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// Chapter 5 //

=Lesson 1: Motion Characteristics for Circular Motion=


 * Speed and Velocity Sneak Their Way Into the Party Again**

Since unit one, speed and velocity have been in every single one of our lessons and in everyone of our equations. Circular motion is no different as speed and velocity still play a huge role. As cars travel in a circle, or a cart goes up a roller coaster, we see examples of circular motion every single day. You can even get uniform circular motion, where you remain at a constant speed around a perfect circle with a constant radius. To find the speed of an object traveling in a circle, it's still the simple distance divided by time equation, yet this time, distance is the circumference of the circle (2 * pie * radius). You then divide that by the period and find your speed. We can make conclusions from simply looking at this equation, as a circle with a larger radius will have a greater speed. The radius of the circle and the speed are actually directly proportional. Although objects can have a constant speed in circular motion, they can never have a constant velocity because the direction will always be changing. Since velocity is a vector quantity, it deals with direction, and therefore an object in circular motion will NEVER have a constant velocity. The direction of this velocity vector is always tangential. All in all, there's no surprise that speed and velocity made it back into another lesson.


 * Circular Motion is Acceleration's Best Friend**

It's a common misconception to think that an object in circular motion is never acceleration because there's constant speed. This is wrong though, for the object is ALWAYS accelerating due to changes in direction. To find acceleration you must subtract the initial velocity from the final velocity and divide it by the time that passed. To find the velocities, you need to draw two tangent lines to find the final and initial velocities. Once doing this you can find the acceleration! You can also simply find the acceleration by using an accelerometer, a very useful device. Acceleration, like speed and velocity, has found another way to sneak into another lesson, and this time, circular motion is acceleration's best friend.


 * Centripetal Forces are Center Seeking Missles**

Acceleration always points to the center of a circle in circular motion, and according to Newton's second law, an object experiencing an acceleration must also be experiencing a net force. This means there also has to be an inward force pointing towards the center. This force pointing towards the center is known as a centripetal force. The law of inertia states that an object will stay in motion unless an unbalanced force acts upon it, and since objects in circular motion are accelerating, then unbalanced force act upon them.


 * Centripetal, Not Centrifugal!**

It's common to believe that centrifugal forces are needed to put an object in circular motion, but its is a common misconception. Circular motion requires an inward force, also known as a centripetal force. Centrifugal means an outward force, and without a centripetal force, an object would stay in a straight line. Many people believe that centrifugal force cause circular motion because they feel like they're being thrown outward from the circle when in motion. This is not true though, as you actually aren't being thrown outward. Remember that centriFugal is the forbidden F-word, and never say it again.


 * Putting it All Together**

You need to know three primary equations for circular motion: speed, acceleration, and force. Speed is basically distance divided by time, where the distance is the circumference of the circle. Acceleration can be found by squaring the velocity and then dividing that by the time. The equation for the centripetal force is the force pointing towards the center of the circle equals mass times acceleration. There are pictures below to demonstrate this.



=Lesson 2: Applications of Circular Motion=


 * What's Newton's Second Law Again?**

Newton's second law states that the acceleration of an object is directly proportional to the net force acting upon the object and is indirectly proportional to the mass. F=ma is used to find the missing variables, but this changes to F=mv^2/r because of the new centripetal acceleration formula. This is used in all types of problems, and in circular motion its not different. If you're looking for a force that isn't centripetal, then you can use different axes to find it.


 * Everyone Loves the Amusement Park!**

Roller coasters are so much fun because they accelerate us in crazy ways that no other thing can do. On these machines, there are types of motions known as loops, dips, and banked turns. It isn't the speed that makes us enjoy our rides, but the weightiness and weightlessness that we experience. We feel this at the top of a loop, where there is less normal force than your weight. You feel light here, but opposed to this, at the bottom of a loop you feel heavier since there is more normal force than your weight. This dips and loops you experience and actually circles, so even during dips if u don't feel like you're going in circular motion, you in fact are. The clothoid loops are where you experience this circular motion and all the types of acceleration that make you go wild.




