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Mass and Inertia For almost 2000 years, people thought that an object's natural state was to be at rest, and any time an object was put into motion, it would struggle to return to its natural state, which is why all observed objects slow down to a stop. Someone or something would have to continually apply a force to keep an object in constant motion. These ideas were put forth around 320 B.C. by Aristotle. He also said that heavier objects fall faster than lighter ones. Why did they last so long? Fear. If you said anything that contradicted such knowledge, you could be put to the death. Imagine, there had to be some people who in the privacy of their own home dropped different sized objects and observed that they hit the ground at the same time. That had to be pretty hard to accept. Not for Galileo. He was the first person to actually challenge Aristotle's notions around 1634. Everyone is familiar with the story of Galileo dropping different sized objects from the Leaning Tower of Pisa. Whether that actually happened is debatable. But, he did advance our knowledge of accelerated motion, and basically began the dawn of science. One of the things Galileo changed was this notion of natural state. It did not seem reasonable to him that the only natural state for an object is at rest. Consider the picture below.  He reasoned that if a ball started on a slope from a certain height and rolled to another slope on the opposite side, it would come to a stop when it reached its initial height. That is illustrated in the first two pictures. Notice on the middle picture the slope on the right is less than the slope on the left. This means the ball would have to actually travel a longer distance to attain the same initial height. So, imagine making the slope on the right smaller and smaller. That would mean the ball would travel greater and greater distances to attain the initial height. Therefore, what happens if there is no slope on the right side making it completely flat? When would the ball stop? We can answer that question with another question: will the ball ever attain its initial height? The answer is obviously no. Therefore, the ball would continue indefinitely! Galileo knew that the only thing hendering the ball from doing that is friction, not the ball's "desire" to return to its natural state of rest. Inertia is that property of an object that makes it want to keep going, as argued in the last example. However, it is also the property of an object that causes it to just sit at rest. These two statements can be summed up nicely by saying inertia is the reluctance of an object to change states of motion. The states of motion refer to being at rest or in uniform motion, which means constant speed and direction. The more inertia an object has, the more difficult it is to make it move from rest, or slow it down. Mass is the amount of matter an object contains. The matter is the atoms that make up that particular object. So, the more mass an object has, the more atoms or the heavier the atoms are that make up that object.  Mass and inertia are directly related. The more mass an object has, the more inertia it will have. This makes sense. If you consider a bowling ball and a soccer ball at rest, you would much rather kick the soccer ball because the bowling ball has more mass and inertia, which means it has more resistance to start moving from rest. This translates to more pain to your foot. Likewise, you would rather stop a rolling soccer ball with your foot as opposed to the bowling ball because again due to the higher intertia, the bowling ball has more resistance to stopping. Sir Issac Newton Issac Newton comes on the scene about 50 years after Galileo revised the old way of thinking. Newton was the first person to formalize a set of laws for motion, in which he used Galileo's work as a spring board to develop his framework for moving bodies. The rest of this chapter will deal with Newton's Three Laws of Motion, which is the cornerstone of physics. Newton's work also includes a comprehensive theory of gravity (covered in chapter seven). It is interesting to note that the mathematics that existed in the mid 1600's was not sufficient in providing a theoretical bases for gravity and the motion of planets. That inspired Newton to discover a more powerfull mathematics that could handle it, known as calculus today. However, he was not the only person to discover it. In fact, Gottfried Willhelm Leibniz developed the same thing at the same time as Newton, and neither of them knew of the other's work. Unfortunately, Newton is given the credit even though we use Gottfried's notation and terms. Newton published his theories in 1687 as Mathematica Principia. This piece of work would become the foundation of physics for over 200 years. Einstein's Theory of Relativity published in 1905 and 1915 would be the first revision of Newton's framework. This revision is only necessary when dealing with objects traveling at near the speed of light, and intense gravitational fields. So, Newton's mechanics are still used today when dealing with everyday motion and normal gravitational situations because amazingly, Einstein's equations reduce to the Newtonian equations when dealing with everyday situations. Therefore, the term classical physics refers to Newton's theories and the physics before 1900, and modern physics refers to the physics that Newton's mechanics cannot handle. This encompasses Relativistic and quantuam physics. (These topics will be covered in