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I just returned from a calc group session at school with my friends and our calculus teacher. My friend, in an attempt to make Taylor Polynomials and series less of a burden, brought along her little dog. Ironically, as I was sitting there, the pup inspired what I am afraid will be my final blog post of my AP Physics C year. Well, my friend had gotten up from her seat, and the dog, which was tied by a leash to the chair, wanted a change of scenery. As a result, she attempted to jump onto the very chair which she was tied onto. However, as soon as her paws came in contact with the chair, she skid across the surface of the chair and nearly fell off the opposite side. So, what did the little doggy fail to consider in her take off towards the chair? Well, there are a few factors. First off, when the dog took off from her hind legs, she made an angle with the floor; she had both horizontal and vertical components to her velocity. As a result, when she hit the peak of her trajectory path, hence landing on the chair, her vertical velocity was zero, but her body continued to move in the horizontal direction due to the horizontal component of her velocity. In addition, because the surface of the chair is slicker than most surfaces, resulting in a lower coefficient of friction, there was little frictional net force present in order to decelerate her horizontal velocity. Ideally, in order to prevent any skidding, the dog would simply have jumped completely vertical and landed on the chair, hence having zero horizontal velocity (this application is not ideal, however, because it would involve the dog jumping through the solid seat of the chair, which is impossible and would hurt, to say the least). However, a large angle with the horizontal would increase the sine component of her velocity and minimize her horizontal velocity, and therefore skidding.

As a kid, I was always at my neighbor's house because they always had the newest and coolest trampoline. Turns out that this cool contraption requires many physics concepts in order to work. The energies required for a spring are kinetic energy, gravitational potential energy, and spring potential energy. When you bend your knees in order to take your first jump, you are using your gravitational force downward in order to compress the spring in the trampoline (work from your knees is transferred into spring potential energy). Then the spring releases, and the potential energy transfers into work done on your body, hence shooting you up into the air!

Since I am a ballet dancer, it would be fair to mention one of the most impressive ballet moves performed: the grand jete. For non-dancers, this move can be described as a "split like jump"; the dancer takes off by extending one leg into the air and taking off into a projectile type motion. In the best case scenario, the ballerina hits a perfect split at the peak of her parabolic path, creating a split second mesmerizing image for the audience to enjoy. In order to complete this leap, several physics ideas must be thought of and considered carefully. First off would be the gathering of energy. It is impossible, no matter how strong your legs are or how much experience you have, to perform a grand jete from a static position. A dancer usually performs a series of quick moving steps across the floor in order to gain the momentum and more importantly, kinetic energy. Hence, when the dancer extends his or her leg at an angle to the horizontal stage (creating projectile motion) this gathered kinetic energy is transferred into potential energy, allowing the dancer to follow a parabolic path. What allows the dancer to hit a peak of their motion in a split and to appear almost frozen in mid air would be the lack of gravitational acceleration downward. For a split second, at the top of the dancer's path, their upward vertical acceleration has been reduced to zero and they are yet to experience a gravitational force downward. Finally, it is important to note the purpose of inertia and center of mass during the execution of this step. When taking off for the grand jete, a dancer must work to keep their torso (primary center of mass) moving in the direction of their anticipated projectile; in other words, the dancer must anticipate the jump. If not, their inertia will resist this upward change in motion, which will limit the success of the grand jete. Crazy that so much physics goes into this ballet step.

A very useful device for many instrumentalists and musicians, in particularly string players, is a metronome. A mechanical metronome is a box like object that produces a steady beat. A musician sets this beat based on the tempo marking of the piece which they are practicing, and then the beats produced by the metronome help the musician to play at a steady pace and to avoid rushing or slowing. So, how does a metronome work? Well, from the outside, a metronome actually appears like an upside down pendulum; at the top there is a weight which is attached to the bottom of the box by a long rod. The musician can adjust the speed of the beats produced by lowering or raising this top weight. The way the metronome works is that at the bottom of this rod (and usually hidden from view) is a weight that acts like the bottom bob of a pendulum. So, when the instrumentalist lowers or raises the upper weight, they are in essence shortening or lengthening the length of the pendulum, hence increasing or decreasing the frequency of the simple harmonic motion and the tempo of the produced beats.

