A Much Needed Ode to Circles

This unit we learned about circular motion and gravitation. We started with circular motion, since it was important to understand these concepts before moving on to the more complex gravitation.

We started with the basics. The first thing we covered was velocity with circles. In the past velocity has always been distance divided by time. However, in those cases we were going in a straight horizontal line. This time we were going around a curve, so the distance became a circumference, or 2πr, r being the radius. The time became the period, which is the time it takes the object to complete one full rotation or revolution. We also learned how the period was the inverse of the frequency, or the rotations per unit of time, usually a second (Hz), and that frequency was the inverse of the period. One important thing that we had to understand was that the velocity was not constant, since the direction of the of the motion changed constantly, and that the velocity was tangential to the circle that the object was moving in. Since the object is changing directions, it is accelerating. Acceleration in circular motion is called centripetal acceleration, because the acceleration is always pointed toward the center, and is perpendicular to the velocity. On that note we learned about centripetal force. We know that Net force = mass times acceleration. Well, with centripetal force it is exactly the same, except that for centripetal acceleration we use  the square of velocity divided by radius, and multiply that by mass. You must never, EVER, confuse centripetal force with centriFugal force, which is actually not a force at all and more a sensation caused by inertia, and is an "F-word" in physics. But the centripetal force doesn't just happen. It has to be provided by another force, which we shall call the centripetal force requirement. Examples of this are the friction between the tires of a car and the road it is on, the tension in the string swinging a stone in a circle, and the gravity acting between a lonely moon and a planet in a galaxy far, far away. These are all forces that act on an object and cause it to move in a circle in the first place. 

After learning these basics, we learned about motion in a vertical circle. This was not much different, except now we were dealing with things like tension and the force on gravity and finding their values. For example, at the highest point in the circle, the centripetal force is equal to  the tension plus the gravity,  which is equal to mass times acceleration.  At the top of the circle, both forces are pointing downward, and so is the acceleration. When the object is at the bottom of the circle, the tension points up, along with the acceleration, and the weight points down, making the centripetal force equal to the tension minus  the gravity, which is equal to mass times acceleration. It was actually easier to understand than it may sound like at first. 

After this we journeyed onward to learn of universal gravitation. This dealt with a lot of gross nasty scientific notation that we had to get our hands dirty in, since many things whose gravity we solved for were larger than your average basket ball. Our dear friend, Isaac Newton, created the inverse square law, which stated that "the gravitational force varies inversely with the square of the distance between two objects", which contributed to the law of universal gravitation, which said that everything in the universe attracts everything else in the universe that has a force which "varies directly with the product  of their masses and inversely with the square of the distance between the  centers of the two masses."

And from these great laws was born a equation for FG, that G (a universal constant that is equal to 

6.67 X 10E-11)Nm/kg2 times the masses of the two objects was divided by the radius squared, an equation which comes in handy when trying to figure out the weight of a planet. 

And finally we reached the end of gravitation, and in fact the end of our unit on dynamics, gravitational acceleration. This excludes finding out the gravitational acceleration for anything on the surface of Earth, since that value is the same everywhere on Earth's surface. We know that FG= mg, so mg is equal to the equation in the paragraph above. The m's cancel out, leaving you with a formula for g, which is equal to G times the mass of object to which everything else is gravitating toward divided by the radius squared. Like I said before, this equation is great for finding the acceleration due to gravity on any planet you choose, which can also help in finding the mass or weight of any object on that planet. 

Of course, as easy as it all may appear to be for me now, it was hard to begin with. The hardest part was keeping track with all the formulas that needed to be used. This became easier as I did classwork and used these formulas, and they were all soon pretty familiar. Another difficulty was simply getting it. Physics isn't something that comes naturally to me. I have to really look at it a few times to figure it out. There were troubles with my calculator that sent me into fits of rage when my slippery fingers hit the wrong buttons for scientific notation, giving me the wrong answers. Once I was up until the wee hours of the morning making sure I really understood all the classwork on gravitation, because at first I didn't. It was difficult at first, but as everything else does, it all fell into place eventually.

One of my problem-solving skills was taking the initiative on my own problems. There was one lab that we did where we hung a pig from string on the ceiling. It was not a real pig, and in fact it flew in circles. We recorded data such as mass and radius (length of the string) and frequency, which we later converted to period. We then had to figure out which equations we were going to use to eventually solve for the net force, assuming that the pig was flying at a constant velocity in a perfect circle. Now, the net force is equal to the centripetal force, but of course when things are stated in a different way than they're meant I get super confused. I sat there scratching my head for at least thirty minutes on my bed, staring at the lab and going through equations and just panicking. My dad was out of town and there was nobody at home to help me. I called a friend but she didn't know, and many hadn't gotten to that part of the lab yet. So the good news was: I was much farther than most of my friends. The bad news was: that seemed to me the farthest I was going to get. Suddenly, I realized that I was not going to pass physics by sitting there feeling sorry for myself. What the heck was I doing? If I couldn't find help from people, I had to turn to my other sources. I had to take the initiative to help myself! So I googled how to find the net force for circular motion. When I found one answer I checked with a few more, just to make sure. It was all suddenly clear to me. Of course I needed to find centripetal force! F=ma and Fc=mac, so why didn't I see that they were the same thing before? Because I was so convinced that I didn't know what the net force was. I didn't give myself time to stop and think and draw connections. If I had I would have figured it out much sooner. But since I couldn't figure it out, I had to figure it out myself, which proved to be the most important problem-solving skill of the unit. For once I didn't ask my dad, for once I didn't save it for the next day when I would do it in a hurry at lunch. For once I did it all by myself, which made me so proud because for once, my problem-solving skill wasn't turning to my father for help.