 * Athletes Can be Nerds Too**

Circular motion, believe or not, is common in every single sport, whether a car is driving around a race track, or a basketball player does a 360 dunk. Turns are the most common types of circular motion seen in sports, which involves circular motion principles. Even if it s just a quarter of a turn, there are still inward forces and centripetal accelerations. When an athlete turns, they lean into it, causing there two be horizontal and vertical forces, and the contact force from below balances the downward force of gravity and meets the centripetal force requirement for an object in uniform circular motion. Even if athletes aren't making full circles while they compete, they are turning, which causes circular motion, and they are leaning, which causes horizontal and vertical forces.



=Lesson 3: Universal Gravitation=


 * Gravity's Like Family**

Literally everyone knows what gravity is, from an elderly man, to a youngster. When things go up into the air, gravity causes them to come back down. The force of gravity is the force that exists between the Earth and all object around it. The acceleration of gravity is when something is up in the air and then comes back down, and gravity speeds it up. The acceleration of gravity happens when gravity is the only force acting upon the object. The value of this 9.8 m/s^2.


 * Kepler and Newton Discover Gravity!**

Kepler came up with three laws to describe the motion of planets around the sun:

The Law of Ellipses- The paths of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. The Law of Equal Areas- An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time The Law of Harmonies- The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun.

There was no explanation for these though, and Newton wanted to find one. He knew the planets needed to have an inward force and an unbalanced force since they were traveling in circles. He came up with the notion of universal gravitation, and he said that if you shot a cannon ball with enough speed it would start to orbit the Earth. He believed that the gravity objects experienced on Earth translated to the solar system, explaining the celestial motions. Finally, the force of gravity is inversely proportional to distance.


 * Universal Gravitation is Universally Accepted**

Newton discovered that gravitation is universal, and ALL objects attract each other with a gravitational force. The force of this attraction is directly proportional to the masses of the objects its affecting and inversely proportional to the square of the distance that separates the centers.



Another breakthrough was added to this equation though, making it more valid. Henry Cavendish discovered the universal gravitation constant, and the value of it being 6.67 x 10^-11. By knowing this constant, we are able to calculate the force of gravitational attraction between any two objects if mass and distance are given.


 * "G"avendish**

Canvendish discovered this constant of proportionality by using a torsion balance, one of the few things that Newton never discovered about gravity. He measured the relationship between the angle of rotation and the amount of torsional force. By placing two large metal spheres by the two smaller spheres, he waited for torsional forces to balance the gravitational forces. After, he measured the masses, distance, and the force of gravity, leaving him with that constant of proportionality.


 * G is Valued at More Than You'd Think**

9.8m/s ^2 has always been the value of gravity we have been using this whole year, but that is only actually at sea level. The value depends on its location, and if the location is further than the Earth, say its orbit, than the value of gravity would be larger than 9.8. This goes against everything we previously knew. You can look at the equation for the force of gravity to figure this out, because if distance increases, so does the denominator, resulting in a bigger value.

=The Clockwork Universe (1-4)=


 * New Ideas**

Nicholas Copernicus broke the previous beliefs that the Earth was the center of the universe, thus starting the heliocentric belief. A heliocentric universe is where the Earth orbit the sun and this put the Church against that idea. Galileo was punished for supporting Copernicus's ideas, and eventually murdered. Keplar added upon this idea and said that the planets moved in elliptical orbits around the Sun. Although he believed it, Keplar had no evidence to support his claim.


 * New Era in Math**

During this time, there were several new math discoveries, starting with Descartes's idea that any geometry problem can be recast as an algebra problem. To do this you simply need a coordinate plane, and by drawing shapes and lines onto it, you can create an algebra problem. This is coordinate geometry and the scientists used this to delve even deeper into circular motion by making newer equations. They found out the equation of a circle during that time period as well, helping them even further.