Unless you are living under a rock, you would know that March Madness and the beloved basketball season are officially coming to a close. As sad as this end may be for some die hard basketball fans, it should be noted that the sport of basketball (like most other things in our world) is possible only due to the presence of physics. While there are many possible applications of physics, from the friction between the shoes of the players and the court, to the tension (or lack thereof) in the strings of the basketball hoop, I would like to draw attention to the most important part of the game: shooting a hoop. When a basketball player takes a shot, they are in essence sending the basketball into a path that is very similar to that of a projectile. However, the basketball player must take into account their distance from the hoop and the height of the hoop when taking their shot. As a result, some shots seem to go higher (due to a larger angle of projection), while other shots seem to be more horizontal in nature due to a smaller angle of projection. Why? Well, when the angle of projection is increased, the y-component of the ball's velocity increases due to the sinecomponent of the angle, so the ball goes faster and further in the y-direction. The opposite is true for smaller angles of projection.

At first glance, this blog post may appear to be about the physics behind a large civil structure on which vehicles and human beings move across. However, that is not the case. This post is about the importance and purpose of a bridge in the structure and function of a violin, as well as the impact a mute has on a violin's performance. On the violin, the bridge is a wooden structure perpendicular to the rest of the violin; it sits atop the wooden face of the violin, and the four strings lie across the top of it. The purpose of the bridge is to transmit the vibration of the four strings into sound. Let me explain. The various strings on a violin (and any stringed instrument for that matter) vibrate, hence creating sound waves, when some type of work is done to them (in the case of the violin, the most common way to get this vibration is to pluck the string or play it by applying pressure with a bow). However, these vibrations do not directly translate into beautiful music. So, the bridge serves to transmit vibrations to the structural part of the violin, which gives the vibrations more space to vibrate on and throughout; instead of vibrating a thin string, the vibrations ring throughout the entire structure, causing an increase in volume and projection of music. Another component worth mentioning would be the impact of a mute on the bridge and this transmittance of vibrations and sound. A mute is placed directly on the bridge whenever a musician wishes to dampen their sound. How does it work? Well, the mute reduces the vibration that occurs on the bridge during the transfer of sound from the strings to the wooden structure of the violin. As a result, the amount of vibrations transferred to the main structure are significantly reduced, resulting in a less vibrations throughout the entire instrument and a noticeably softer sound.

One of my favorite activities as a young kid was to play on the playground; I loved the monkey bars and slides, but one of my all time favorite thing to do would be to swing on a swing. Swings give the sensation of flying, which is probably why I loved them so much. Ironically, the way a swing works happens to revolve a lot around the conservation of energy. Think about it: In order to start swinging, someone has to either give the swing a push, or the swinger must kick themselves off of the ground. As one swings back and forth, they must continue to pump their legs because otherwise they will lose the battle to air resistance and slowly but surely come to an eventual stop. As one swings and gains elevation, all of the energy from the leg pumping becomes potential energy, which varies directly with the height of the swing from the ground. For a split second, when all of the kinetic energy is converted into potential and the swing is at its peak, the swing stops (as a kid and admittedly now, that is my favorite part). Then, as the swing accelerates towards the ground, the potential energy becomes kinetic, resulting in maximum speed at the bottom of swinging motion. In fact, a swing is very much like a pendulum in terms of how energy works to keep the participant swinging and how energy is transferred and conserved.

Yesterday, as I climbed into bed, bundled up in blankets and a heavy sweatshirt, I reached across by bed to grab a final blanket. All of the sudden, out of the darkness I saw a spark, which was followed by a stinging feeling in my finger. Had I not learned about electrostatics, I probably would have screamed and thought that there was something wrong with me or that the house was on fire. However, physics helped me to understand that I was not dying and that what had happened was simply an attempt of two objects to reach electrostatic equilibrium. As I was bundling up, I was unknowingly rubbing my heavy sweater against my blankets; as a result, electrons were transferred from that blanket onto the sweater (and therefore me), causing me to become a charged object. When I reach for another object, in this case the other blanket, which was uncharged, some of the electrons on me "jumped" onto the blanket in order to reach equilibrium. This transfer of electrons was both seen in the milisecond spark and felt through the hand that I had used to grab the blanket.

