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Newton’s Laws of Motion We have now studied how to describe the motion of an object. This is the field of kinematics. However, we have not talked about why objects move. This is dynamics, and is the next subject that we shall look at. In order to study the dynamics of an object, indeed, even to study the kinematics of an object, we must choose a coordinate system. Remember that coordinates are man made systems that we use to help us convert a physical situation to a mathematical model. Unfortunately, there are infinitely many coordinate systems that we could chose. Is there some way of consistently choosing a useful one? The answer is yes, but in order to fully understand why, we must define an inertial frame of reference. An inertial frame of reference is one in which the object that the coordinate frame is attached is at rest or is moving at a constant velocity. Thus, a car speeding up to get on the freeway is not an inertial reference frame, but once the car is on the freeway and moving with a constant velocity it is. Thus, in order to choose the proper coordinate system, we usually want to choose one which is inertial and simplifies the vector description of the system. For example, if a block is sliding down an inclined plane, the simplest coordinate system is one which is lined up with the incline. Definition of Force Once we have our coordinate system, we are ready to introduce one of the central concepts in physics, the concept of force. Whenever an object undergoes an acceleration, its motion is changing. What is causing this change? The simplest answer is to say that the acceleration is caused by another vector, which we will say points in the same direction as the acceleration and is proportional to it. F = ca We will call the new vector the force. We must still determine the constant of proportionality. As with all proportionality constants, we must perform experiments to determine this one. Consider two balls, each with a different mass. In between them is a compressed spring. The whole system is held in place by a taunt string. When the string is cut, the spring expands and the two balls are pushed away at different speeds. By repeating this experiment with different springs, we would find that the ratio of the speed of the balls is inversely proportional to the ratio of the masses of the balls. Thus, we are lead to the conclusion that the constant of proportionality is the mass of the object being acted upon. So we can write F = ma (5.1) From this, we see that force has the units of kg-m/s2. Since force occurs so frequently, we give this combination a special name, the Newton (N). 1 N = 1 kg-m/s2 Newton’s Three Laws We are now ready to state Newton's laws of motion. Until the turn of the 20th century, these laws were the driving force behind almost all of physics. Newton based his laws on observations made by Galileo and others before him. Newton's first law states that when the vector sum of the forces acting on a body is zero, if it was originally at rest it will remain at rest, and if it was moving at a constant velocity it will continue at that constant velocity. Mathematically, it becomes F 0 v constant (5.2) This is the condition of translational equilibrium. Basically, it says that once a body is in motion, it does not require additional force to keep it in motion if there are no other forces acting on it. This went against the conventional wisdom of the time, which thought that any object left to itself would come to rest. It took time and experiment to show that the first law was actually correct. The second law states that when the vector sum of the forces acting on a body is not zero, then the body will experience an acceleration in the direction of the vector sum of the forces. This acceleration is proportional to the resultant force and inversely proportional to the mass of the body. This is just our original statement F = ma (5.3) Notice that the mass acts as a measurement of the inertia of the body. Inertia is the body's resistance to change its velocity. For this reason, the mass in (5.3) is sometimes called the inertial mass. Another type of mass was gravitational mass. This is the mass of a body that affects how it is attracted gravitationally. It wasn't until the theories of relativity were established that it was shown that inertial mass and gravitational mass were indeed the same. The third law of motion states that for every action there is an equal and opposite reaction. Thus, for any force acting between two objects, it doesn't matter if we saw the force originated at the first object and acted on the second one, or if it originated at the second object and acted on the first one. The two forces are equal in magnitude and opposite in direction: F12 = -F21 (5.4) An easy, intuitive example of this is a book on a table. Obviously the book is at rest on the table top. But we know that if the table were not there the book would fall under the influence of gravity. Thus the book is experiencing a force equal to F = mg Since the book is sitting on a table, it exerts a force on the table equal to its weight. Newton's third law tells us that the table then exerts a force back on the book with a magnitude equal to the weight of the book and directed upward: F' = -mg This is the reaction force of the table to the book. We encounter this type of reaction force so frequently that we give it a special name, the normal force. We call it this since the table pushes on the book in such a way that the reaction force is perpendicular, or normal, to the surface of the table. It should be emphasized that the normal force always acts at right angles to the surface, regardless of the orientation of the surface and the direction of the other forces. Another special force is that caused by the acceleration due to gravity. The force equation (5.3) becomes F = mg We call this force the weight of the object. It depends on the rest mass of the object as well as the local acceleration due to gravity. Since this acceleration varies at different parts of the Earth, the weight of an object is different at different places. It is important to realize that the weight of an object acts on it all the time, even when there is no reaction forces acting against it.