the special topics section of this site but they will not be completed until this section is finished). Most introductory physics courses deal exclusively with classical physics, which will be the focus for the rest of this book. Newton's First Law Newton's First Law is a formal way of expressing Galileo's reasoning behind inertia and the object's tendency to keep moving. It can be expressed as follows: An object at rest will continue to be at rest, and an object in motion will continue to be in uniform motion unless acted upon by an outside influence. This seems simple, but it needs to be stated because it actually explains many physical phenomenon that will be discussed in this chapter, and in chapter six. So, let's break down the first law. An object will stay at rest unless acted upon by an outside force simply means an object will not move by itself. Again, the obvious nature of this statement is not that impressive. But, consider some of the applications. Why do you lean back in your car when you take off from rest? There is nothing in front of you pushing you back right? The reason: your body is at rest and wants to stay at rest. So, when the car moves, your body "leans" back because it is "trying" to stay at its original position. Unfortunately for your head and neck, whiplash occurs when the car is suddenly jerked from rest. Technically, your head does not snap back. Your body moves forward because the seat provides the "outside" force, but your head stays at rest, which gives the illusion that it snaps back.  This concept also explains how a magician can pull a table cloth out from underneath a dinning set. The dinning set is at rest at wants to stay at rest. The "trick" is to pull the cloth with a quick jerk so that the force is not sustained long enough to overcome the inertia of the dinning set pieces. Now, of course you need to have a very slick table cloth and it does take practice. I do not recommend you taking your grandma's homemade cotton table cloth and trying this with your best china. Start small. You can put a full glass of water on top of a piece of paper. Just make sure you use a quick jerk and pull the paper horizontally and not partly up and partly out. An object in motion will continue in uniform motion means it will continue in a straight line and a constant speed, which is constant velocity. Again, this has some interesting applications. Why do you "lean" forward when a car comes to a stop? The reason: your body is in motion and wants to stay in that motion. When the car is moving, your body is moving at the same speed. This is why seatbelts are detrimental to the safety of anybody who drives a car. If you are traveling 60 mph and hit a tree, your body will fly toward the dash board at 60 mph! Also, If anybody has ever ridden a bike and happened to come to a sudden stop, they know that over the handle bars they will go! Notice from the two parts of the First Law, you only feel the jerk or leaning when the velocity changes either from rest to moving, or moving to moving faster or slower. This of course means you are accelerating. What happens when you aren't accelerating? Do you feel anything? The answer is no! But you might be thinking that you feel the bumps in a car while traveling at a constant speed. Those bumps are making the car accelerate, just not in the horizontal direction. The cool thing is, if you were in a car traveling at a constant velocity with no bumps and no windows to look out, you would not be able to tell if you were moving or at rest. It would feel exactly the same. Accelerating is the only time you "feel" motion. So, in a sense, the First Law is a description of motion that cannot be felt or noticed by the moving object. Ask yourself this question: do you feel any movement while sitting at rest in a chair? Of course not. But that does not mean you aren't moving. Remember, you are traveling around the sun at over 66,000 miles per hour! Not to mention the fact that you are traveling at over 1000 miles per hour due to the earth's rotation. You cannot feel anything because for the most part, there is no acceleration. (I say most part because I am neglecting all impercievable accelerations due to rotations and gravity). Before moving on to the Second Law, if anybody has a problem with believing the ball in Galileo's argument would continue forever without friction or the above paragraph consider this. The space shuttle in orbit is traveling at thousands of miles per hour! When the astronauts step away from the shuttle, they are not instantly left behind. It looks as if they aren't even moving. This is due to two reasons. One, there is no air resistance, so there is no outside influence to slow them down. Two, since the astronauts were traveling the same speed as the shuttle when they were on the shuttle, they are going to keep traveling at that speed when they step off. So, the astronauts can not tell that they are traveling at such high speeds, except when they look down and see the continents move by. Force According to the First Law, all objects will maintain a constant velocity, which can be zero, unless acted upon by an outside influence. That influence is called a Force . So a force is any influence that changes the velocity of the object. The unit for force is called the Newton denoted by a N. It is convenient to talk in terms of Net Force, which means the total force acting on an object. The reason for this is because Force is a vector quantity. Consider the picture below and determine the net force on that box.  