 * Newton Comes to the Rescuse**

With new inventions and discoveries, the physics from the past with Aristotle was becoming outdated, and there needed to be new ideas in physics. Issac Newton came and provided this new physics. To make people understand the universe better, Newton created a set of laws to explain everything. Newton paid more attention to the deviation of motion other than constant motion straight. Once this deviation occurred, Newton would look for anything that caused it to move that way. Finally, Newton made a quantitative link between force and the deviation of motion to create his Law of Universal Gravitation.


 * Newton the Genius**

Newton was able to expand on Keplar's ideas that the planets traveled in an elliptical orbit around the Sun by using his new single law for gravity. By putting gravity and his laws of motion together, Newton was able to get this. Mechanics, the study of force and motion was born, and it suggest the world was like a clock. They also believed the future development was predictable, and this belief was known as determinism. Many people believed in free will, meaning they were the ones that were going to determine their future and live free. All in all, Newton was able to extend Keplar's ideas and put sense to them.

=Lesson 4(a-c): Planetary and Satellite Motion=


 * Keplar's, Not Newton's, Three Laws**

Keplar had three planetary motion laws that described how the moved

The Law of Ellipses- The paths of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. The Law of Equal Areas- An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time The Law of Harmonies- The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun.

His Law of Equal Areas states that as a planet orbits the Sun, it will have different speeds at different points of orbit. It travels faster closest to the Sun, and slower furthest from the Sun. This happens because the Sun is a foci of an ellipse. The third law then allows you to make comparisons with several different planets of their characteristics.


 * Say Hi From Space**

A satellite is any object orbiting the Earth, Sun, or any other massive body, and there are two types of satellites. Natural satellites are ones like the Earth orbiting the Sun, or the moon orbiting us. A man-made satellite would be one we put to orbit the Earth for communication, weather, and so on. The only force acting upon satellites is gravity, making them projectiles. A satellite is always falling towards Earth, but since Earth curves, it never actually gets closer.The velocity of a satellite is tangent to any point on the circle, and the acceleration and net force point towards the center. The centripetal force is also acted upon by gravity. In the end, the satellite ends up orbiting the planet in an ellipse.




 * Satellite Motion for Dummies**

Newton's laws still come in to play in elliptical motion, and they are applied to many equations to help understand this motion. This formula is used to arrive to the force of gravity and you just need G, the two masses of the object, and the distance from the two centers.
 * Fgrav = ( G • Msat • MCentral ) / R2.**

Velocity can be found from this equation below.



Finally, acceleration can be found through this equation.

=Lesson 4(d-2): Planetary and Satellite Motion=


 * Light as a Feather**

Before learning about weightlessness, you need to know about contact forces and action-at-a distance forces. Contact forces are when you are actually touched by the force and action-at-a distance forces are when you aren't touched, such as gravity. Weightlessness is when you have no contact forces acting upon you, exerting a push or a pull. A scale is a measure of the upward force applied by the scale to balance the downward force of gravity, instead of the measure of weight. THE SCALE DOESN'T MEASURE YOUR WEIGHT. Contact forces can go away in an elevator, so as you go up, the reading on the scale will probably change. You don't have any contact forces at the top of a hill on a roller coaster, and that is why you feel weightless on them.


 * The Life of a Satellite**

Satellites have a tangential velocity to the Earth causing them to fall as the Earth curves. The gravity is proportional to the tangential velocity so it doesn't affect the inward force affecting the satellite. There is a force on the satellite that does work though, making it move at certain speeds.



The work-energy theorem find the mechanical energy is defined by the equation below.
 * KEi + PEi + Wext = KEf + PEf**

TME stands for the amount of total mechanical energy, while Wext are the external forces. K stands for kinetic energy, PEi stands for energy of position, and by plugging numbers into these, you can find the mechanical energy.