In a previous blog post I wrote about how the lower coefficient of friction due to ice causes a decrease in rotational motion and an increase in skidding when someone is driving. I want to extend on that topic only because a few weeks ago, when the snow was really bad, I first hand experienced the horror of making a turn without a strong centripetal force present. It was rather snowy and icy, and my friend was driving. However, as they went to make the turn, they turned too sharply, and we skid all over the road. Luckily, we gained control of the car and neither of us were injured; either way, it was a terrifying experience. Why did we skid? Well, part of it does have to do with the reduced coefficient of friction in ice because, in a turn, friction acts as the centripetal force that keeps the object moving in a circle. However, another notable point would be the width of the radius of the turn. When my friend went to make the turn, they cut the corner kind of tightly, resulting in a much tighter centripetal radius. Because centripetal force is equal to mv^2/r, a decrease in the length of the radius means that the centripetal force to keep an object moving in that circular direction must increase. Coupled with the reduced force of friction caused by the slippery ice, it is no wonder that the car strayed from the circular path which it was taken around the bend!

For winter break, I traveled with my friend to southern Florida in order to escape the chilling winds and snow. Yesterday, as we returned from our sunny break and descended into our final destination, I applied physics in a kind of unique way. As the plane got closer and closer to the ground, the turbulence became increasingly worse (the pilot had warned us that this was expected due to strong winds). As the descend continued and the bumpiness of the ride worsened, I heard a little boy in front of me grow more and more uneasy and scared. At one point, he asked his dad "What if we just drop out of the sky?". I thought about it, and I realized that the boy had no need to worry of the plane simply falling out of the sky. Why? Well, if you simplify it, a plane descending is simply an application of kinematics and projectile motion. Even if the pilot were to completely cut the engines, the plane would still continue to go at the same horizontal speed because there is no acceleration in that plane (granted, as I did mention, there was some wind, which could cause a resisting drag force, but the wind did not have nearly enough force to decelerate the plane's velocity to a magnitude of zero instantaneously). Even if the plane were to go into freefall, it would continue to travel in the forward motion, but the acceleration due to gravity would increase, hence the planes downward displacement would be increasing faster in comparison to its displacement in the y direction. Even so, given the conditions, the little passenger in front of me had no need to worry; we were not going to simply fall out of the sky, and we safely landed on the runaway.

Like I said in my last blog post, I love spy movies, and I think I am starting to love them more and more because of all the physics applications in them. While Bond movies are awesome, I would say my all time favorite film would be Johnny English starring Rowan Atkinson, most known for his role of playing Mr. Bean. In the film, Atkinson plays a mock agent, and pretty much no one takes him seriously or believes that he can be successful at anything. However, after several mess ups, Johnny manages to save the day by preventing the villain, Pascal, from being crowned the new King of England. The way he accomplishes this task is rather unorthodox, and it applies the physics concepts of a pendulum. As the Bishop of Canterbury is about to place the crown on Pascal's crown at the alter of Westminster Abbey, English, who is in the balcony, grabs hold of a free cable, and in an almost "Tarzan-like" manner, swings from the balcony and grabs the crown. While his foil of the coronation is successful, English then has a slight problem: he cannot get off of the swinging rope. See, when he swung, English did not understand the principle of conservation of energy as well as air resistance. For starters, he reaches for a pole on the opposite side of the swing that is above where he swung from; by conservation of energy, he would not have enough energy to get to a height higher than where he started from because that would require more energy to be converted into potential. As a result, English misses the grab and is hopelessly swinging back and forth. Then, thanks to air resistance, the distance of the swings becomes shorter, and he is forced to come face to face with Pascal, who by that point has a gun aimed at him. But, in English's clumsy fashion, he survives and saves the day!

