Ch5_KwarkR

toc Chapter 5

=Summary: Lesson 1 (a-e)= a)A Way to find Speed and Velocity of an Object in Circular Motion. Finally! Objects in uniform circular motion (motion at constant speed) can be described using kinematics. Basically, the speed of the object is the amount of distance traveled over time: Since Circumference=2πr, this equation becomes To find the direction at a single point, you can find the slope of the line tangent to the point. Since the direction changes constantly, the velocity of the object changes constantly.

b)The Misconception of Circular Acceleration Many have the misconception that objects in circular motion do not have acceleration. But they do, and the acceleration points inward. Using the definition of acceleration (the change in velocity over time), we can pick any two moments and find the change in velocities, then figure out its orientation using vector addition; the result is an acceleration pointing inwards. This can also be seen when corks immersed in fluid are spun in a circular motion; the less massive end of the cork points inward, indicating that there is an acceleration inward.

c)Centripetal Force: REQUIRED!! Centripetal forces are forces that are center-seeking. In circular motion, there must be a constant force allowing for the object to continue changing its velocity. One example is a car making a turn; the friction between the tires and road allows for the car to change direction (b/c it is an unbalanced force), but the passengers inside feel like they are being accelerated away from the direction the car turned. This is due to inertia; you tend to keep moving the way you did before the car changed direction. This makes you think that the force is outward, but in fact the centripetal force points inward. This centripetal force that enables objects to keep moving in a circular path is not a new type of force, but another descriptive term for any of the four mechanical forces we have learned already.

d)The Word Centrifugal Has NO Place in Circular Motion! Centrifugal (outward-seeking) forces dont exist in circular motion. The feeling we get when we go around in a circular motion that we are being "thrown" outward is not due to a force; it is a result of Newton's law of Inertia that says that objects tend to do whatever they have been doing. That feeling you get is not because of a centrifugal force, but because you tend to keep going straight while the object changed velocities due to a centripetal force.

e)Speed, Force, Acceleration are Essential! We can solve many circular motion problems by finding speed, force, and acceleration. We can find these by using the following equations.

Speed: Acceleration: Net Force:

=Summary: Lesson 2= a)Newton's Law Still Exists - Despite Circular Motion! We solve problems by first drawing a FBD, then creating F=ma equations derived from the forces acting upon the object. Since it is in circular motion, F=ma becomes F=mV^2/r, since centripetal acceleration is V^2/r. If there are other forces that need to be found, we can simply solve for other axes or use systems of equations. Be warned - we may need to solve for components of a force.

b)Rollercoasters? No sweat. On rollercoasters, there are three types of centripetal motion; loops, dips, and banked turns. Loops like the one below have changing acceleration (both in direction and in speed. This clothoid loop also shows that the radius of the circle changes.

At the top of the loop, there is less normal force due to your weight. At the bottom, there is more normal force, which creates the feeling of pressure in your stomach. This is the same for dips and hills; at the top of the hill there is less normal force than at the bottom of the dip. Remember, every one of these loops or dips at some point are shaped in a circular path.

c)Athletics ALSO deal with Circular Motion! All sports deal with circular motion, whether it be track and field, baseball running or ice-skating. Whenever someone or something is making a turn, that is dealing with circular motion.

For example, a skater making a turn on the ice needs to angle his legs towards the center about which he is turning: There are components from both legs that are directed toward the center.

= Summary: Lesson 3 = a)Gravity is Huge! Gravity causes things that go up to come down. It is more than the 9.8 m/s value of the acceleration of gravity that causes things to slow as they jump, eventually fall back down to Earth. It is more than the Force of gravity as well. It has much further-reaching implications than that, on a much larger scale.

b)Kepler and Newton's Discoveries Kepler analyzed data and came up with three laws: But although these provided laws for how planets moved, they did not provide an explanation for why. Newton reasoned that the paths of these bodies would be similar to if a cannonball were launched with greater and greater speed until it had enough to orbit the Earth. With that, he realized that this applied not just to objects on earth like an apple, but objects in the heavens as well, like the moon. He realized that the force of gravity was diluted with distance, and eventually found out that the force of gravity was inversely proportional to the square of the distance between the objects. c)The Universality of GravityGravity exists not only between planets and heavenly bodies, but between everyday objects! Newton's law of universal gravitation says that ALL objects attract each other with a force of gravitation, even people and books! But those forces are too small to be noticed until the masses get large. And speaking of mass, distance is not the only thing that affects gravitational force; so does distance. Gravity is directly proportional to the product of the two masses. When you put this together with the information on distance, the following relationship forms: But after Newton's death, Henry Cavendish discovered another part to this relationship that truly made it an equation; the universal gravitational constant. This constant isG = 6.673 x 10-11 N m2/kg2 . Thus, the full equation for the force of gravity between two objects is:
 * 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 Ellipses)
 * 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 Equal Areas)
 * 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. (The Law of Harmonies

d)Cavendish and his "G" After Newton realized that gravitational force was directly proportional to the product of the masses while being inversely proportional to the square of the distance between the objects, Lord Cavendish found the value for the universal gravitational constant. He did this by using a Torsion balance and measuring the relationship between the angle of rotation and the amount of torsional force. He then placed two large metal spheres near the smaller spheres until the torsional forces balanced the gravitational forces (the rod and balls came to rest). Cavendish measured m1, m2, d, and Fgrav, he managed to find the value of g, which was very similar to the currently accepted value of 6.67259 x 10-11 N m2/kg2.  e)What is "G" good for?We have always used 9.8 m/s as the accepted value for the acceleration of gravity; however, this is only true at sea level. The further or closer you get to the earth, the more this value increases. This is derived from the equation g= (G*M)/d 2. As the distance increases, the bigger the denominator, which results in a smaller value for g. Because mass in the numerator, the bigger the planet, the larger the value for g. For other planets, the acceleration value g can be found using the following formula:

= = =Notes: The Clockwork Universe - Mechanics and Determinism = Part 1)The Beginnings of Astrophysics. In 1543, Nicolas Copernicus rejected the popular Earth-centered model of the universe and suggested that the solar system moved in a heliocentric pattern, in which the Earth moved around the sun.Galileo became a proponent of this theory as well. Kepler took this idea one step further, and, using the best observational data at the time, determined that the planets moved in elliptical movements around the sun instead of the perfectly circular movements suggested by Copernicus. Unfortunately, he did not know why the planets moved in such patterns, though he guessed there was //some// influence from the sun that enacted such a pattern.

Part 2)A New Type of Math. Around this time, many new discoveries in mathematics were occurring. Chief among them was Descartes' discovery that geometry can be recast as problems in algebra by simply using a coordinate grid and placing shapes and lines on it. This branch of math became known as coordinate geometry, which represents geometrical shapes by equations and establishes geometrical truths by combining and rearranging those equations. By using this method, many mathematicians and scientists were able to go much further than ever before.

Part 3)Newton: The Right Man at the Right Place at the Right Time. Newton was fortunate to be in his prime when the discoveries by Kepler were still unexplained and when coordinate geometry was discovered. Because of this, he was able to produce an underlying framework that would help many others understand the laws of the universe. He did so using a single set of laws. Some key points were that 1)He looked not at motion, but a deviation from motion. 2)Whenever deviation occurred, he looked for forces that could cause it. 3)He then made a quantitative link between force and deviation from steady motion, which became the basis for his Law of Universal Gravitation.

Part 4)Mechanics and Determinism. Newton provided a single law for gravity that worked for all matter in the Universe. Due to that discovery, he was able to explain Kepler's results and go beyond them. Newton's discovery gave birth to Mechanics, a study of force an motion, which regarded the universe as something predictable and precise according to mathematical and scientific laws, like a clock. Determinism is the principle that once the universe, like a clockwork, has been set, its future motion and actions are predictable. This brought much controversy over free will and the indication of God, as people are also made of matter.

=Notes: Lesson 4 (a-c)= a)New Scientific Discovery: Kepler's Laws! Kepler proposed three rules for planetary motion using Tycho Brahe's extensive data. His first law, the Law of Ellipses, was that all planets orbit the sun in an elliptical path, with the sun being located at one of the foci of that ellipse. His second law, the Law of Equal Areas, said that the area swept out for a given amount of time is equal to the area swept out at another position for the same amount of time. By that definition it can be concluded that the speed at which the planet moves is different at different positions in its orbit; that a planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Finally, Kepler's third law, the Law of Harmonies, compares the orbital period and the average radius of orbit to those of other planets. it states that for every planet in the solar system, r^3/T^2 is constant.

b)Satellites and Planets Move Too! Satellites are objects that orbit around a massive object, and it can be man-made or natural (like a satellite in orbit around earth and the Moon). These satellites are projectiles, which mean they are only affected by gravity; however, if launched at great speeds, the rate at which they fall equals the rate at which the massive object curves, so it remains at the same distance above the surface, never crashing into the ground. The velocity vector is tangential to the orbit, while the Force and acceleration vectors point toward the center of the object being orbited. In an elliptical path, the central body being orbited is at one focus of the ellipse. The force of gravity, however, varies in direction at various position in the orbit, which causes a change in the speed (and velocity) that the projectile travels at.

c)There's Math behind Motion? Of Course! We can calculate the Force of gravity between two objects by using their masses, the Universal constant G, and the distance between them in the formula **Fgrav = ( G • Msat • MCentral ) / R2.** Acceleration can be derived from this formula, which results in:

Velocity can be derived from the gravitational force equation and the equation for Net force, resulting in:

d)Weightlessness – Do YOU Know What It Means?

Contact forces are forces where there must be actual contact between the objects, while action-at-a-distance forces (like gravity) affect objects at distance, without contact. Weightlessness occurs when there is no contact/normal force. A scale reading only measures the contact/normal force which balances with gravity, allowing you to find your weight; but if you are on a moving elevator the amount of contact force to balance weight may differ, resulting in different weights. In orbit, astronauts are weightless because there are no contact forces acting upon them, even though gravity(an action-at-a-distance force) is.

e)Satellites do Work?

When a satellite is orbiting at a constant radius with the same speed, the inward Force and the tangential velocities are perpendicular to each other, so they can’t affect each other. However, in elliptical orbits, the two vectors are not perpendicular, so some Force can do work in slowing the object down and speeding it up during orbit.

The work-energy theorem states that the initial amount of total mechanical energy (TMEi) of a system plus the work done by external forces (Wext) on that system is equal to the final amount of total mechanical energy (TMEf) of the system.

** KEi + PEi + Wext = KEf + PEf **

However, the W(ext) only measures energy done by outward forces, and gravity is an internal/conservative force, so the W(ext) is 0, and the entire process is conserved. In circular orbits, KE and PE remain constant, so the TME (total mechanical energy) remains the same, too. In elliptical orbits, the KE and PE value change; as the satellite gets closer to the focus and speeds up while losing height (distance from the planet), the KE value increases and the PE value decreases, but the TME stays the same.