways with some velocity, the ball is also moving sideways with the same velocity. Therefore, if Paul aims for the white box, he will end up missing it because the ball will not travel straight but slightly sideways due to Paul's velocity in the horizontal direction. Therefore, Paul should aim to one side of the white box so that the ball actually hits the white box and has a chance of going in. However, playing with Paul turns into an hour of pain and frustration.

## Biking

Biking is one of the most electrifying activities out there. Picking up speed as you approach a jump, wondering how much air you'll get and then being launched into the air. Not many people, however, know all of the physics behind just simply going off a jump. It can be thought of in terms of kinematics by knowing the bikers initial velocity, but then one neglects how the biker obtained that initial velocity. Rather we can consider work and energy to talk about the correlation between the force of the bike, the distance the biker accelerates, their final velocity and the height they get off of the jump. Since work is equal to the bikes force multiplied by the distance the force is acting for, and since work equals the change in kinetic energy, the greater the force or the greater the distance, the greater the bikers kinetic energy. When looking at kinetic energy of the biker, we can look at linear and rotational, but for simplicity we'll just focus on linear. Since linear kinetic energy is a function of speed and mass, the speed of the biker increases, because the bikers kinetic energy increases and the bikers mass is constant. Finally, if the biker has no potential energy before the ramp, and no kinetic energy at his maximum height, we can set kinetic energy equal to potential energy, a function of height, mass and the acceleration due to gravity. Therefore, the height a biker gets is dependent on the bikers speed, but since his speed is dependent on the bikes force and distance that the force acts, the height the biker ultimately attains is dependent on the bikes force and distance that the force acts.

## Rotational motion (don't forget friction)

In a recent lab done in my physics c class, my group was experimentally determining the moment of inertia of six different objects. We set up a ramp for the objects to roll down at an angle of 3.325 degrees. We rolled the objects down the ramp, recorded the time for each object and then found each objects linear acceleration, radius, angular acceleration, mass, net torque and finally moment of inertia. When we checked our answers with our teacher they were horribly wrong, like an average of 200% error. This was because we neglected ed to include for unction in our calculations for torque.  This was a major mistake seeing as friction is the only force on the objects that provide a net torque. We went back and fixed our equation for net torque, which previously was the radius of the object multiplied by the x component of the force of gravity, to be the x component of the force of gravity subtracted by linear acceleration and the objects mass, this result was multiplied by the radius to get the net torque. Our lab then produced less massive percent errors, so my advice to you if you are doing a rotational motion lab is don't forget friction!

## Cereal

You may have wondered why it seems that all of your cereal clumps together in the middle of the bowl, even when you only have a few bits left to eat. The fact that cereal accumulates toward the center is due to something scientists have called "The Cheerios Effect." In 2005, the effect was mathematically proven. The surface tension between the milk and the bowl causes the milks surface to cave in slightly toward the middle of the bowl. Similar to the cohesive and adhesive properties of water, these properties in milk cause a concave surface of the milk. Clearly once one understands that the surface of the milk is caved toward the middle of  the bowl, it is clear that there is a component of the force of gravity pushing the cheerios together into the middle of the milk's surface as the friction between the milk and the cheerio isn't enough to counteract the component of the force of gravity.

My family and I were making bread the other night and my mother had to teach us how a flat piece of dough could turn into a delicious, golden brown loaf. All she knew was that heat made the dough rise, but there is so much more physics involved in making bread rise. In terms of energy, as heat from the oven goes into the dough, the heat energy is turned into mechanical energy in the molecules of the bread, mostly in the form of kinetic energy. This conversion from heat energy to kinetic energy causes the molecules to increase their speed and begin colliding with one another. As the rate of collision's increase, the molecules "look" for more room as to not collide with so many other molecules. This causes the bread to rise as the molecules push outward to avoid hitting other molecules. The conversion of energy combined with molecular movement causes the dense dough to become a fluffy loaf of bread.

## Pancakes

My sister Abby loves to make pancakes for breakfast. She makes three small pancakes at a time using one pan. How does this cook all of the pancakes evenly? This is where physics comes into the equation. The flame is concentrated in the middle of the pan, so wouldn't that be the only place where the pancakes would be able to be cooked? One would assume so, but due to energy and particle movement, the entire pan is able to cook a pancake, even though the flame is not directly under that spot. The flame heats up the molecules in the pan directly above it, causing the heat energy to be converted into kinetic energy. As the molecules then move rapidly, bouncing off one another, the collisions with other molecules in the pan transfer energy from one molecule to another, transferring energy across the whole pan. The kinetic energy in each of the molecules and collisions cause the entire pan to heat up. This is why it is possible to make three pancakes by using just one pan.

## Fall

Fall is by far the best season. It's not too hot, not too cold and the leaves falling all around create beautiful views any way you turn. Physics is also all around during fall. To pick one example, falling leaves illustrate many principles of physics. One could pretend air resistance doesn't exist and see a leaf fall 9.8 m/s^2 in a straight line to the ground, but that would take away from the beauty of the leaf falling. One would have to include air resistance, measured by either bv or cv^2, where b and c are constants and v represents velocity of the leaf. Even the inclusion of air resistance, however, wouldn't totally explain the nature of the leaf falling. It would describe the leaf speeding up as it falls, eventually reaching a terminal velocity until it stops on the ground. The irregular shape of the leaf is what needs to be taken into account to truly define the nature of the falling leaf with physics. The irregular shape is what makes the leaf move side to side, accelerating at different rate throughout its fall. If we were to consider a ball falling, air resistance would be easy to calculate, but due to the irregularity of the leaf, the nature of its fall is difficult to explain in terms of physics. It is amazing how complex the physics is behind an object as simple as a falling leaf.

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