I love spy movies, so its no surprise that when my family received Casino Royale (a James Bond movie), for Christmas, that I was glued to the T.V. In the film, James' female counterpart, Vesper, is kidnapped by evil gamblers. Being the smart guy that he is, James quickly figures out their plot, hops into his Aston Martin and speeds after the kidnappers. However, his kidnappers are actually after Bond, so they tie up Vesper and place her directly in the road; Bond swerves to avoid Vesper, his Aston Martin does a few 360s, and he ends up crashing and in the hands of the evil guys. While I would love to go on about the rest of the plot, the car crash is what I really need to discuss because physics plays a big role in why Bond's sweet ride goes out of control. When Bond is speeding in the car, the mass and therefore inertia of the combined system (Bond and the car), is moving in a straight line. However, when Bond swerves in order to avoid hitting Vesper, he is abruptly changing his path of motion. Because inertia is the measure of an object to resist a change in motion, and the inertia of Bond and the car, due to their combined large mass, is rather high, the inertia causes Bond and the car to continue in the original direction and resist the change in motion. As a result, Bond's car flips through the air and he is taken prisoner; but not to worry, in his smooth fashion, he manages to escape and save Vesper!

I spent this summer internship working in a remote sensing lab. My job was to analyze Landsat satellite imagery in order to analyze the impact of wildfire on vegetation growth in Akagera National Park in Rwanda. I was able to do this analysis because of the data provided by the satellite. Satellite orbits are possible because of the strong gravitational field of the Earth and the relative masses of the satellite and the Earth. For example, for the Landsat 8 imagery that I analyzed, the force of gravity felt by both the satellite and the Earth was equal to G(mass of the earth)(mass of Landsat 8)/(radius of Landsat's orbit)^2. Thanks to Newton's third law, both the Earth and the satellite felt this same force due to the other's existence. However, the mass of the Earth is exponentially greater than that of the satellite, so the satellite orbits in the Earth instead of the Earth orbiting the satellite. This orbit takes the form of centripetal motion, where the centripetal force is equal to the force of gravity which is equal to (mass of Landsat)(velocity)^2/(radius of orbit). This centripetal motion keeps the satellite constantly changing direction and therefore constantly accelerating. Thanks to gravity and the laws of motion, scientists can view the Earth from above, and I was able to complete my research this summer!

We all learn the importance of friction in how the world works at a very young age. For me, a Bill Nye video revealed the power and importance of this contact force. Without it, we wouldn't be able to stand still in one place, driving would be a nightmare, and about everything else in our lives would be completely out of whack, from our electronics to our daily routines. Today, however, as I was doing homework, I had a realization of friction that, although simplistic, was something I had to share in order to continue the emphasis of friction's importance in how the world works and the life of a physics student. Without friction, erasers and pencils, and therefore homework and school, would not work or be possible! I will start with the simpler of the two: erasers. It seems pretty clear, that erasers work because the force of friction between the paper, eraser, and graphite on the paper causes the eraser to do work on the graphite (hence erasing), and the paper and graphite to do work on the eraser (resulting in eraser shavings and discoloration). But the more interesting of the two is the pencil. A pencil, mechanical in this case, is made of a piece of graphite that is held together by some molecular attractive forces determined by composition. The only reason that a pencil can write is because the friction between the graphite and the page is greater than the attractive forces that hold the graphite together. The force of friction "wears away" at the graphite, which allows students to write. In fact, if you observe the paper closely as the pencil writes, you will see tiny shavings that are the paper, which are a result of the force of friction doing work on the pencil. Crazy, that without friction even the simplest of things, such as doing homework, would not be possible!

A few days ago, my sister and I were leaving school. As our hands were both full, she used her body weight in order to push on the main door in order to exit the building. However, the door would not budge. I quickly realized that my sister, in her attempt to open the door, was not applying torque to her favor in this situation. Torque, which is the rotational equivalent of linear force, is dependent on three factors: the angle at which the force is applied, the magnitude of the force, and the radius at which the force is being applied. To simplify my sister's exiting scenario, however, we will only consider force magnitude and radius, because she applied her force at a direction perpendicular to the door, hence applying the whole component of her force. When my sister went to open the door, the force she applied using her body was concentrated at a location that was somewhere in between the midpoint of the door and the hinge. Hence the radius at which my sister applied her force to was about 1/4 the length of the door, causing her torque to be of a significantly less magnitude and the door to not move. All she needed to do was to apply that same force to the end of the door opposite the hinge, which would produce a radius four times bigger than originally, and hence a torque of equally greater magnitude as well. After this quick switch, the door did indeed open, and we were on our way into the brisk cold.