If you said zero you are correct. There are a total of four forces acting on the box at the same time. Each force has another force with the same magnitude and opposite direction which mean they cancel. Will that box move? Since there is no outside influence, because they all cancel, the box will not move. Consider the next picture and determine the net force.  The forces acting in the vertical direction still cancel, but the horizontal forces do not this time. Therefore, if we make right positive and left negative, the net force will be a + 5 Newtons. In fact, this picture can be redrawn as,  That is equivalant to the picture above because forces that cancel are as if they aren't even there. We have already established that when the net force on an object is zero, the object will not move. This is the first part of Newton's First Law. What happens if the object is moving at a constant velocity and the net force is zero? Does anything happen to the object's velocity? Consider the picture below.  The block is moving, say in space, at a constant velocity, and it has ropes attached to both ends. Let's say you pull up next to the block in a spaceship and reach out and pull the ropes at the same exact time with the same exact force. Will the box slow down, speed up, or stay the same?  If you said the block would maintain its constant velocity, you are correct. Remember, if the forces cancel, the picture can be redrawn as:  It's as if they aren't even there, so how could they affect the block's motion? Therefore, the object will continue in its uniform motion which is the second part of Newton's First Law. Consequently, the First Law applies to every situation where the net force is zero, which means there is no acceleration. This can be summarized below. The Fn means net force.  Newton's Second Law It turns out that there is a direct relationship between force and acceleration. Force is directly proportional to the acceleration which can be written as  Do not confuse a proportionality with equality. Just because the force and acceleration increase and decrease by the same amount does not mean they equal each other. They are not equal! In order to convert a proportionality into an actual equation, you have to multiply by a constant. It is called the constant of proportionality. A constant is some quantity that does not change for a given situation. It turns out that the constant needed is the mass of the object. Therefore, Newton's Second Law can be expressed in equation form as  Suppose the net force on a 5.0-kg block is 10.0 N, what would be its acceleration. Obviously, the answer is 2.0 m/s^s. The equation would become, Solving for a gives the answer of 2.0 m/s^2. We can now move on to a little more challenging problem. For this next example, we are going to see how to set up the net force equation when there is more than one force to consider. Weight Let's consider a man hanging from a rope as pictured below (don't laugh!).  In order to find the net force in this situation, we need to consider the concept of weight. Weight is really the force the earth exerts on objects. Since it is a force, it can be expressed as,  Where w is the symbol for weight. We can even make this equation more specific, because what is the acceleration associated with gravity? That's right, little g, which is equal to 9.8 m/s^2.  Weight is a force that always acts towards the earth. So, when considering the forces that act on the man, they can be drawn as shown. Step One:  Drawing in the forces is the first step in setting up the net force equation. It is crucial that all forces are drawn so that the direction and nature of the forces can be determined. In this example there are only two forces: weight, and the force exerted by the rope labeled as tension. Notice, the directions have already been assigned: up is positive, and down is negative. Step Two: The next step is to now determine which forces are positive and which are negative. In this example, it is obvious that weight is negative and tension is positive. So, adding the two forces will result in the net force side of the Second Law equation. Step Three:  If the person is just hanging from the rope, the acceleration is zero and the equation becomes  Let's say the person has a mass of 80.0-kg. Therefore, the equation becomes, Which becomes, T obviously has to equal 784 Newtons. The fact that the tension has to equal the weight of the person is obvious, but the practice of setting up the net force equation is not always simple, so it is good to practice on an easy situation. Now let's say the maximum tension the rope can support is 800 Newtons. With what acceleration can the man pull himself up the rope without breaking it? Now that acceleration is not zero, the equation becomes,  Now all we need to do is plug in for the tension and mass and we have, Solving for a gives an answer of .20 m/s^s. That is the maximum acceleration up the rope. If the man went up any faster, the rope would snap. Let's consider this problem conceptually. We know that when the person just hangs, the acceleration is zero and the net force is zero, which means the rope exerts a force equal to the man's weight. In order for the person to pull up and accelerate in the upward direction, the net force must be pointing in the upward direction. (That is because an object will always accelerate in the direction of the net force). This can only happen if the rope exerts more force, because the rope is the only thing able to supply that extra force.  This picture can be redrawn with just the net force left.  |